# Thermodynamic properties for saturated air, an engineering correlation.

INTRODUCTION

All engineering calculations required to design an HVAC system start with estimations of the air-water vapor mixture properties that are the basis for heat and mass balances. Some time ago, engineers relied on hand calculations, and tables were their main tool for properties estimation. Tables were prepared from very precise calculations, such as those of Hyland and Wexler (1980), which are iterative and lengthy.

In modern engineering, except for rough initial estimates, most calculations are made with the aid of computers, and equations are needed to calculate properties. These equations may be as simple as the ideal gas model or as complicated as virial equations of state.

Although it would be expected that values obtained from a computer should have very high precision, this is not necessarily so. This fact may be easily verified, as an example, for the Psychrometric Analysis CD (ASHRAE 2002) results, which are obtained from the ideal gas formulation. The reason for this is probably that the calculations must be made fast enough to display results in real time as the cursor moves around the psychrometric charts, which would preclude the use of complicated models.

Even though computer time is increasingly cheaper and computers are faster, there are still instances in which computing time is a premium commodity. This is true of real-time system simulation and control applications. For these cases, simple and accurate equations are needed.

This work started in response to readers' comments regarding precision of the ideal gas model in Chapter 6 of the ASHRAE Handbook--Fundamentals (ASHRAE 2005). In particular, the comments were about errors in the [W.sub.s] and [h.sub.ms] predictions, but they led to identification of causes and possible solutions and finally to development of the model presented in this work.

IDEAL AND REAL GAS MODELS

Model Description

Two models set the range of precision and complexity of calculations. On one hand, there is the ideal gas model for air-water vapor mixtures (ASHRAE 2005) and on the other the real gas model presented by Hyland and Wexler (1983a, 1983b) and Nelson and Sauer (Sauer et al. 2001; Nelson et al. 2001b).

Because of its simplicity and first principle foundation, results of the ideal gas model set the minimum for both acceptable precision and complexity of calculations. In this model, the specific volume is calculated from the ideal gas PVT relation and enthalpy and entropy from ideal mixture laws.

The real gas model provides a standard for comparison because it yields results of very high precision (as shown in the Hyland and Wexler work) but with considerable complexity and calculation effort. This real gas model is based on the use of a virial equation of state for specific volume, Beattie's formulation (Beattie 1949) for enthalpy and entropy, and the so-called enhancement factor equation for the interaction of air-water vapor molecules (Hyland and Wexler 1973a,1973b; Hyland 1975).

To show the gas real model complexity and as background for model development, the Hyland and Wexler (1980) model will be briefly described. The air and water vapor molar fractions at saturation are related to the enhancement factor through the following equations:

[x.sub.as] = [[P - f[P.sub.w]]/P][x.sub.ws] = [[f[P.sub.w]]/P] (1)

ln(f)=[[(1+[kappa][P.sub.w])(P-[P.sub.w])-[kappa](P-[P.sub.w.sup.2])/2]/[RT]][v.sub.c] + ln[gamma](1 - k[x.sub.as]P) + ([[x.sub..sup.2]P]/[RT])[B.sub.ww] - ([2[x.sub.[as].sup.2]P]/[RT])[B.sub.aw] - ([P - [P.sub.w] - [x.sub.[as].sup.2]P]/[RT])[B.sub.ww] + [[3[x.sub.[as].sup.3][P.sup.2]]/[(RT).sup.2]][C.sub.aaa] + [[3[x.sub.[as].sup.2](1 - 2[x.sub.as])[P.sup.2]]/[2[(RT).sup.2]]][C.sub.aaw] - [[3[x.sub.[as].sup.2](1 - [x.sub.as])[P.sup.2]]/([RT.sup.2])][C.sub.aww] - [[(1 - 2[x.sub.as])[(1 - [x.sub.as]).sup.2][P.sup.2] - [P.sub.w.sup.2]]/[2[(RT).sup.2]]][C.sub.www] - [[[x.sub.[as].sup.2](1 - 3[x.sub.as])[(1 - [x.sub.as]).sup.2][P.sup.2]]/[(RT).sup.2]][B.sub.aa][B.sub.ww] - ([3[x.sub.[as].sup.4][P.sup.2]]/[2[(RT).sup.2]])[B.sub.[aa].sup.2] - [[2[x.sub.[as].sup.2](1 - [x.sub.as])(1 - 3[x.sub.as])[P.sup.2]]/[(RT).sup.2]][B.sub.[aw].sup.2] - [[[P.sub.w.sup.2] - (1 + 3[x.sub.as])[(1 - [x.sub.as]).sup.3][P.sup.2]]/[2[(RT).sup.2]]][B.sub.[ww].sup.2] (2)

