A rigorous calculation of the critical point from the fundamental equation of state for the water + ammonia mixture

international journal of refrigeration 32 (2009) 95–101

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A rigorous calculation of the critical point from the fundamental equation of state for the water D ammonia mixture Ryo Akasaka* Faculty of Humanities, Kyushu Lutheran College, 3-12-16 Kurokami, Kumamoto 860-8520, Japan

article info

abstract

Article history:

The critical point of the water þ ammonia mixture was calculated directly from the Helm-

Received 6 March 2008

holtz free energy formulation. The calculation was performed according to the critical

Received in revised form

point criteria expressed in terms of the derivatives of the Helmholtz free energy with

13 May 2008

respect to mole numbers. Smooth critical locus linking between the critical points of

Accepted 17 May 2008

pure water and ammonia was obtained. The critical locus showed a good agreement

Published online 26 May 2008

with the most reliable experimental data. Simple correlations for the critical temperature, pressure, and molar volume for a given composition were developed. The information ob-

Keywords:

tained in this study is helpful for design and simulation of the cycles using the water

Absorption system

þ ammonia mixture as working fluid. ª 2008 Elsevier Ltd and IIR. All rights reserved.

Ammonia/water Calculation Critical point Equation of state Aqueous solution

Calcul rigoureux du point critique a` partir de l’e´quation d’e´tat pour un me´lange ammoniac/eau Mots cle´s : Syste`me a` absorption ; Ammoniac/eau ; Calcul ; Point critique ; E´quation d’e´tat ; Solution aqueuse

1.

Introduction

A rigorous calculation of the critical point is performed for the water þ ammonia mixture. Although many calculations of the vapor–liquid equilibrium (VLE) and single-phase properties have been extensively carried out using equations of state or

correlations for the mixture, no attempt to calculate the critical point has been reported yet. The water þ ammonia mixture is regarded as the most important working fluid for absorption-refrigeration cycles and power cycles for utilizing low temperature heat sources. For design and simulations of the cycles, several thermodynamic models for the mixture have been proposed in the past.

* Tel.: þ81 96 343 1600; fax: þ81 96 343 0354. E-mail address: [email protected] 0140-7007/$ – see front matter ª 2008 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2008.05.007

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international journal of refrigeration 32 (2009) 95–101

Greek symbols a reduced Helmholtz free energy d inverse reduced molar volume s inverse reduce temperature

Nomenclature A A Cv H G L M N p R S T V V x

Helmholtz free energy (J) molar Helmholtz free energy (J mol1) isochoric molar heat capacity (J mol1 K1) molar enthalpy (J mol1) molar Gibbs free energy (J mol1) determinant used in the critical point criteria determinant used in the critical point criteria mole number (mol) pressure (Pa) universal gas constant (8.314471 J mol1 K1) molar entropy (J mol1 K1) temperature (K) volume (m3) molar volume (m3 mol1) mole fraction of ammonia

Subscripts 0 reference state 1 water 2 ammonia c critical point i component i red reducing parameter Superscripts ideal gas or ideal-gas mixture part r residual part 

Although the critical point of the mixture lies between 11 MPa and 23 MPa, most of the models are applicable only in a restricted range below 20 MPa. The only model applicable over the critical region (up to 40 MPa) is the formulation developed by Tillner-Roth and Friend (1998). The formulation represents the Helmholtz free energy of the mixture as a function of temperature, molar volume, and composition. Since experimental data in the critical region for the water þ ammonia mixture is limited, the Tillner-Roth and Friend formulation in the critical region is based on the VLE prediction by a modified Leung-Griffiths model (Rainwater and Tillner-Roth, 1999) which was developed simultaneously with the formulation. Rainwater and Tillner-Roth (1999) concluded that their model is a reasonable representation of the high-pressure phase boundary for the mixture. However, until now an assessment of the critical point calculated from the formulation has not been made. This study calculates the critical point of the water þ ammonia mixture directly from the Tillner-Roth and Friend formulation. The calculation result is compared with reliable experimental data, and the capability of the formulation for the prediction for the critical point is discussed. Equations required for the calculation are described in detail. Simple correlations to estimate the critical temperature, pressure, and molar volume for a given composition are presented.

