(PDF) Vaporization Enthalpy of Pure Refrigerants: Comparative Study

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Ind. Eng. Chem. Res. 2010, 49, 5018–5026

Vaporization Enthalpy of Pure Refrigerants: Comparative Study of Eighteen Correlations A. Mulero,*,† M. I. Parra,‡ K. K. Park,§ and F. L. Romań| Departamento de Fı´sica Aplicada and Departamento de Matema´ticas, UniVersidad de Extremadura, AVda. de ElVas s/n, Badajoz, Spain, CP 06071, School of Mechanical and AutomotiVe Engineering, Kookmin UniVersity, Seongbuk-gu, Seoul, Republic of Korea, 136-702, and Departamento de Fı´sica Aplicada, UniVersidad de Salamanca, EPS de Zamora, AVda. Cardenal Cisneros 34, Zamora, Spain, CP 49022

In this work, the performance and accuracy of 18 correlations for the vaporization enthalpy of 48 pure substances is studied. Forty-six of these substances are refrigerants, and two (hexane and heptane) could be used in refrigerant mixtures. Seven of the correlations include 2 adjustable coefficients for every fluid, whereas the other 11 have three coefficients. Some of these correlations are classical expressions, but others have been proposed recently or have been slightly modified or extended for this study. In all the cases, we consider the normal boiling point as a constraint, and then the input parameters are the critical temperature, and the temperature and vaporization enthalpy at the normal boiling point. The data and input properties were taken from both the DIPPR (1968 data) and the NIST databases (10 650 data). Thus, as a first step, we have studied the possible differences between these data. The results showed that the correlation recently proposed by Park gives the lowest mean average percentage deviations for both two-coefficient and three-coefficient models. Introduction The enthalpy of vaporization is an essential property for the analysis of refrigeration cycles. Moreover, since the enthalpy of vaporization is involved in most of the correlations for the prediction of phase change heat transfer, its calculation also affects the evaluation of the heat transfer coefficients. For these practical purposes, accurate expressions are needed in order to calculate the vaporization enthalpy at different temperatures along the whole vapor-liquid equilibrium curve. Some of these expressions are based on the corresponding-states principle, and they can be considered as predictive, because they have fixed coefficients and thus only require the values of certain properties of the fluid as input.1-9 Nevertheless, it is well-known that their accuracy for certain families of fluids is only limited,9 so that, in order to obtain better accuracy, one needs to consider correlation expressions including at least two or three adjustable coefficients. A plethora of correlations have been proposed in order to fit the enthalpy of vaporization of pure fluids as a function of the temperature.10-14 Some of them include coefficients that must be calculated for each substance,15-24 or the value of the enthalpy at a given temperature.25-34 The choice of a specific equation depends on several factors, mainly, the available number of experimental data, the temperature range over which one wants to apply the correlation, and the set of substances chosen for testing the equation. These factors and the desired or expected accuracy of the predicted values will also determine the number of adjustable parameters or coefficients in the correlation under consideration. Recently, Park35 made a comparative study of the accuracy and performance of several correlations when applied to 22 pure refrigerants. In all of the cases, the boiling temperature and * To whom correspondence should be addressed. E-mail: mulero@ unex.es. † Departamento de Fı´sica Aplicada, Universidad de Extremadura. ‡ Departamento de Matema´ticas, Universidad de Extremadura. § Kookmin University. | Universidad de Salamanca.

vaporization enthalpy, as well as the critical temperature, were considered as inputs, and coefficients were obtained by a fitting procedure to data from ASHRAE.36 Three previous proposals including two adjustable coefficients were studiedsthose of Somayajulu,19 Majer et al.,13 and Xiang;22 the mean average absolute deviations (MAADs) obtained for the 22 fluids being 0.27%, 0.18%, and 0.18%, respectively. Moreover, a new proposal was introduced, which reduced the MAADs to 0.14%. Similarly, four previous correlations including three adjustable coefficients were also consideredsthose of Guermouche and Vergnaund15 (a modified version was used) with MAAD ) 0.08%; Radosz and Lydersen,16 MAAD ) 0.08%; Aarebrot,17 MAAD ) 0.13%; and Somayajulu,19 MAAD ) 0.08%. The proposal of Park reduced the MAAD to only 0.05%. Nevertheless, the Park study does not consider some other recent interesting proposals, such as those of Chen et al.,28 Romań et al.,31 Velasco et al.,32 and Al-Shorachi and Hashim,33 and that used in the DIPPR database,37 which is due to the work of Chueh and Swanson.38 Also recently, Jovanovic and Grozdanic24 have tested 27 correlations, including a new one proposed by them, using 1958 data for 162 compounds. The new equation, with 4 adjustable coefficients, gave an overall percentage deviation of 0.27% for the whole temperature range for 157 fluids, and of 0.59% for temperatures above 0.9 times the critical temperature (only for 19 fluids). Nevertheless, no comparison was made with models published after 1997. Moreover, the temperature range considered for many substances was not very large, and some other correlations give similar results. For that reason, it is interesting to test this proposal over the whole temperature range and compare it with the results obtained by using some other recent proposals. In the present work, the Park study is extended by including two different sources of data (DIPPR37 and NIST39), one of them with values covering the whole temperature range and some other recently proposed correlations cited above. Moreover, some of those correlations, which were originally proposed by including only fixed (instead of adjustable) coefficients, have been extended in order to include two or three adjustable