The isothermal compressibility . ([Pa.sup.-1]), saturation pressure [P.sub.W] (Pa), condensed phase specific volume [v.sub.c] ([m.sup.3), Henry's law constant, [kappa]([Pa.sup.-1]), and the virial coefficients are functions of temperature only, and equations for their calculation are given in Hyland and Wexler (1980).

An iterative solution of Equations 1 and 2 to obtain the enhancement factor and, hence, the saturation air and water vapor molar fractions and the humidity ratio is the first step in the real gas model. From these equations we may also conclude that, albeit complex and implicit, the enhancement factor is a function of temperature and pressure. This was recognized by Nelson et al. (2001a), who used sixth-degree polynomials for f as a function of T in their routine calculations of air-water vapor mixture properties at high temperature. They employed one polynomial for each investigated pressure.

The following virial equation of state is used for the mixture volume:

[v.sub.ms] = [[RT]/P][1 + [[B.sub.ms]/[v.sub.ms]] + [C.sub.ms]/[v.sub.[ms].sup.2]] = [[RT]/P][Z.sub.ms] (3)

In this equation, the mixture virial coefficients are related to the saturation air and water vapor molar fractions and to the components' virial coefficients through the following:

[B.sub.ms] = [x.sub.[as].sup.2][B.sub.aa] + 2[x.sub.as][x.sub.ws][B.sub.aw] + [x.sub.[ws].sup.2][B.sub.ww] [C.sub.ms] = [x.sub.[as].sup.3][C.sub.aaa] + 3[x.sub.[as].sup.2][x.sub.ws][C.sub.aaw] + 3[x.sub.as][x.sub.[ws].sup.2][C.sub.aww] + [x.sub.[ws].sup.3][C.sub.www] (4)

Equations 3 and 4 constitute the second iterative loop of this model and solution results, in particular the compressibility factor, are functions of temperature and pressure.

Enthalpy and entropy are calculated from:

[h.sub.ms] = [x.sub.as]([5.summation over (i = 0)][a.sub.i][T.sup.i] + [H.sup.a]) + [x.sub.ws]([5.summation over (i = 0)][b.sub.i][T.sup.i] + [H.sub.w]) + RT[([B.sub.ms] - T[d[B.sub.ms]]/[dT])[1/[v.sub.ms]] + ([C.sub.ms] - [1/2]T[d[C.sub.ms]]/[dT])1/[v.sub.[ms].sup.2]] [s.sub.ms] = [x.sub.as]([4.summation over (i = 0)][c.sub.i][T.sup.i] + [c.sub.5]lnT + [s'.sub.w]) (5)

- Rln(P/[101,325]) + [x.sub.as]Rln([Z.sub.ms]/[x.sub.as]) + [x.sub.ws]Rln([Z.sub.ms]/[x.sub.ws]) - R[([B.sub.ms] + T[d[B.sub.ms]]/[dT])[1/[v.sub.ms]] + ([C.sub.ms] + [1/2]T[d[C.sub.ms]]/[dT])1/[v.sub.[ms].sup.2]] (6)

Expressions in the first two parentheses of Equations 5 and 6 are curve fits of the ideal gas properties (enthalpy and entropy, respectively) plus reference state values (primed quantities) for the mixture components. Hyland and Wexler (1980) give the required constants.