2.

Calculation method

2.1.

Critical point criteria

and

vN1

 A12  vL  ¼ 0; vN 2

T;V;N2

(2)

T;V;N1

and A12 ¼

v2 A : vN1 vN2

The criteria include only the partial derivatives of the Helmholtz free energy with respect to mole numbers. Different criteria including derivatives of the Helmholtz free energy with respect to volume have been more commonly used by many attempts, e.g., Spear et al. (1971); Teja and Rowlinson (1973); Peng and Robinson (1977); Hicks and Young (1977); and Baker and Luks (1980). The advantage of the criteria given by Eqs. (1) and (2) over the commonly used criteria was described by Akasaka (2008). As shown later, this study calculates A11, A22, and A12 analytically from the Tillner-Roth and Friend formulation. The derivatives of L(T, V, N1, N2) with respect to mole numbers appearing in Eq. (2) are estimated numerically, because an analytical derivation of the derivatives is enormously involved. Using the 7-point formula, the value for (vL/vN1) is estimated as follows:   vL vN1 T;V;N2 ¼

The method used here is based on the approach presented by Akasaka (2008). For the critical point of binary mixtures, the approach employs the criteria   A A12  ¼0 (1) L ¼  11 A12 A22    A11 M ¼  vL

where A11, A22, and A12 are defined as ! ! v2 A v2 A A11 ¼ ; A ¼ ; 22 vN21 vN22

45L½ þ 1  45L½  1 þ 9L½  2  9L½ þ 2 þ L½ þ 3  L½  3 ; 60DN (3)

where L[k] (k ¼ 3, 2, ., þ3) indicates L(T, V, N1 þ kDN, N2) and DN is the step size of numerical differentiation. This study set DN to 105 mol.

2.2. Analytical derivation of the derivatives in the critical point criteria The Helmholtz free energy A can be split into the ideal-gas mixture part A and the residual part Ar. Therefore,

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international journal of refrigeration 32 (2009) 95–101



A ¼ A þ Ar

(4)

The derivatives of A with respect to mole numbers can also be split into two parts as 

A11 ¼ A11 þ Ar11 

(5)

Ar22

(6)

A22 ¼ A12 þ Ar12

(7)

A22 ¼ A22 þ and 

Mathematical expressions for the derivatives of the idealgas mixture part and the residual part are individually derived.

2.2.1.

sðT; xÞ ¼



The derivatives of the ideal-gas mixture part A 11, A 22, and A 12 can be calculated only from the thermodynamic relations. For binary ideal-gas mixtures, A can be written as

dðV; xÞ ¼

Ar11

Ar22

Qi ðTÞ ¼

T

T0



Cv;i dT þ H0  T

Z

T

T0



Cv;i dT þ S0 T

¼





v2 A vN21





A22 ¼

v2 A vN22

! ¼ T;V;N2

! ¼ T;V;N1

! ði ¼ 1; 2Þ:

RT ; N1

(9)

RT ; N2

(10)

! T;V;N1

(16)

!# "   var v 2 ar þN ; ¼ RT 2 vN2 vN22

(17)

 2 r   r   r  v2 Ar va va v a þ þN ¼ RT vN1 vN2 vN1 vN2 vN1 vN2

(18)

The derivatives (var/vN1), (v2ar/vN21), and (v2ar/vN1vN2) occurring in Eqs. (15), (16), and (18) can be transposed into the following expressions:        r va vs vd vx ¼ ars þard þarx ; (19) vN1 T;V;N2 vN1 T;N2 vN1 V;N2 vN1 N2

v 2 ar vN21

! ¼ arss T;V;N2



vs vN1

2

þardd



vd vN1

2

þarxx



vx vN1

2

þars

v2 s vN21

!

! !    v2 d v2 x vs vd r þ ax þ 2arsd þ 2 2 vN1 vN1 vN1 vN1       vs vx vd vx þ 2ardx ; þ 2arsx vN1 vN1 vN1 vN1 (20)

and

v2 A ¼ ¼ 0: vN1 vN2

2.2.2.