10.1021/ie901015f  2010 American Chemical Society Published on Web 04/14/2010

Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010

coefficients. In all cases, the models have been conveniently modified in order to include the normal boiling temperature and vaporization enthalpy at this temperature as input parameters. Sources of Data The data and properties used as inputs in the model were taken from both the DIPPR37 and the NIST39 databases. These are well-known databases readily accessible to everyone and are different in some respects, as will be explained below. The DIPPR37 (Design Institute for Physical Property Data) database gives single values for property constants and single correlations for temperature dependent properties. Data are collected from recognized compilations, review articles, handbooks, and primary sources. The original sources are available in the database, but the listed value is the single best value as determined by the project staff and steering committee. The accepted values may be derived from many experimental determinations or, when no reliable experimental data are available, estimated from tested correlations and prediction methods. Quality codes indicating the deviation of data with respect to a regression are used as reference instead of the experimental uncertainty of the accepted data. The evaluation procedure is at the heart of the project, and the procedures and practices are rigorous and thorough to ensure that the best value is indeed selected based on available information.40-43 In particular, the version of this database used includes accepted data for the vaporization enthalpy for more than 1500 substances. These accepted data include experimental as well as predicted values. The prediction method used in the database is the Clapeyron equation. To apply this equation, DIPPR used the values of the derivatives of vapor pressure with respect to temperature obtained from the vapor pressure correlation in a differentiated form; liquid volumes were calculated from the liquid density correlation; and the vapor volume was calculated by the Lee-Erbar-Edmister equation.44 When the Clapeyron estimation resulted in a poor curve or disagreed with experimental data, prediction by the Lee-Kesler method45 was used. As a last resort, the Watson method25 was used if there is at least one point of reliable vaporization enthalpy datum. In this work, we considered only the values accepted by the DIPPR team in the 2003 version of the database. Sometimes there are two similar accepted values for the same or a very similar temperature. In particular, slightly different values were accepted for the vaporization enthalpy at the boiling point. Then we considered as valid the vaporization enthalpy obtained by using the accepted value of the boiling temperature in the particular correlation proposed by DIPPR. Moreover, we would note that for some fluids the triple-point temperature was estimated as the melting temperature. In previous analyses, we have found that some of the accepted data do not agree with most of the rest of the data for the same fluid, i.e. there are accepted data with values that are clearly higher or lower than all the other values (see refs 8 and 9 for examples). The disagreement may be due to mistakes or because different sources give very different values.8,9 These data were not considered in our comparison, and a list of them is available upon request. The other database used is the NIST Standard Reference Database 69: NIST Chemistry WebBook.39 It provides access to data compiled and distributed by NIST under the Standard Reference Data Program. It contains thermophysical property data, as well as thermochemical data, reaction thermochemical data, etc. In particular, accurate thermophysical property data

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are available for 75 fluids, including vaporization enthalpy, phase change densities, critical, boiling and triple points, etc. In this work, the data used for the enthalpy of vaporization are not real experimental data, but are values calculated using reliable equations of state applying the equilibrium condition at a given saturation temperature or at a given saturation pressure. References relevant to the calculation are available on the site, but the accuracy of the data is not given Since the data were generated through a model, a very large number of data points with well-defined regular behavior were collected. The value of the vaporization enthalpy in kilojoules per kilogram was calculated as the difference between the enthalpy of saturated vapor and the enthalpy of saturated liquid at a given temperature. Five significant digits were maintained throughout the calculation. We searched for data for several refrigerants or fluids that can be used as pure or mixture refrigerants. We then selected the 48 fluids in Table 1 for which data are available in both databases. Indeed, some of those substances are not used as refrigerants but as foam blowing agents, propellants, or solvents. Some of them may no longer be used as refrigerants because of their toxicity or high ozone depletion and/or high globalwarming potential. Comparison of Data The most important difference between the DIPPR and NIST databases is that since the DIPPR data are from different sources one can not always expect regular behavior or a wide range of temperatures, while the NIST data are regular with well-defined behavior. As a prior step to the fitting procedure to the models, we checked the degree of agreement between the two sets of data. First, for some fluids we did not consider various data qualified as accepted by the DIPPR database. In particular, these excluded data were as follows: one datum for ethane at the boiling temperature, because it is clearly different from the other DIPPR and NIST data; one datum for R-13 at 298.15 K, and one datum for propane and R-14 at the boiling temperature, because all of them are very different from other data at similar temperatures; for R-14 and R-23, a source of predicted data (seven and six data, respectively), because the data do not extend to the correct critical point; for R-22, two data from the same DIPPR source for the boiling temperature and 298.15 K, which are clearly different from the other NIST and DIPPR sources; two DIPPR data for R-32, each being the only datum from a different source which clearly does not match the other data. We found excellent agreement, with practically the same values for the vaporization enthalpy, between the DIPPR and NIST sets for 23 fluids (ordered as in Table 1): R-11, R-13, R-23, methane, R-113, R-115, R-123, R-125, R-141b, R-142b, R-143a, ethane, R-236ea, butane, ethene, propene, R-C318, helium, ammonia, water, nitrogen, oxygen, and dinitrogen monoxide. It needs to be borne in mind that the temperature range where DIPPR data are available is not as wide as in the case of the NIST data. In the particular case of helium, only NIST data are available at low temperatures, but it is noticeable that, as can be seen in Figure 1, the behavior of the data is completely different from that observed in the other fluids. In the case of refrigerant R-236ea, there are data at low temperatures from both databases, but only NIST data are available near the critical point. As can be seen in Figure 2, the main drawback in this case is that the vaporization enthalpy from the NIST data does not tend to zero at the critical point. As can be seen, there is a clear step between the last datum and the zero value.