Real Gas Model Predictions

The first step to evaluate the ideal gas (or any other) model was to program the real gas model equations. A spreadsheet was implemented to generate comparison databases, and predictions of this program at 0.1 MPa (14.5 psia) were evaluated with respect to those of ASHRAE RP-216 (Hyland and Wexler 1980). This report proved to be a very useful debugging aid since their tabulations include not only the virial coefficients and all properties but values of the various terms of each equation used in calculations. They are available at 10[degrees]C (5.55[degrees]F) intervals for temperatures of -100[degrees]C to 90[degrees]C (-148[degrees]F to 194[degrees]F) for 0.1 MPa (14.5 psia), -100[degrees]C to 150[degrees]C (-148[degrees]F to 302[degrees]F) for 0.5 MPa (72.5 psia), -100[degrees]C to 170[degrees]C (-148[degrees]F to 338[degrees]F) for 1.0 MPa (145.03 psia), and -100[degrees]C to 200[degrees]C (-148[degrees]F to 392[degrees]F) for 5.0 MPa (725.18 psia).

Percent error listings of the spreadsheet predictions of the saturated mixture volume [v.sub.ms], enthalpy [h.sub.ms], and entropy [s.sub.ms] at 0.1 MPa (14.5 psia), as compared with those of the ASHRAE report, are given in Table 1. From these results it may be concluded that our real gas model program may be used to generate reference values of acceptable precision, since the maximum and minimum errors are 6.4 x [10.sup.-4%], and -1.09 x [10.sup.-3%]respectively.

Ideal Gas Model Predictions

Ideal gas model errors (100 x [ideal gas prediction - real gas value] / real gas value) are shown in Figures 1, 2, and 3. Figure 1 shows mixture volume error as a function of temperature for 75 and 101.325 kPa (10.87 and 14.695 psia). These results show that the ideal gas model predicts the air-water vapor mixture volume within the desired precision ([+ or -]0.2%).

[FIGURE 1 OMITTED]

As shown in Figure 2, errors in enthalpy are very small for atmospheric pressure and low temperatures and then they become very large as enthalpy approaches zero. As pointed out by Gatley (2005) this is because absolute errors, which are small, are divided by the very small enthalpy values. Finally, as temperature goes up, enthalpy errors become bigger and go beyond the set precision objective. The same problem occurs for 75 kPa (10.87 psia) but with slightly greater errors.

[FIGURE 2 OMITTED]

Figure 3 shows that for entropy, although tendencies are similar, errors are larger and, therefore, we may conclude that this model does not meet our specified precision limit.

ENGINEERING CORRELATIONS MODEL

Correlation Objectives

The main objective was to develop simple yet accurate correlations to calculate the air-water vapor mixture properties v, h, and s. Early in the project we detected that restriction of the range was important and selected to encompass ASHRAE psychrometric charts for normal and low temperatures (No. 1 and 2) at sea level as well as the normal temperature charts for altitudes up to 2250 m (7282 ft) above sea level (No. 5, 6, and 7).

Therefore, the range of temperature for the correlations was -40[degrees]C to 50[degrees]C (-40[degrees]F to 122[degrees]F) and the range of pressure was from 77.059 to 101.325 kPa (11.175 to 14.695 psia). To provide some leeway for extrapolation, the study was performed for temperatures from -60[degrees]C to 70[degrees]C (-76[degrees]F to 158[degrees]F) at 5[degrees]C (2.78[degrees]F) intervals while pressures varied from 75 to 105 kPa (10.87 to 15.22 psia) at 5 kPa (0.73 psi) intervals.

For precision, our goal was to obtain results with errors below ([+ or -]0.2% of the exact model predictions.

Model Description

Our first model was based on the use of polynomial correlations of the enhancement factor to obtain better predictions of the air and water vapor molar fractions. Hence, the ideal gas model would then provide better predictions of the properties. Although this simple option did indeed raise the precision of the enthalpy and entropy predictions, the errors as h and s approach zero were not eliminated and the mixture volume prediction errors were larger than those of the ideal gas model for the higher investigated temperatures.