T;V;N2

!# "   var v 2 ar þN ; ¼ RT 2 vN1 vN21

ard



A12

¼

Ar12 ¼

and 

v2 Ar vN22

!

and

Here, T0, V0 , H0 , and S0 are the temperature, molar volume, molar enthalpy, and molar entropy at the reference state, and  Cv;i is the ideal-gas isochoric heat capacity of pure components. Partial differentiation of Eq. (8) with respect to mole numbers results in

A11 ¼

(14)

v2 Ar vN21

ð8Þ

where N ¼ N1 þ N2, V ¼ V=N, and Z

Vred ðxÞ ; V

where Tred and Vred are the reducing parameters. They are functions of composition. The first and second partial derivatives of Ar with respect to mole numbers are calculated as follows:   r    r  vA vðNRTar Þ va ; (15) ¼ ¼ RT ar þ N Ar1 ¼ vN1 T;V;N2 vN1 vN1 T;V;N2



A ðT; V; N1 ; N2 Þ ¼ N1 Q1 ðTÞ þ N2 Q2 ðTÞ   V N1 N2  N2 ln ;  N1 ln  RT N ln N N V0

(13)

and

Derivatives of the ideal-gas mixture part 

Tred ðxÞ T

(11)

Derivatives of the residual part

Generally, the residual part of the Helmholtz free energy Ar is formulated in terms of its dimensionless form ar ¼ Ar/(NRT ). For the water þ ammonia mixture, Tillner-Roth and Friend (1998) expressed ar as Ar ðT; V; N1 ; N2 Þ ¼ ar ðs; d; xÞ NRT ¼ ð1  xÞar1 ðs; dÞ þ xar2 ðs; dÞ þ Daðs; d; xÞ;

         v2 ar vs vs vd vd vx vx þ ardd þ arxx ¼ arss vN1 vN2 vN1 vN2 vN1 vN2 vN1 vN2  2   2   2  v s v d v x þ ard þ arx þ ars vN1 vN2 vN1 vN2 vN1 vN2          vs vd vs vd vs vx þ þ arsx þ arsd vN1 vN2 vN2 vN1 vN1 vN2          vs vx vd vx vd vx r þ þ adx þ vN2 vN1 vN1 vN2 vN2 vN1 (21) Here,

(12)

where x is the mole fraction of ammonia (x ¼ N2/N ), ari is the residual part of the pure component equation of state, Da is the departure function, and s and d are dimensionless temperature and volume defined as

arX ¼

 r va ; vX

arXX ¼



 v2 ar ; vX2

and arXY ¼

v2 ar ; vXvY

where X and Y stand for s, d, or x. The expressions for (var/vN2) and (v2ar/vN22) are obtained from Eqs. (19) and (20) by exchanging the subscript 1 for 2. All partial derivatives appearing in

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international journal of refrigeration 32 (2009) 95–101

650

Table 1 – Critical temperatures, pressures, and molar volumes of the water D ammonia mixture calculated from the formulation by Tillner-Roth and Friend (1998) pc [MPa]

Vc [dm3 mol1]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

647.096 639.834 631.062 621.797 611.759 601.300 590.909 580.393 569.650 558.648 547.378 535.833 523.988 511.791 499.156 485.977 472.183 457.941 446.048 430.271 405.400

22.064 22.412 22.294 22.068 21.798 21.563 21.359 21.153 20.931 20.684 20.408 20.095 19.731 19.295 18.750 18.044 17.127 15.992 14.962 13.676 11.360

0.05595 0.05691 0.05761 0.05752 0.05565 0.05360 0.05273 0.05237 0.05224 0.05224 0.05234 0.05254 0.05281 0.05317 0.05362 0.05424 0.05522 0.05717 0.06542 0.07433 0.07569

a Mole fraction of ammonia.

Eqs. (19)–(21) are derived from the formulation. The derivations are presented in Appendix A.

3.

