Thermodynamic properties of ammonia water mixtures for power-cycle applications

Energy 24 (1999) 525 536 www.elsevier.com/locate/energy Thermodynamic properties of ammonia water mixtures for power-cycle applications Feng Xu a, D. Yogi Goswami b,* a Donlee Technologies, Inc., 693 North Hills Road, York, PA 17402, USA b Department of Mechanical Engineering, University of Florida, Gainesville, FL 32611, USA Received 28 July 1997 Abstract Ammonia water mixtures have been used as working fluids in absorption refrigeration cycles for several decades. Their use as multi-component working fluids for power cycles has been investigated recently. The thermodynamic properties required are known or may be calculated at elevated temperatures pressures. We present a new method for these computations using Gibbs free energies empirical equations for bubble dew point temperature to calculate phase equilibria. Comparisons of calculated experimental data show excellent agreement. 1999 Published by Elsevier Science Ltd. All rights reserved. 1. Background Many studies have been published on vapor liquid equilibrium (VLE) the thermodynamic properties of ammonia water mixtures, including p t x y data caloric properties. For enthalpy data, see Refs. [1 3]. Ref. [4] published new values of enthalpy entropy from 70 to 370 F pressure up to 300 psia using experimental data from [2,3,5]. Ref. [6] created tables of VLE caloric properties that were used by other researchers to propose computational models [7 9]. In Ref. [10], measured data from [11] were used to give correlations for pressures of 0.2 to 110 bar temperatures of 230 to 600 K. Refs. [12 16] also presented models for calculating the thermodynamic data at elevated temperatures pressures. In the present study, a method that combines the Gibbs free energy method for mixture properties bubble dew point temperature equations for phase equilibrium is used. This method * Corresponding author. Fax: 1-352-392-1701; e-mail: solar@cimar.me.ufl.edu 0360-5442/99/$ - see front matter 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S03 60-544 2(99)00007-9

526 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 combines the advantages of the two avoids the need for iterations for phase equilibrium by the fugacity method. 2. Gibbs free energy equation for a pure component The Gibbs free energy of a pure component is given by T P T G h 0 Ts 0 C p dt v dp T (C p /T) dt, (1) T 0 P 0 T 0 where h 0, s 0, T 0 P 0 are the specific enthalpy, specific entropy, temperature pressure at the reference state. Use of empirical relations for v C p [9] leads to the following equations. For the liquid phase: G L r h L r,o T r s L r,o B 1 (T r T r,o ) (B 2 /2)(T 2 r T 2 r,o) (B 3 /3)(T 3 r T 3 r,o) B 1 T r ln(t r /T r,o ) B 2 T r (T r T r,o ) (B 3 /2)(T 2 r T 2 r,o) (A 1 A 3 T r A 4 T 2 r)(p r (2) P r,o ) (A 2 /2)(P 2 r P 2 r,o). For the gas phase: G g r h g r,o T r s g r,o D 1 (T r T r,o ) (D 2 /2)(T 2 r T 2 r,o) (D 3 /3)(T 3 r T 3 r,o) D 1 T r ln(t r /T r,o ) D 2 T r (T r T r,o ) (D 3 /2)(T 2 r T 2 r,o) T r ln(p r /P r,o ) C 1 (P r (3) P r,o ) C 2 (P r /T 3 r 4P r,o /T 3 r,o 3P r,o T r /T 4 r,o) C 3 (P r /T 11 r 12P r,o /T 11 r,o 11P r,o T r /T 12 r,o) (C 4 /3)(P 3 r/t 11 r 12P 3 r,o/t 11 r,o 11P 3 r,ot r /T 12 r,o). Here, the superscripts are L for liquid g for gas, while subscript o is for the ideal gas state. The reduced (subscript r) thermodynamic properties are T r T/T B, P r P/P B, G r G/RT B, h r h/rt B, s r s/r v r vp B /RT B. The reference values for the reduced properties are R 8.314 kj/kmol K, T B 100 K P B 10 bar. The constants in Eqs. (2) (3) are given in Table 1. 3. Thermodynamic properties of a pure component The molar specific enthalpy, entropy volume are related to Gibbs free energy, in terms of reduced variables, by h RT B T r 2 (G T r /T r, (4) r ) P r