On these bases we decided that a better model should include correlations of the enhancement and compressibility factors as functions of pressure and temperature, to be used with Equation 3 and simplified versions of Equations 5 and 6. The proposed model is composed of the following equations.

f = [g.sub.1](T,P)[Z.sub.ms][g.sub.2](T,P) (7)

[v.sub.s] = [[[R.sub.a]T]/(P - f[P.sub.w])][Z.sub.ms] (8)

[x.sub.as] = [[P - f[P.sub.w]]/P][x.sub.ws] = [[f[P.sub.w]]/P] (9)

[h.sub.s] = ([g.sub.3](T) + [H.sub.a]) + [[x.sub.ws]/[x.sub.as]]([g.sub.4](T) + [H.sub.w]) + [v.sub.h] (10)

[s.sub.s] = ([g.sub.5](T) + [s'.sub.a]) + [[x.sub.ws]/[x.sub.as]]([g.sub.6](T) + [s'.sub.w]) - [[R.sub.a]/[x.sub.as]]ln(P/[101,325]) + [[R.sub.a]/[x.sub.as]]ln[gamma]([Z.sub.ms]/[x.sub.as]) + [[x.sub.ws]/[x.sub.as]][R.sub.a]ln[gamma]([Z.sub.ms]/[x.sub.ws]) + [v.sub.s] (11)

The v terms in Equations 10 and 11 represent the virial contribution to properties and were introduced in an attempt to eliminate the "near zero" indetermination noted in the ideal gas model results. It was possible to calculate them as a function of pressure.

[v.sub.h] = [g.sub.7](P)[v.sub.s] = [g.sub.8](P) (12)

To fulfill our objectives, [g.sub.2] to [g.sub.2] should be the simplest possible equations.

Model Functions

Various curve fits for f and Z as a function of temperature led us to conclude that fifth-degree polynomials were very good ([R.sup.2] > 0.9993) and further tests showed that pressure effects were approximately linear. A curve fit with fifth-order dependence in T and first-order in P, which will be named model 1, produced the following equations (with [R.sup.2] values of 0.9843 and 0.9693 for f and Z, respectively).

[g.sub.1] = 2.2770286 - 0.02406584T + 1.8213945X[10.sup.[ - 4]][T.sup.2] - 6.8894708X[10.sup.[ - 7]][T.sup.3] + 1.297668X[10.sup.[ - 9]][T.sup.4] - 9.7078508X[10.sup.[ - 13]][T.sup.5] + 3.9945654X[10.sup.[ - 8]]P (13)

[g.sub.2] = 1.9208388 - 0.018226313T + 1.4278754X[10.sup.[ - 4]][T.sup.2] - 5.5526317X[10.sup.[ - 7]][T.sup.3] + 1.073933X[10.sup.[ - 9]][T.sup.4] - 8.2740746X[10.sup.[ - 13]][T.sup.5] + 3.9755302X[10.sup.[ - 9]]P (14)

To obtain equations as simple as possible for property predictions, regressions with lower-order dependence on T were also investigated. The simplest equation (model 2) is of second order in T and first order in P (with [R.sup.2] values of 0.9686 for f and 0.7766 for Z).

[g.sub.1] = 1.0398885 - 0.028774113T + 5.2013515X[10.sup.[ - 7]][T.sup.2] + 3.9753593X[10.sup.[ - 8]]P (15)

[g.sub.2] = 0.97165359 - 2.0355266X[10.sup.[ - 4]]T + 3.6608098X[10.sup.[ - 7]][T.sup.2] + 4.0079476X[10.sup.[ - 9]]P (16)

A constant specific heat model was used to calculate the ideal gas enthalpies [g.sub.3] and [g.sub.4]. The following equations were obtained by least-squares minimization of percent errors of enthalpy values with respect to those of the Hyland and Wexler (1980) ideal gas predictions.

[g.sub.3] = 1.0041923(T - 273.15)[g.sub.4] = 1.8642569(T - 273.15) (17)

Within the range of temperatures used for this estimation,--60[degrees]C to 90[degrees]C (-76[degrees]F to 194[degrees]F), errors in enthalpy were from -0.187% to 0.115% for air and from -0.022% to 0.003% for vapor.

Constants [h'.sub.a] and [h'.sub.w] were calculated to obtain zero enthalpy values at the reference states, 0[degrees]C (32[degrees]F) and 101325 Pa (14.696 psia) for air and the steam triple point. With Equation 17, their values are zero and 2700.7876, respectively.