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Table 1. Fluids and Data Considered and the Best Results (Lowest AADs) from Modelsa DIPPR fluid

ASHRAE code

ND

trichloromonofluoromethane dichlorodifluoromethane chlorotrifluoromethane tetrafluoromethane dichlorofluoromethane chlorodifluoromethane trifluoromethane difluoromethane fluoromethane methane 1,1,2-trichloro-1,2,2-trifluoroethane 1,2-dichloro-1,1,2,2-tetrafluoroethane chloropentafluoroethane hexafluoroethane 2,2-dichloro-1,1,1-trifluoroethane 1-chloro-1,2,2,2-tetrafluoroethane pentafluoroethane 1,1,1,2-tetrafluoroethane 1,1-dichloro-1-fluoroethane 1-chloro-1,1-difluoroethane 1,1,1-trifluoroethane 1,1-difluoroethane ethane octafluoropropane 1,1,1,2,3,3,3-heptafluoropropane 1,1,1,2,3,3-hexafluoropropane propane butane isobutane pentane hexane heptane ethene propene octafluorocyclobutane decafluorobutane

R-11 R-12 R-13 R-14 R-21 R-22 R-23 R-32 R-41 R-50 R-113 R-114 R-115 R-116 R-123 R-124 R-125 R-134a R-141b R-142b R-143a R-152a R-170 R-218 R-227ea R-236ea R-290 R-600 R-600a R-601

24 43 25 31 29 24 36 29 31 60 42 72 31 32 38 59 29 35 30 38 31 51 73 25 29 26 101 79 66 105 40 37 59 83 21 37

Tmin (K)

NIST Tmax (K)

2 coeff

ND

3 coeff

min. AAD

model

min. AAD

model

0.23 0.25 0.09 0.28 0.32 0.11 0.07 0.17 0.15 0.18 0.11 0.29 0.24 0.20 0.16 0.16 0.07 0.08 0.29 0.09 0.03 0.13 0.38 0.62 0.54 1.10 0.17 0.14 0.13 0.21 0.31 0.70 0.16 0.22 0.09 0.25

P2 X97 X97 CHC RW38 X97 P2 X97 MSP RW40 P2 X97 CHC RW40 X97 X97 X97 P2 X97 X97 X97 X97 RW40 CHC CS P2 P2 X97 X97 X97 P2 P2 X97 CHC RW40 CS

0.14 0.08 0.08 0.27 0.24 0.11 0.06 0.14 0.11 0.04 0.11 0.24 0.20 0.16 0.04 0.10 0.08 0.07 0.11 0.04 0.06 0.03 0.10 0.24 0.09 0.76 0.13 0.10 0.11 0.09 0.15 0.26 0.07 0.10 0.08 0.08

ER40 ER38 ER38 ECHC RL ER38 ER40 P3 ER40 MGV S88 S88 ERλ ER38 S88 ER38 P3 S88 S88 S88 ER38 S88 ER38 GV MGV ECHC P3 ER38 P3 ER40 GV MGV P3 GV ER40 ER40

0.12 0.25 0.20 0.13 0.22 0.19 0.28 0.17 0.06 0.08 0.07 0.15

RW40 RW38 RW40 X97 MSP X97 MSP RW40 P2 CS P2 RW40

0.11 0.22 0.03 0.13 0.22 0.05 0.05 0.03 0.05 0.05 0.07 0.04

ERλ ERλ P3 ER40 Erλ P3 GV RL RL P3 P3 ER40

organic compounds

R-1150 R-1270 R-C318 R-3-1-10

223.15 194.91 135.51 93.53 175.15 168.02 149.82 143.13 136.67 90.69 243.71 187.39 178.68 180.22 174.46 142.09 174.81 175.61 178.66 150.17 166.97 154.56 90.35 208.43 148.74 197.74 85.47 134.86 148.55 181.95 214.00 223.00 104.00 94.00 243.15 151.94

413.15 383.15 271.80 204.75 406.42 332.37 298.71 316.13 285.68 188.78 438.52 410.93 317.84 263.52 440.00 368.15 305.25 340.00 430.96 393.15 311.29 383.15 301.30 313.15 337.35 371.14 369.60 420.00 390.00 465.00 488.71 513.15 266.50 360.00 383.15 353.15

311 271 211 131 253 256 183 217 169 102 253 226 180 122 292 290 169 207 310 270 186 234 217 222 231 196 287 293 296 328 332 360 181 267 157 206

b b b

b b b b

b

b b

inorganic compounds hydrogen helium ammonia water neon nitrogen oxygen argon carbon dioxide dinitrogen monoxide sulfur dioxide krypton total number of data points

R-702 R-704 R-717 R-718 R-720 R-728 R-732 R-740 R-744 R-744A R-764 R-784

22 17 45 40 30 54 31 30 32 13 38 15 1968

14.00 3.00 197.15 274.15 25.09 63.15 57.28 85.57 216.58 184.67 204.22 115.77

32.00 5.00 393.15 645.65 39.96 124.00 139.12 135.77 304.00 309.25 403.15 206.00

194 154 212 376 201 128 102 136 177 130 235 189 10650

a First, the fluids are classified as organic and inorganic compounds. Then they are sorted by the ASHRAE R-number. The number of data (ND) and minimum and maximum temperature for DIPPR data are given. For NIST, the data covered the full temperature range, except for 10 fluids, for which the range is from the critical point to a certain temperature above the triple point (see ref 39 for details). b Tmin is above the triple-point temperature.