Result and discussion

For a given set of (N1, N2), T and V satisfying the critical point criteria given by Eqs. (1) and (2) are the critical temperature Tc and critical volume Vc. The iterative procedure to find Tc and Vc using the Newton–Raphson method is presented elsewhere (Akasaka, 2008). In this study, initial estimates for Tc and Vc were given by the correlations presented later. The critical pressure pc was calculated with the formulation from Tc and Vc. The calculated Tc, Vc ¼ Vc =ðN1 þ N2 Þ, and pc for various compositions are listed in Table 1. Table 2 summarizes available experimental critical point data for the water þ ammonia mixture. Measurements of Tc and pc were reported by Postma (1920); Rizvi and Heidemann (1987); and Sassen et al. (1990). The measurements of Tc are consistent with each other within experimental uncertainties. However, there are some differences in measurements of pc

600

Critical temperature, Tc [K]

Tc [K]

550

447 500

446

445 0.89 400

0.6

0.8

1

26 REFPROP 8 Tillner−Roth and Friend

Postma Rizvi and Heidemann Sassen et al. Sakabe et al.

24

22

20

18

15.1

16 15.0 14

10

1

0.4

between the data of Rizvi and Heidemann and of Sassen et al. Van Poolen and Rainwater (1998) assessed available experimental VLE data in the critical region, and concluded that the VLE data by Sassen et al. (1990) is the most reliable. Therefore, the measurements of pc by Sassen et al. can be considered to be more accurate. Experimental data for Vc is very

Author

3 9 7

0.2

0.91

Fig. 1 – Critial temperatures calculated from the formulation by Tillner-Roth and Friend (1998) and experimental values (Postma, 1920; Rizvi and Heidemann, 1987; Sassen et al., 1990; Sakabe et al., 2007).

12

Tc and pc Postma (1920) Rizvi and Heidemann (1987) Sassen et al. (1990) Tc, pc and Vc Sakabe et al. (2007)

0

0.90

Mole fraction of ammonia, x

Table 2 – Experimental data for the critical point of the water D ammonia mixture No. of points

REFPROP 8 Tillner−Roth and Friend

450

Critical pressure, pc [MPa]

xa

Postma Rizvi and Heidemann Sassen et al. Sakabe et al.

14.9 0.89 0

0.90 0.2

0.91 0.4

0.6

0.8

1

Mole fraction of ammonia, x Fig. 2 – Critial pressures calculated from the formulation by Tillner-Roth and Friend (1998) and experimental values (Postma, 1920; Rizvi and Heidemann, 1987; Sassen et al., 1990; Sakabe et al., 2007).

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international journal of refrigeration 32 (2009) 95–101

1.40

1.000

0.995

δ

Table 3 – Coefficients ai, bi, and ci of Eqs. (22)–(24)

1.35

i

ai

bi

ci

1.30

0 1 2 3 4 5

0.999656 0.202431 0.444892 0.647487 0.371859 –

1.00235 0.123541 1.27823 2.19357 1.52625 –

1.00389 0.541515 4.90462 11.9654 12.5743 5.32934

1.25 0.990

τ [−]

1.15 0.985

τ

δ [−]

1.20

1.10 1.05

0.980

1.00 0.975

0

0.2

0.4

0.6

0.8

1

0.95

Mole fraction of ammonia, x Fig. 3 – Changes of s and d on the critical locus.

limited. Only one measurement has recently been published by Sakabe et al. (2007). Figs. 1 and 2 show the calculated critical locus plotted on xTc and xpc diagrams, as well as the experimental data. The experimental errors in Tc and pc of the measurements by Sassen et al. were reported to be  0.5 K and  0.02 MPa. For comparison, an estimation by REFPROP (Lemmon et al., 2007) is also plotted on the diagrams. The estimation probably comes from a particular correlation (the method used in REFPROP is undocumented). The calculation represents well all experimental values on the xTc diagram, and that shows better agreement with the measurements by Sassen et al. on the xpc diagram. The average absolute derivations between

Critical molar volume, Vc [dm3/mol]