F. Xu, D.Y. Goswami/Energy 24 (1999) 525 536 527 Table 1 Coefficients of Eqs. (2) (3) Coefficient Ammonia Water A 1 3.971423 10 2 2.748796 10 2 A 2 1.790557 10 5 1.016665 10 5 A 3 1.308905 10 2 4.452025 10 3 A 4 3.752836 10 3 8.389246 10 4 B 1 1.634519 10 +1 1.214557 10 +1 B 2 6.508119 1.898065 B 3 1.448937 2.911966 10 2 C 1 1.049377 10 2 2.136131 10 2 C 2 8.288224 3.169291 10 +1 C 3 6.647257 10 +2 4.634611 10 +4 C 4 3.045352 10 +3 0.0 D 1 3.673647 4.019170 D 2 9.989629 10 2 5.175550 10 2 D 3 3.617622 10 2 1.951939 10 2 h L r,o 4.878573 21.821141 h g r,o 26.468873 60.965058 s L r,o 1.644773 5.733498 s g r,o 8.339026 13.453430 T r,o 3.2252 5.0705 P r,o 2.000 3.000 s R G r T r P r (5) v RT B P B G r P r T r. (6) 4. Ammonia water liquid mixtures The Gibbs excess energy for liquid mixtures allows for deviation from ideal solution behavior. The Gibbs excess energy of a liquid mixture is expressed by the relationship proposed in [9], which is limited to three terms is given by: G E r [F 1 F 2 (2x 1) F 3 (2x 1) 2 ](1 x), (7) where x is the ammonia mass fraction F 1 E 1 E 2 P r (E 3 E 4 P r )T r E 5 /T 4 E 6 /T 2 r,

528 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 F 2 E 4 E 8 P 4 (E 9 E 10 P r )T r E 11 /T r E 12 /T 2 r F 3 E 13 E 14 P r E 15 /T r E 16 /T 2 r The constants for Eq. (7) are given in Table 2. The excess enthalpy, entropy volume for the liquid mixtures are given as: h E RT B T r 2 (G T E r /T r, (8) r ) P r, x s E R GE r T r P r, x (9) v E RT B P B GE r. (10) P r T r, x In addition, the enthalpy, entropy volume of a liquid mixture are given by: h L m x f h L a (1 x f )h L w h E, (11) s L m x f s L a (1 x f )s L w s E s mix, (12) s mix R[x f ln(x f ) (1 x f ) ln(1 x f )] (13) v L m x f v L a (1 x f )v L w v E, (14) Table 2 Coefficients of Eq. (7) E 1 41.733398 E 9 0.387983 E 2 0.02414 E 10 0.004772 E 3 6.702285 E 11 4.648107 E 4 0.011475 E 12 0.836376 E 5 63.608967 E 13 3.553627 E 6 62.490768 E 14 0.000904 E 7 1.761064 E 15 24.361723 E 8 0.008626 E 16 20.736547