Figure 1. Vaporization enthalpy versus temperature for helium from DIPPR37 and NIST39 databases.

For other fluids (R-12, R-14, R-22, R-114, R-116, R-124, R-134a, R-152a, R-218, R-227ea, hexane, heptane, and R-3-110; argon, krypton, and R-764), we found slight differences between the data for particular temperature ranges or points.

Figure 2. Vaporization enthalpy versus temperature for R-236ea from DIPPR37 and NIST39 databases.

For hydrogen, slight differences are found near the critical point, and for CO2 near the triple point. For pentane, there are slight differences between different DIPPR sources at intermedi-


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Table 2. MAPD’s for Two-Coefficient Models by using DIPPR and NIST39 Databases MAPDs (%)

Figure 3. Vaporization enthalpy versus temperature for propane from DIPPR37 and NIST39 databases. NIST data are shown as a line in order to more clearly distinguish the data, but calculations were made with data. DIPPR2 are DIPPR data from a source giving only values near the critical point. DIPPRe is for a datum at the boiling temperature, which was not considered in our calculations.

two-coeff. models

eq

DIPPR

NIST

CS (1973) MSP (1989) X97 (1997) CHC (2003) RW38 (2005) RW40 (2005) P2 (2007)

7 5 2 3 4 4 6

0.60 0.53 0.50 0.56 0.54 0.51 0.48

0.69 0.50 0.34 0.58 0.57 0.41 0.31

boiling point as a constraint, and then the input parameters were the critical temperature and the temperature and vaporization enthalpy at the normal boiling point. Once the inputs were introduced, 7 correlations included 2 adjustable coefficients, and the other 11 correlations included 3 adjustable coefficients. Some of these correlations are classical expressions and have been considered in previous studies,20-22,24,35 but others have been proposed recently or have been slightly modified or extended for this study. We first define a dimensionless temperature as t ) (T - Tb)/(Tc - Tb) ) (Tr - Tbr)/(1 - Tbr)

Figure 4. Vaporization enthalpy versus temperature for R-41 from DIPPR37 and NIST39 databases.

ate temperatures, although the general agreement with the NIST data is good. In the case of isobutane, the data are very similar at high and intermediate temperatures, but not at low temperatures, and indeed the DIPPR team does not consider as acceptable some other data available near the triple point. For propane, DIPPR uses 10 different sources of data, with the result that one can find two data for the same temperature. One datum at the normal boiling temperature was not considered (see Figure 3). The NIST and DIPPR data are very similar except near the critical point, as can be seen in Figure 3. For R-21, the data are clearly different only at low temperatures, where DIPPR uses two different sources which do not match exactly. Also, there are no available NIST data at very low temperatures or DIPPR data near the critical point. For R-32, the data are clearly different only at intermediate temperatures. For Ne, the agreement is only acceptable. Finally, as can be seen in Figure 4, there the DIPPR (from different sources) and NIST data for R-41 are different in the whole temperature range. The Clapeyron equation, the LeeKesler method, and the Watson equation were used by DIPPR, but only the data from the Lee-Kesler method are considered as acceptable. This may be because the accepted values for the boiling point from other sources agree better with the data predicted by this last method. Correlations The 18 correlations for the vaporization enthalpy selected are described below. In all the cases we considered the normal

(1)

where Tc and Tb are the critical and boiling temperatures, and Tr ) T/Tc and Tbr ) Tb/Tc are the reduced temperatures. Two-Adjustable-Coefficient Correlations (2C Models). We considered seven correlations with two adjustable coefficients (2C models). A list of the abbreviations used for each correlation and the corresponding equation is given in Table 2. The first expression is that proposed by Xiang,22 which we call X97, and which is a modification of the Torquato and Stell18 model, and originally included two fixed exponents and three adjustable coefficients. In his study of 30 pure substances, Xiang concludes that his proposal fit the data better than existing threeand even four-coefficient correlations.22 According to Park,35 X97 gives an MAAD ) 0.18% for 22 refrigerants. Very recently, Jovanovic and Grozdanic24 have obtained an overall deviation of 0.35% for this expression when used for 157 fluids. When the boiling temperature is used as a constraint, the X97 expression22 obtained is n(Tbr /Tr)β + m(1 - t)δ ∆h ) (1 - t)β ∆hb 1 + (n + m - 1)(1 - t)

(2)

where ∆hb is the vaporization enthalpy at the normal boiling point, β ) 0.325 and δ ) 0.51 are fixed exponents which represent the universal behavior in the critical region, and n and m are adjustable coefficients. More recently, Chen et al.28 (CHC) proposed a new correlation including the boiling point as a constraint, as follows: ∆h/∆hb ) (1 - t)m+n|1-