For the constant specific heat model, the ideal gas entropy functions [g.sub.5]and [g.sub.6] are

[g.sub.5] = 1.0041923ln[gamma](T),[g.sub.6] = 1.8642569ln[gamma](T) (18)

with reference values [s'.sub.a] = -5.63354 and [s'.sub.w] = -3.661889.

The error functions [g.sub.7] and [g.sub.8] depend on values of the enhancement and compressibility factors, but, fortunately, they do not differ much when either model 1 or model 2 is used. This is consistent with previous work (Gatley 2005), which states that the use of an average value of the enhancement factor yields results of acceptable precision.

The largest deviations occur for entropy when model 2 is used for calculation, and Figure 4 shows values of [v.sub.s] as percent of the exact values of entropy, which would be an error if this quantity was not included.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

A comparison of these errors with those of the ideal gas model (Figure 3) resulted in two observations. The first one is that errors near zero are still present (in fact they tend to infinity as the function tends to zero) and the second is that errors are lower elsewhere. The latter, when analyzed together with the values of the functions and their absolute errors, led to a very simple means for error calculation.

Absolute values of [v.sub.s] are shown in Figure 5, where it may be noted that, except for a few points at the large end of the investigated temperatures, a small constant value of [v.sub.s] at each pressure would be a good representation of error. If this fact is analyzed together with the values of ([s.sub.s] - [v.sub.s]), shown in Figure 6, it may be concluded that the effect of adding a small constant is negligible at the higher temperatures.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Therefore, to eliminate the large percent errors in enthalpy and entropy as their values approach zero, the constant (with respect to temperature) to be added was found by least-squares minimization of the calculated percent errors. This procedure was followed at each investigated pressure, and results were curve-fitted as functions of pressure to obtain, for model 1:

[g.sub.7] = - 4.1281106X[10.sup.[ - 18]][P.sup.3] + 1.8191667X[10.sup.[ - 12]][P.sup.2] - 3.0990568X[10.sup.[ - 6]]P + 0.28844128 (19)

[g.sub.8] = 1.3323852X[10.sup.[ - 23]][P.sup.4] - 6.0489891X[10.sup.[ - 18]][P.sup.3] + 1.0524313X[10.sup.[ - 12]][P.sup.2] - 8.8970830X[10.sup.[ - 8]]P + 3.2465229X[10.sup.[ - 3]] (20)

and for model 2:

[g.sub.7] = - 4.1877292X[10.sup.[ - 18]][P.sup.3] + 1.8317367X[10.sup.[ - 12]][P.sup.2] - 3.0980589X[10.sup.[ - 3]]P + 0.28801940 (21)

[g.sub.8] = 1.3900054X[10.sup.[ - 23]][P.sup.4] - 6.5157721X[10.sup.[ - 18]][P.sup.3] + 1.1762918X[10.sup.[ - 12]][P.sup.2] - 1.0177460X[10.sup.[ - 7]]P + 3.6530569X[10.sup.[ - 3]] (22)

It should be mentioned that the order of the [g.sub.7] and [g.sub.8] correlations was the minimum to obtain results within the prescribed precision (in fact, [R.sup.7] was 1.0000000). It should also be noted that in Equations 13 to 22, eight-digit accuracy must be used. This was defined by analyzing the effect of the number of digits on the f and Z predictions with the aid of a commercial curve-fitting program.

CORRELATION RESULTS

An error analysis of model 1 and 2 results, based on predictions for all investigated temperatures and pressures as defined in the objectives, is shown in Table 2. By comparison, it may be concluded that the advantage of model 1 is very small for all the properties so that model 2 would be the preferred choice because it is simpler.

Although enthalpy errors in Table 2 are a little above the prescribed [+ or -]0.2%, they occur only at the highest temperature included in the study (70[degrees]C [158[degrees]F]). If the analysis is restricted to the range of the psychrometric charts (50[degrees]C [122[degrees]F]), the set goal for precision is met.

As shown in Figures 7 to 9 for volume, enthalpy, and entropy, the variables are predicted with the specified precision within the prescribed range with the simpler model.