F. Xu, D.Y. Goswami/Energy 24 (1999) 525 536 529 where subscripts a w refer to ammonia water, respectively subscript f refers to the saturated liquid condition. 5. Ammonia water vapor mixture Ammonia water vapor mixtures are often assumed to be ideal solutions. The enthalpy, entropy volume of the vapor mixture are computed by: h g m x g h g a (1 x g )h g w, (15) s g m x g s g a (1 x g )s g w s mix (16) v g m x g v g a (1 x g )v g w. (17) 6. Vapor liquid equilibrium At equilibrium, binary mixtures must have the same temperature pressure. Moreover, the partial fugacity of each component in the liquid gas mixtures must be equal: L fˆ a fˆg a, (18) L fˆ w fˆg w, (19) where fˆ is the fugacity of each component in the mixture at equilibrium. The fugacities of ammonia water in liquid mixtures are given by [17]: L fˆ a a f 0 ax a (20) L fˆ w w f 0 w(1 x) w, (21) where is the activity coefficient, f 0 is the stard-state fugacity of the pure liquid component corrected to zero pressure, is the Poynting correction factor from zero pressure to saturation pressure of the mixture x is the ammonia mass fraction in liquid phase. Assuming an ideal mixture in the vapor phase, the fugacities of the pure components in the vapor mixtures are given by g fˆ a a Py (22)

530 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 g fˆ w w P(1 y), (23) where is the fugacity coefficient y is the ammonia mass fraction in vapour phase. Eqs. (18) (19) are used to calculate the boiling dew point temperatures given the pressure ammonia concentration in the liquid mixture. However, these two equations must be solved iteratively to produce the VLE properties of ammonia water mixtures. Alternatively, the bubble dew point temperatures can be calculated using the explicit equations developed in Ref. [14]. 7. Bubble point dew point temperature equations Eqs. (24) (25), developed in [14], determine the start end of the mixture phase change compute the mass fractions of ammonia water in the liquid vapor phases, respectively. This avoids the complicated method of calculating the fugacity coefficient of a component in a mixture to determine the bubble (T b ) dew point (T d ) temperatures. 7 T b T c i 1 10 (C i j 1 C ij x j )[ln(p c /P)] i (24) 6 4 T d T c (a i A ij [ln(1.0001 x)] j [ln(p c /P)]) i, (25) i 1 j 1 where 4 T c T cw a i x i, (26) i 1 8 P c P cw exp( i 1 P in psia T in F. b i x j ), (27) 8. Results In this study, the Gibbs free energy method is used to calculate the properties of pure ammonia water [Eqs. (2) (6)]. The properties of the ammonia water mixture are also calculated from the Gibbs free energy method using Eqs. (7) (17). In order to determine the phase quilibrium, bubble dew points are calculated using the alternative method of Eqs. (24) (27) instead of the conventional method of equating the fugacities [Eqs. (18) (23)]. Using the alternative method

F. Xu, D.Y. Goswami/Energy 24 (1999) 525 536 531 avoids the iterative solution necessary to solve Eqs. (18) (23), thereby reducing the computational time. The property data generated in this study have been compared with available experimental theoretical data in the literature. 9. Comparison of bubble dew point temperatures Fig. 1 shows that the bubble dew point temperatures generated by this study compare favorably with the data from Ref. [6]. The differences between our computed values the data are less than 0.3%. Refs. [9,10] are reported to have differences of up to 2% from these data. 10. Comparison of saturation pressure at constant temperature Figs. 2 3 show the saturation pressures of ammonia water mixtures as compared with the data from Ref. [11]. For temperatures less than 406 K, the computational results fit the experimental data well, except at saturated liquid pressures. At higher temperatures, our computed values are within 5% of the data even at pressures higher than 110 bar, while Ref. [9] has reported a difference of more than 15%. Ref. [10] reported an error of less than 5% under 110 bar higher errors over 110 bar. Fig. 1. Bubble dew point temperatures at a pressure of 34.47 bar.

532 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 Fig. 2. Saturation pressures of ammonia water mixtures at 333.15 K. Fig. 3. Saturation pressures of ammonia water mixtures at 405.95 K.