√1-t|

(3)

where m and n take fixed values (0.360 and 0.044, respectively). Twenty-four refrigerants were considered, with the mean deviation being 0.87%. We considered this proposal again but obtained new values for coefficients m and n for every fluid studied. Romań et al.31 have proposed a new almost-universal model for the vaporization enthalpy of fluids, based on the use of the triple point data as reference. As a first step, the model can be used with only two fixed coefficients with physical meaning, and then without using adjustable coefficients. In order to obtain

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a better fit, we considered the model again, but we included two adjustable coefficients taking the boiling point as reference. Indeed, the two coefficients can be considered in two different ways. One is to consider a fixed value for the critical exponent λ and then to include the two adjustable coefficients in the exponential term, as follows: ∆h/∆hb ) exp[a1t + a2t2](1 - t)λ

(4)

with a1 and a2 being the adjustable coefficients, and with λ having a fixed value near 0.38. We considered this model here by using both λ ) 0.38 and λ ) 0.4. These models are referred to as RW38 and RW40, respectively (see Table 2). Another way in Roman et al.’s model is to consider the critical exponent as an adjustable coefficient and then to use only one term in the exponential function: ∆h/∆hb ) exp[a1t](1 - t)λ

(5)

In particular, when a1 ) 0 and λ is considered as an adjustable coefficient, then we recover the well-known Watson25 proposal, considered more recently by Liley23 and Malagoni et al.30 Indeed, this model is numerically the same as the one proposed by Majer et al.,13 so that we call it MSP. More recently, this model has been used by Velasco et al.32 in order to predict the vaporization enthalpy of metals and metalloids, but without using adjustable coefficients for every fluid. Very recently, Park35 has proposed a new model for the vaporization enthalpy of refrigerants. When only two adjustable coefficients, m and n, are considered, the model (P2) can be written as follows: ∆h ) n(1 - t)m(Tr /Tbr)2-m + (1 - n)(1 - t) ∆hb

(6)

As mentioned above, Park has shown that this proposal gives more accurate results for 22 refrigerants (MAAD ) 0.14%) than other previous proposals, including the Xiang and MSP expressions, eqs 2 and 5, of those cited in this section. Finally, we consider here the expression commonly used in the DIPPR database in order to fit the data for a great number of fluids. When only two adjustable coefficients are used, which is usual in the DIPPR database, the expression is the same as that proposed by Chueh and Swanson38 (CS). Later Meyra and co-workers29,46 use the same expression, but with the triple point temperature as reference. Here the boiling point is considered as a constraint and, then, this expression becomes ∆hV ) ∆hb(1 - Tr)d1+d2Tr /(1 - Tbr)d1+d2Tbr

(7)

where di are the adjustable coefficients. Three-Adjustable-Coefficient Correlations (3C Models). We considered eleven correlations containing three adjustable coefficients (3C models). A list of the used abbreviation for each correlation and the corresponding equation is given in Table 3. The first correlation is again the expression used in the DIPPR database, but included three adjustable coefficients as proposed by Guermouche and Vergnaund15 (GV). We write it here as follows: ∆hV ) ∆hb(1 - Tr)d1+d2Tr+d3Tr /(1 - Tbr)d1+d2Tbr+d3Tbr 2

where d1, d2, and d3 are the adjustable coefficients.

2

(8)

Table 3. MAPD’s of Three-Coefficient Models by Using DIPPR37 and NIST39 Databases MAPDs (%) three-coeff models

eq

DIPPR

NIST

GV (1974) RL (1980) S88 (1988) A-SH (2007) P3 (2007) JG (2008) MGV (this work) ECHC (this work) ER38 (this work) ER40 (this work) ERλ (this work)

8 9 10 13 14 15 16 11 12 12 12

0.44 0.43 0.44 0.76 0.43 0.70 0.44 0.51 0.44 0.45 0.46

0.27 0.22 0.24 0.91 0.17 0.96 0.23 0.49 0.23 0.23 0.30

The second expression is due to Radosz and Lydersen16 (RL) and has a less straightforward analytical expression, because it includes four fixed exponents. The expression is ∆h ) n(1 - t)1/3 + m(1 - t)2/3 + l(1 - t)5/3 + ∆hb (1 - n - m - l)(1 - t)2 (9) where n, m, and l are the adjustable coefficients. Somayajulu19 (S88) proposed a similar expression, also including four fixed exponents, which can be written as ∆h ) n(1 - t)3/8 + m(1 - t)11/8 + l(1 - t)19/8 + ∆hb (1 - n - m - l)(1 - t)27/8 (10) where n, m, and l are the adjustable coefficients. Park (2007) found an MAPD of 0.08% with these last two expressions when they are used to fit data for 22 refrigerants. In order to evaluate the performance of the Chen et al.28 model, given by eq 3, when another adjustable coefficient is added, we have extended the exponent term as follows: ∆h/∆hb ) (1 - t)m+n|1-

√1-t|+l(1 - √1 - t)2

(11)

with n, m, and l being the adjustable coefficients. We call this model ECHC. Similarly, we can extend the Romań et al.31 model by adding an extra coefficient in the exponential term: ∆h/∆hb ) exp[a1t + a2t2 + a3t3](1 - t)λ

(12)

In particular, we consider two possibilities: (i) λ ) 0.38 or 0.4, and then three adjustable coefficients (ER38 and ER40 models); (ii) a3 ) 0, and then a1, a2, and λ being adjustable coefficients (ERλ model). The following 3C model studied here is that proposed recently by Al-Shorachi and Hashim33 (A-SH): ∆h/∆hbn ) m(1 - t)λ