0.080 Sakabe et al. REFPROP 8 Tillner−Roth and Friend

0.075

the data of Sassen et al. and the calculation with the formulation are 0.43% for Tc and 2.0% for pc. The REFPROP estimation for Tc agrees well with all experimental values, but that for pc deviates from both data of Sassen et al. and of Rizvi and Heidemann in the range of x from 0.3 to 0.8. A small local maximal and minimal pattern on the calculated critical locus on the xTc and xpc diagrams is observed around x ¼ 0.9. Fig. 3 explains this observation. This figure shows the changes of s and d along the critical locus. To represent a strong molecular interaction in ammonia-rich mixtures, the formulation uses exponential composition terms in the equations for the reducing parameters. Therefore, as shown in Fig. 3, s and d exhibit a steep gradient around x ¼ 0.9. This behavior causes the unnatural pattern on the critical locus. Fig. 4 shows the calculated critical locus plotted on an x  Vc diagram, as well as the measurement by Sakabe et al. (2007) and the estimation by REFPROP. Although the measurement lies on the calculated critical locus, additional experimental values for Vc are desired to further assess this behavior of the critical locus on the x  Vc diagram. The following polynomial correlations reproduce the critical point calculated using the formulation within 0.16% for Tc, 0.28% for pc, and 1.1% for Vc : 4 Tc X ¼ ai xi ; T i¼0

(22)

4 pc X ¼ bi xi ;  p i¼0

(23)

and 5 Vc X ¼ ci xi ;  V i¼0

0.070

(24)

where T* ¼ 647.096 K, p* ¼ 22.064 MPa, and V ¼ 0:055948 dm3 mol1. The coefficients ai, bi, and ci are presented in Table 3. Eqs. (22) and (24) were used for giving initial estimates of the iterative procedure. The average absolute deviations between the experimental data of Sassen et al. and the calculation with the correlations are 0.12% for Tc and 0.26% for pc.

0.065

0.060

0.055

0.050

0

0.2

0.4

0.6

0.8

Mole fraction of ammonia, x Fig. 4 – Critial molar volume calculated from the formulation by Tillner-Roth and Friend (1998) and experimental values (Sakabe et al., 2007).

1

4.

Conclusions

The critical point of the water þ ammonia mixture was calculated from the formulation developed by Tillner-Roth and Friend (1998). A smooth critical locus linking between the critical points of pure water and ammonia was obtained. This

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international journal of refrigeration 32 (2009) 95–101

critical locus showed a good agreement with probably the most reliable experimental data. Simple polynomial correlations for the critical temperature, pressure, and molar volume were developed. The correlations are helpful for design and simulation of the cycles using the water þ ammonia mixture as working fluid. The result obtained in this study is at a satisfactory level, but additional experimental data for the critical point is desired for more accurate assessment of the result.

Appendix A Derivations of the partial derivatives appearing in Eqs. (19)–(21)



vd vN2

 ¼ V;N1

    1 dVred vx : Vred þ N dx V vN2

(A.7)

Using d ¼ Vred =V, Eqs. (A.6) and (A.7) can be rearranged to      vd d x dVred (A.8) ¼ 1 vN1 V;N2 N Vred dx and      vd d 1  x dVred : ¼ 1þ dx vN2 V;N1 N Vred

(A.9)

The second partial derivatives of d are ! ! 2 v2 d dx2 d Vred ; ¼ vN21 N2 Vred dx2 V;N

(A.10)

2

According to the definition, the partial derivatives of x with respect to mole numbers are     vx x vx 1x ¼ ; ¼ ; vN1 N2 N vN2 N1 N v2 x vN21

! N2

2x ¼ 2; N

v2 x vN22

! N1

2x  2 ¼ ; N2

! 2 dð1  xÞ2 d Vred ; dx2 N2 Vred

(A.11)

! 2 v2 d dxð1  xÞ d Vred : ¼ 2 dx2 vN1 vN2 N Vred

(A.12)

v2 d vN22

! ¼ V;N1

and

and

For the reducing parameters Tred and Vred , Tillner-Roth and Friend (1998) used the functional forms

v2 x 2x  1 ¼ : vN1 vN2 N2

Tred ðxÞ ¼ ð1  xÞ2 Tc;1 þ x2 Tc;2Þ þ 2xð1  xk ÞTc;12

The first partial derivatives of s with respect to mole numbers can be written as          vs 1 vTred 1 dTred vx sx dTred ¼ ¼ ¼ vN1 T;N2 T vN1 N2 T dx vN1 N2 NTred dx (A.1) and     vs sð1  xÞ dTred ¼ dx vN2 T;N1 NTred Similarly, the second partial derivatives of s are !    v2 s 1 v dTred vx ¼ dx T vN1 vN1 vN21 T;N2 "  !#  2 sx dTred d Tred ¼ 2 2 ; þx dx dx2 N Tred

and   Vred ðxÞ ¼ ð1  xÞ2 Vc;1 þ x2 Vc;2 þ 2x 1  xl Vc;12 ;