F. Xu, D.Y. Goswami/Energy 24 (1999) 525 536 533 11. Comparison of saturated liquid vapor enthalpy 1. Saturated liquid enthalpy. The saturated liquid enthalpy of this work is compared with the data from Ref. [6], as shown in Fig. 4. The differences are less than 2% for all the data. 2. Saturated vapor enthalpy. The saturated vapor enthalpy at constant pressure is shown in Fig. 5. The agreement with the data is within 3%. Ref. [10] reported a 5% maximum difference. The mass fraction of ammonia vapor shown in this figure is the ammonia liquid mass fraction when the mixture reaches a saturated state. So, in order to compute the saturated vapor enthalpy, the ammonia vapor mass fraction must be determined first. 12. Comparison of saturated liquid vapor entropy The value of entropy is very important in predicting the performance of a turbine in a power cycle. Entropy data are also essential to the second-law analysis of thermal systems. Ref. [4] published saturated liquid vapor entropy data based on experimental data from [2,3,5]. Ref. [16] published calculated entropy. The entropy data from the present study are compared with the experimental data in Ref. [4] the computational data of Ref. [16]. 1. Saturated liquid entropy. Fig. 6 shows saturated liquid entropy data compared with those of Fig. 4. Saturated liquid enthalpy of ammonia water mixtures at 34.47 bar.

534 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 Fig. 5. Saturated vapor enthalpy of ammonia water mixtures at 34.47 bar. Fig. 6. Entropy of saturated liquid at 310.9 K.

F. Xu, D.Y. Goswami/Energy 24 (1999) 525 536 535 Fig. 7. Entropy of saturated vapor at 310.9 K. Ref. [4]. Our data agree with the experimental data of [4] much better than the data generated by the method of Ref. [16]. 2. Saturated vapor entropy. Fig. 7 shows an excellent agreement of our computed values of saturated vapor entropy with the data of Ref. [4]. Data computed by Ref. [16] are consistently lower. Since it was very difficult to identify saturated vapor entropy data from Ref. [16], we did not compare our results with them. 13. Conclusion Different methods for calculating the properties of ammonia water mixtures are studied. A practical accurate method is used in this study. This method uses Gibbs free energy equations for pure ammonia water properties, empirical bubble dew point temperature equations for vapor liquid equilibrium. The iterations necessary for calculating the bubble dew point temperatures by the fugacity method are avoided. Therefore, this method is much faster than using the fugacity method. The computational results have been compared with accepted experimental data in the literature show very good agreement. References [1] Jennings BH, Shannon FP. Refrig Eng 1938;44:333. [2] Zinner KZ, Gesamt Z. Kalte-Ind 1934;41:21.

536 F. Xu, D.Y. Goswami/ Energy 24 (1999) 525 536 [3] Wucherer J, Gesamt Z. Kalte-Ind 1932;39:97. [4] Scatchard G, Epstein LF, Warburton J, Cody PJ. Refrig Eng 1947;53:413. [5] Perman EP. J Chem Soc 1901;79:718. [6] Macriss RA, Eakine BE, Ellington RT, Huebler J. Research bulletin no 34. Chicago (IL): Chicago Institute of Gas Technology, 1964. [7] Gupta CP, Sharma CP. ASME paper 75-WA/PID-2. New York (NY): ASME, 1975. [8] Schulz SCG. Proc XIIth Int Cong Refrig 1972;2:431. [9] Ziegler B, Trepp C. Int J Refrig 1984;7:101. [10] Ibrahim OM, Klein SA. ASHRAE Trans 1993;99:1495. [11] Gillespie PC, Wilding WV, Wilson GM. AIChE Symp Ser 1987;83:97. [12] Kalina AI. ASME paper 83-JPGC-GT-3. New York (NY): ASME, 1983. [13] Herold KE, Han K, Moran MJ. ASME Proc 1988;4:65. [14] El-Sayed YM, Tribus M. ASME special publication AES 1. New York (NY): ASME, 1985:89. [15] Kalina AI, Tribus M, El-Sayed YM. ASME paper 86-WA/HT-54. New York (NY): ASME, 1986. [16] Park YM, Sonntag RE. ASHRAE Trans 1992;97:150. [17] Walas SM. Phase equilibria in chemical engineering. Stoneham (MD): Butterworths, 1985.