(13)

Originally they fixed the values of the parameters as m ) 45.129353, n ) 1.06402, λ ) 0.3913. We tried to reproduce their results, but found that the dimensions used for the enthalpy could possibly be incorrect, so that the values of these parameters should be taken with caution. We therefore considered these coefficients as adjustable for every fluid. Almost simultaneously with the foregoing model, Park35 proposed (P3):


∆h ) n(1 - t)m(Tr /Tbr)l-m + (1 - n)(1 - t) ∆hb

(14)

which is the same as eq 6, but using three adjustable coefficients (l, m, and n). He showed that this new expression gives a MAAD of only 0.05% when data for 22 refrigerants are fitted. The latest model published to date is due to Jovanovic and Grozdanic24 (JG). Using the boiling point as reference,it can be written as

( ) (

ln

∆h 1 1 )B ∆hb Tr Tbr

)

(Tr - Tbr) + +C (1 - Tr)(1 - Tbr) Tbr(1 - Tr) D ln Tr(1 - Tbr)

(

)

(15)

where B, C, and D are the adjustable coefficients. As mentioned above, the original expression, including four adjustable coefficients instead of the ∆hb data, gives an overall deviation of 0.27% when tested with data for 131 fluids. Nevertheless, only for 19 fluids were data near the critical point considered. Finally, we consider the GV model15 again but using the dimensionless temperature, t, as follows: 2 ∆h ) (1 - t)n+mt+lt ∆hb

(16)

where n, m, and l are the adjustable coefficients. We call this model the modified GV (MGV), which was already studied by Park,35 who found an MAAD of 0.08% for 22 pure refrigerants. Results Each correlation model was fitted separately to the available DIPPR37 and NIST39 data using the Mathematica computer program. In particular, we used the function NonlinearRegress, which also provides fitting diagnostics, minimizing the χ2 merit function given by the sum of squared residuals. Since the optimization methods used by NonlinearRegress are iterative, starting values are required for the parameters. A careful choice of these values may be necessary to avoid falling into a local minimum of the χ2 merit function. This was the case with Park’s model with three parameters which needed as starting values 2, 0.4, and 0.4 for l, n, and m, respectively, to obtain the best fit. In the case of the JG model, it was necessary to apply an appropriate logarithmic transformation in order to achieve adequate convergence for all the fluids studied. In the end, 864 fitting procedures were performed for each one of the databases, generating 2256 × 2 ) 4512 adjustable coefficients. The values of these parameters and their corresponding standard errors are available upon request, but they can be obtained easily using the same procedure with the same or with different databases. We calculated the percentage deviation (PD) of the values obtained with each correlation model (m) for each datum, ∆hm(Ti), from those accepted by the databases, ∆hi, as follows: PDim ) 100(∆hm(Ti) - ∆hi)/∆hi

(17)

Then we calculated the average absolute deviation (AAD) for each fluid j: NDj

∑ |PD

m

i

AADj ) m

i)1

NDj

|

(18)

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where NDj is the number of available data for each fluid. Finally, we calculated the mean absolute percentage deviation (MAPD) for the number of available data (ND ) ∑jNDj), as follows: ND

∑ |PD

m

i

MAPD ) m

i)1

ND

|

(19)

Tables 2 and 3 list the MAPD’s obtained for each model and each data source. Comments on the overall and individual results are given below. Comments on the Overall Results. An observation of Tables 2 and 3 shows that the models with the lowest MPADs were the same for both databases, i.e., which data were used did not influence the choice of the best model. Thus, good overall performance with respect to the NIST database with its regular data over the whole temperature range was predictive of good overall performance with respect to the DIPPR database with its irregular data sometimes covering only a limited temperature range. For some particular fluids, however, differences can be observed, as will be detailed in the following paragraphs. When two adjustable coefficients are used, the Park model, eq 6, gives the lowest MAPD with respect to both sources of data (Table 2). When the full range is considered, i.e., with the NIST database, the difference with respect to the other 2C models is clearer because most of them give a MAPD higher than 0.5%. The model of Xiang, eq 2, also gives very good overall results, but the Park model gives AADs higher than 1% for the NIST data only for R-236ea, whereas the X97 model does so for R-236ea, hydrogen, and helium. When only data accepted by the DIPPR database are considered, we found similar results for all the models, with the difference relative to the Park model being very small. With respect to the recent models of Romań et al. (RW38 and RW40), one observes that the choice of λ ) 0.4 as mean value for the critical exponent leads to slightly lower MAPDs than when the 0.38 value is used or when the critical exponent is considered as a fitting parameter in the MSP model. This is seen more clearly when NIST data are used. As can be seen, the CS model, used by DIPPR, gives the highest MAPD values with respect to both databases. In any case, it must be taken into account that for some fluids the DIPPR database uses more than two adjustable coefficients. One observes in Table 3 that when three coefficients are used, the P3 model also gives the lowest MAPD in both cases, although with only a minor difference with respect to most of the other models. Thus, when the NIST data are used, the P3 model is the only one giving a value below 0.2%, although the improvement is really small because most of the models give MAPDs in the range from 0.2% to 0.3%. With respect to the choice of the critical exponent, the results indicate that it is slightly better to consider a fixed value, but that this fixed value can be 0.38 or 0.4 indistinctly. Finally, it must be noted that, surprisingly, some recent models, in particular ECHC, A-SH, and JG give the poorest results. In the case of the very recent JG model, it seems to be that the model adequately reproduces the data only when intermediate temperature ranges are considered. When the DIPPR data are used, the difference between the 3C models is very small, except for the A-SH and JG models which give the poorest results. The choice of the adequate value for the critical exponent is not important due to the fact that data very near the critical point are available only for some fluids.