(A.14)

where Tc,i and Vc;i are the critical temperature and molar volume of the pure component, and Tc,12 and Vc;12 are defined as Tc;12 ¼

(A.2)

(A.13)

kT ðTc;1 þ Tc;2 Þ 2

and Vc;12 ¼

kV ðVc;1 þ Vc;2 Þ 2

The values for the adjustable parameters k, l, kT, and kV are presented elsewhere (Tillner-Roth and Friend, 1998). From Eqs. (A.13) and (A.14), the first and second derivatives of Tred and Vred with respect to x are

dXred ¼ 2Xc;1 þ 2xðXc;1 þ Xc;2 Þ þ 2 1  xz ð1 þ zÞ Xc;12 dx

(A.3)

(A.15)

and 2

2

v s vN22

! T;N1

"  !#  2 sð1  xÞ dTred d Tred 2 ;  ð1  xÞ ¼ dx dx2 N2 Tred

d Xred ¼ 2ðXc;1 þ Xc;2 Þ  2zxz1 ð1 þ zÞXc;12 ; dx2 (A.4)

and "  !#  2 v2 s sð1  xÞ dTred d Tred 2 : þx ¼ dx dx2 vN1 vN2 N2 Tred

(A.5)

A little more attention needs to be paid to the partial derivatives of d with respect to mole numbers. Since the molar volume V ¼ V=N depends upon the mole numbers, !      vd v Vred 1 vðNVred Þ ¼ ¼ vN1 V;N2 vN1 V=N N2 V vN1 N2     1 dVred vx (A.6) ¼ Vred þ N dx V vN1 and

(A.16)

where X stands for T or V and z for k or l. Tillner-Roth and Friend (1998) adopted the water and ammonia equations of state developed by Wagner and Pruß (2002) and Tillner-Roth et al. (1993), and developed the following expression for Dar: " Dar ðs; d; xÞ ¼ BðxÞ a1 st1 dd1 þ

6 X i¼2

þ DðxÞJ14 ðs; dÞ;

# Ji ðs; dÞ þ CðxÞ

13 X

Ji ðs; dÞ

i¼7

(A.17)

where BðxÞ ¼ xð1  xg Þ;

(A.18)

CðxÞ ¼ x2 ð1  xg Þ

(A.19)

international journal of refrigeration 32 (2009) 95–101

DðxÞ ¼ x3 ð1  xg Þ;

" #   6 13 X X vDard t1 d1 1 ¼ ¼ Bx a1 d1 s d þ Ji;d þ Cx Ji;d þ Dx J14;d ; vx s;d i¼2 i¼7

(A.20)

Dardx

(A.21)

where

and

101

(A.35)

Ji ðs; dÞ ¼ ai sti ddi expð  dei Þ:

The values for the coefficients ai and exponents ti, di, ei, and g are presented elsewhere (Tillner-Roth and Friend, 1998). From Eq. (12), the first partial derivative of ar with respect to x are  r   va arx ¼ ¼  ar1  ar2 þ Darx ; (A.22) vx s;d

Ji;s ¼ ai ti ddi sti 1 expð  dei Þ

(A.36)

and Ji;d ¼ ai ðdi  ei dei Þddi 1 sti expð  dei Þ:

(A.37)

references

where Darx ¼



vDar vx

"



¼ Bx a1 st1 dd1 þ

s;d

6 X

# Ji þ C x

i¼2

13 X

Ji þ Dx J14 ;

(A.23)

i¼7

Bx ¼ 1  xg ð1  xg Þ;

(A.24)

Cx ¼ x½2  xg ð2 þ gÞ;

(A.25)

and Dx ¼ x2 ½3  xg ð3 þ gÞ:

(A.26) r

Similarly, the second partial derivative of a with respect to x are arxx ¼



v2 ar vx2



¼ Darxx ;