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Normally, 3C models would give smaller average AAD’s than 2C models, but we found that the performances of JG and A-SH are not better than any two-coefficient model in Table 1. It is also notable that adopting a nondimensional temperature in MGV only slightly improves the performance of the original GV model. The AAD’s obtained with NIST data would normally be smaller than those with DIPPR data, which has scattered data due to experimental errors. However, this expected reduction of AAD’s does not occur with three 2C models (CS, CHC, and RW38) and two 3C models (A-SH and JG). The performance of these models is relatively poor. Comments on Individual Results. Tables with AAD values for every fluid, model, and database are available upon request. Table 1 lists only the minimum AADs with respect to NIST data. We shall describe in the following some particular details for selected fluids. As one observes in Table 1, none of the models can reproduce the NIST data for the refrigerant R-236ea with AADs below 1% when two coefficients are used and below 0.7% when three coefficients are used (the maximum PDs being higher than 40% and 32%, respectively). This is because NIST data at high temperatures do not clearly tend to zero at the critical temperature (Figure 2). Indeed, there is a sharp step between the last NIST data available and the critical point. Far from the critical point, the fit is completely adequate. Thus, when DIPPR data are used the AADs found are near 0.3% for every 2C model or 3C model. In the case of hydrogen, R-702, we found clear differences between the predictions of models. Thus, for 2C models, the AADs range from 2.56% (GV) to only 0.12% (RW40) with NIST data. For this fluid, the MSP, RW40, and RW38 models reproduce the NIST data better than the models with the lowest MAPDs (P2 and X97). For the DIPPR data, the lowest AADs are found with the MSP and P2 models (around 0.4%), while the RW40 and RW38 models give AADs around 0.8%. In these last models, the values used for the critical exponents seem to be in better agreement with the tendency of the NIST data than with the tendency of the DIPPR data near the critical point. For 3C models, ER40, ERλ, and P3 give the lowest AADs (below 0.15%) with respect to the NIST data, whereas RL, P3, and ERλ do so with respect to the DIPPR data (AADs below 0.3%). For heptane, when DIPPR data are used all the 2C and 3C models give AADs near 0.5%. The NIST data, however, include some temperatures very near the critical point, which increases the calculated AADs. Thus, with 2C models, the lowest AADs are 0.70% for the P2 model (see Table 1) and 0.75% for X97. With 3C models, the AADs are below 0.3% only with the MGV, RL, and P3 models. In this case, the models based on either fixing or fitting the critical exponent give higher AADs, especially when 2C models are used. In the particular case of helium, as we noted previously and as is shown in Figure 1, only NIST data are available at low temperatures, and those data behave completely differently from the other fluids. As a consequence, the CHC and A-SH models give AADs around 7% even with three adjustable coefficients. The CS 2C model gives also a high AAD, which is significantly reduced by using three coefficients. As can be seen in Figure 5, the X97 model gives PDs > 1% for wide temperature ranges. The most accurate 2C model is RW38, with an AAD very similar to that obtained with the best 3C models. As can be seen in Figure 5, the main difference between the P2 and the RW38 models is in the behavior near the triple and the critical

Figure 5. Percentage deviations (PD) for vaporization enthalpy data of helium given by three 2C models and the NIST data.

points. Thus, RW38 is very accurate near the critical point, with a significant difference with respect to the other models. Near the triple point, the P2 model gives lower deviations. All the models except A-SH give AADs below 0.25% with respect to DIPPR data for neon (there are no DIPPR data at temperatures greater than Tr ) T/Tc ) 0.9). If NIST data are used, then CS gives the highest AAD. High AADs are also found using the RW40 and RW38 models. In the MSP model, the critical exponent obtained by the fitting procedure is 0.4654, which is clearly different from the value 0.4 and 0.38 considered in the RW40 and RW38 models. The MSP and surprisingly the CHC models give the best agreement for 2C models. When 3C models are used, ER38, ER40, and S88 give the highest deviations with respect to NIST data. The ERλ model, however, which considers the critical exponent as adjustable (in particular, 0.4661 for this fluid) gives the best results. The use of the MGV 3C model instead of the CS 2C model, both used as reference by the DIPPR database, clearly improves the results. None of the 3C models improves the results obtained for the NIST data with the MSP and CHC models with only 2 coefficients. There are also some other significant differences for individual fluids according to whether the NIST or the DIPPR databases are used. Obviously, these differences are due to the fact that, as explained above, the data behave differently in some temperature ranges. In particular, for propane, the models give very good results for the NIST data (Table 1), with the lowest AADs being 0.17% (P2 model) and 0.13% (P3 model). The AADs for the DIPPR data are, however, higher than 1.2%. As explained above, this is due to the presence of DIPPR data near the critical point that are in disagreement with the NIST data. For R-12, the 271 NIST data can be reproduced with AADs around 0.25% by using the X97 or the Park 2C models, whereas the 41 DIPPR data (which come from two different sources and are more irregular) cannot be reproduced with AADs below 1.1%. As can be seen in Figure 6, the behavior of models at low and high temperatures are different even for models giving similar AADs values. Thus, the X97 models give slightly lower deviations at low temperatures, whereas the P2 model gives significantly lower deviations at high temperatures. The datum at the highest temperature is not shown, because it is out of scale. The PDs for this datum are 17.8% and 11.1% for the X97 and P2 models, respectively. Finally, we would note that for this fluid and some others, the 2C models give a very low AAD, so that the use of 3C models does not significantly improve the results. For R-124, and despite the NIST and DIPPR data only disagreeing slightly at low temperatures, the lowest AADs are