(A.27)

s;d

where Darxx ¼

" #  2 r 6 13 X X v Da t1 d1 a þ Cxx ¼ B s d þ J Ji þ Dxx J14 ; xx 1 i vx2 s;d i¼2 i¼7 (A.28)

Bxx ¼ gð1 þ gÞxg1 ;

(A.29)

Cxx ¼ 2  ð1 þ gÞð2 þ gÞxg ;

(A.30)

and Dxx ¼ 6x  ð2 þ gÞð3 þ gÞxgþ1 :

(A.31)

The derivatives of ars ¼ (var/vs) and ard ¼ (var/vs) with respect to x are  r  vas arsx ¼ ¼  ars;1  ars;2 þ Darsx (A.32) vx s;d and ardx ¼

 r  vad ¼  ard;1  ard;2 þ Dardx : vx s;d

(A.33)

The expressions for ars, i and ard, i are presented in the original papers (Wagner and Pruß, 2002; Tillner-Roth et al., 1993), and Darsx and Dardx are " #   6 13 X X vDars Darsx ¼ ¼ Bx a1 t1 st1 1 dd1 þ Ji;s þ Cx Ji;s þ Dx J14;s vx s;d i¼2 i¼7 (A.34) and

Akasaka, R., 2008. Calculation of the critical point for mixtures using mixture models based on Helmholtz energy equations of state. Fluid Phase Equilib. 263, 102–108. Baker, L.E., Luks, K.D., 1980. Critical point and saturation pressure calculations for multipoint systems. Soc. Petrol. Eng. 20, 15–24. Hicks, C.P., Young, C.L., 1977. Theoretical prediction of phase behavior at high temperatures and pressures for non-polar mixtures – Part 1. Computer solutions techniques and stability test. J. Chem. Soc., Faraday II 73, 597–612. Lemmon, E.W., Huber, M.L., McLinden, M.O., 2007. NIST Standard Reference Database 23 (REFPROP): Version 8.0. Peng, D.-Yu., Robinson, D.B., 1977. A rigorous method for predicting the critical properties of multicomponent systems from an equation of state. AIChE J. 23, 137–144. Postma, S., 1920. Le syste`me ammoniaque–eau. Rec. Trav. Chim. 39, 515–536. Rainwater, J.C., Tillner-Roth, R., 1999. Critical Region Vapor– Liquid Equilibrium Model of Ammonia–Water. In: Thirteenth International Conference on the Properties of Water and Steam Toronto, Canada. Rizvi, S.S.H., Heidemann, R.A., 1987. Vapor–liquid equilibria in the ammonia–water system. J. Chem. Eng. Data 32, 183–191. Sakabe, A., Arai, D., Uematsu, M., 2007. Measurements of the Critical Properties for Ammonia–Water System. In: The Twenty-Eighth Japan Symposium on Thermophysical Properties Sapporo, Japan. Sassen, C.L., van Kwartel, R.A.C., van der Kooi, H.J., de Swaan Arons, J., 1990. Vapor–liquid equilibria for the system ammonia þ water up to the critical region. J. Chem. Eng. Data 35, 140–144. Spear, R.R., Robinson Jr., R.L., Chao, K.-Chu, 1971. Critical states of ternary mixtures and equations of state. Ind. Eng. Chem. Fundam. 10, 577–592. Teja, A.S., Rowlinson, J.S., 1973. The prediction of the thermodynamic properties of fluids and fluid mixtures – IV. Critical and azeotropic states. Chem. Eng. Sci. 28, 529–538. Tillner-Roth, R., Harms-Watzenberg, F., Baehr, H.D., 1993. Eine neue Fudamentalgleichung fu¨r ammoniak. DKVTagungsbericht 20, 167–181. Tillner-Roth, R., Friend, D.G., 1998. A Helmholtz free energy formulation of the thermodynamic properties of the mixture water þ ammonia. J. Phys. Chem. Ref. Data 27, 63–96. Van Poolen, L.J., Rainwater, J.C., 1998. Critical-region model for bubble curves of ammonia–water with extrapolation to low pressures. Fluid Phase Equilib. 150–151, 451–458. Wagner, W., Pruß, A., 2002. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31, 387–535.