5025

any clear improvement of the results, i.e., there is at least one 2C model that gives practically the same results as the best of the 3C models. These fluids are R-13, R-14, R-21, R-22, R-23, R-32, R-41, R-113, R-114, R-115, R-116, R-124, R-125, R-134a, R-142b, propane, butane, isobutane, R-C318; hydrogen, helium, water, neon, carbon dioxide, dinitrogen monoxide, and sulfur dioxide. Finally, for R-143a the best result is obtained with the X97 2C model, all of the 3C models giving higher AADs. Conclusions

Figure 6. Percentage deviations (PD) for vaporization enthalpy data of R-12 given by two 2C models and the NIST data.

Figure 7. Percentage deviations (PD) for vaporization enthalpy data of R-124 given by two 2C models and the NIST data.

around 0.1-0.2% for the 290 NIST data but around 1% for the 59 DIPPR data. As can be seen in Figure 7, although the ER38 and P3 models give practically the same AAD, the comparison with NIST data shows that the ER38 one gives lower deviations at almost every temperature and especially near the critical point. Finally, we shall comment on some of the results about the choice of the appropriate number of coefficients for some particular fluids or models. For this purpose, we shall focus only on the most regular NIST data. Detailed results are given in Table 1, where the lowest AAD values are given for every fluid. In particular, the use of three coefficients in the Park model clearly improves the overall results when compared with the same model using only two coefficients. A clear example is hydrogen, for which the AAD is reduced from 0.86% to only 0.14%. The P2 model gives AADs below 0.3% for 28 fluids, the P3 model does so for 44 fluids, and for some fluids, the P2 and P3 models give practically the same results. It is also interesting to note that for R-227ea the MGV 3C model gives an AAD of only 0.09%, the AAD being higher than 0.5% for all the 2C models. For methane, R-123, ammonia, nitrogen, oxygen, argon, and krypton, the lowest AADs for 2C models are around 0.2-0.3% and are as low as 0.03-0.05% when some 3C models are used. For R-12, ethane, and decafluorobutane (C4F10), the improvement obtained by using a further coefficient is only moderate. For pentane, only the ER38 and ER40 models improve clearly the best result obtained with 2C models. For R-218 and heptane, three coefficients are needed in order to reproduce the NIST data with AADs around 0.25%, these being around 0.7% when two coefficients are used. On the other hand, for 26 fluids, the use a further coefficient does not provide

In this work, we have studied the accuracy of eighteen correlations of the vaporization enthalpy by using data retrieved from both the DIPPR and the NIST databases for forty-eight substances. Comparison of both databases raises a slight difference between them for 23 fluids at a particular temperature range, whereas an excellent agreement is found for other different 23 fluids. Special attention deserves Ne since the DIPPR and NIST data present several differences, and R-41 for which the data are completely different. We have found that the choice of the database does not influence the choice of the best overall model. The correlation recently proposed by Park35 was found to give the lowest mean average percentage deviations for both two- and three-coefficient models. The two-coefficient model proposed by Xiang, and the three-coefficient model by Radosz and Lydersen also gave excellent overall results. With respect to the recent models proposed by Romań et al., eq 4, and the modifications proposed here, eq 12, we found that it is slightly better to consider a fixed value for the critical exponent λ. We also considered results for individual fluids. Thus, we found that the highest AADs with respect to the NIST data are found for H2, heptane, He, Ne, and especially for R-236ea. Finally, we checked how many adjustable coefficients are needed for some particular fluids or models when NIST data are used. We found that for 21 fluids a significant or moderate improvement of the results is obtained by the use of three coefficients instead of only two. For another 26 fluids, the use of three-coefficient models does not necessarily mean a clear improvement over the two-coefficient models. For R-143a, the contrary was the case. As a final conclusion, the use of the models proposed by Park is highly recommended for the refrigerants studied, so that its use could be extended to other similar refrigerants. The Xiang model is a very adequate model if only two adjustable coefficients are used. If accurate data near the critical point are available, the models proposed by Romań et al. or their extended versions are also very good alternatives. Acknowledgment This work was partially supported by Projects FIS200602794/FEDER (A.M. and M.I.P.), FIS2006-03764/FEDER and FIS2009-07557 (F.L.R.), and by Research Program 2009 of Kookmin University in Korea (K.K.P.). Literature Cited (1) Carruth, G. F.; Kobayashi, R. Extension to Low Reduced Temperatures of Three-Parameter Corresponding States: Vapor Pressures, Enthalpies and Entropies of Vaporization, and Liquid Fugacity Coefficients. Ind. Eng. Chem. Fundam. 1972, 11, 509.

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ReceiVed for reView June 23, 2009 ReVised manuscript receiVed March 22, 2010 Accepted March 29, 2010 IE901015F

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