It was mentioned that errors calculated by the ideal gas model become very large because the reference values for enthalpy and entropy approach zero within the range of calculation. To effect a better comparison of the present and ideal gas models and since only the difference of values between two states are meaningful for enthalpy and entropy, errors of enthalpy and entropy differences with respect to real gas predictions were calculated. The reference state was any selected pressure and the lowest temperature for ASHRAE psychrometric chart No. 2, -40[degrees]C (-40[degrees]F).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

The enthalpy difference prediction errors as a function of temperature at two pressures for the ideal gas model and the present model are shown in Figure 10. As may be seen there, at the low end of temperature, the present model has slightly higher absolute errors than the ideal gas model for both pressures; however, these errors are within the desired [+ or -]0.2%. As temperature goes up, errors for the present model diminish, while those for the ideal gas model grow beyond the desired value.

[FIGURE 10 OMITTED]

A similar behavior is in evidence for the entropy difference prediction errors shown in Figure 11.

[FIGURE 11 OMITTED]

CONCLUSIONS

The scope of models for saturated moist air properties prediction was addressed and a real gas model described. Based on this model, a spreadsheet program was implemented, tested, and used to evaluate predictions of the ideal gas model. These were found to have very large percent errors in enthalpy and entropy as properties approach zero. For all investigated pressures, errors were found to be above 0.5% for the high end of the range of temperatures of interest for this work, which was defined as that of ASHRAE psychrometric charts 1, 2, 5, 6, and 7 (-40[degrees]C to 50[degrees]C, -40[degrees]F to 122[degrees]F).

An engineering model for calculation of properties of saturated moist air was tailored after the real gas model. The precision objective, set at [+ or -]0.2%, was met with a model that does not require an iterative solution and is based on simple polynomial correlations of the enhancement and compressibility factors. Two tested submodels, which closely resemble the real gas model, were based on substitution of the virial coefficient contribution to property values in favor of a simple correlation. Within the defined range of temperature and pressure, both had acceptable results and, therefore, the simpler one is to be preferred.

NOMENCLATURE

[a.sub.i] = correlation coefficients

[B.sub.aa] = second virial coefficient of dry air, [m.sup.3]/mol

[B.sub.aw] = second virial coefficient of moist air, [m.sup.3]/mol

[b.sub.i] = correlation coefficients

[B.sub.ms] = second virial coefficient of the mixture air-water vapor, [m.sup.3]/mol

[B.sub.ww] = second virial coefficient of water vapor, [m.sup.3]/mol

[C.sub.aaa]= third virial coefficient of dry air, [m.sup.6]/[mol.sup.2]

[C.sub.aaw] = third cross-virial coefficient of moist air, [m.sup.6]/[mol.sup.2]

[C.sub.aww] = third cross-virial coefficient of moist air, [m.sup.6]/[mol.sup.2]

[c.sub.i] = correlation coefficients

[C.sub.ms] = third virial coefficient of the mixture air-water vapor, [m.sup.6]/[mol.sup.2]

[C.sub.www] = third virial coefficient of water vapor, [m.sup.6]/[mol.sup.2]

[d.sub.i] = correlation coefficients

[e.sub.h] = saturated air enthalpy percent error

[e.sub.s] = saturated air entropy percent error

[e.sub.v] = saturated air volume percent error

[e.sub.h] = saturated air enthalpy difference percent error

[e.sub.s] = saturated air entropy difference percent error

f = enhancement factor

[g.sub.i] = correlation functions

[h'.sub.a] = constant for chosen reference state for air, kJ/kg

[h.sub.ms] = saturated air enthalpy, J/mol

[h.sub.s] = saturated air enthalpy, kJ/kg

[h'.sub.w] = constant for chosen reference state for water, kJ/kg

k = Henry's law constant, [Pa.sup.-1]

P = total pressure, Pa

[p.sub.w] = water vapor pressure, Pa

R = universal gas constant, J/mol*K

[R.sub.2] = coefficient of correlation

[R.sub.a] = air gas constant, kJ/kg*K

[s'.sub.a] = constant for chosen reference state for air, kJ/kg.K

[S.sub.ms] = saturated air entropy, J/mol*K

[s.sub.s] = saturated air entropy, kJ/kg*K

[s'.sub.w] = constant for chosen reference state for water, kJ/kg.K

T = temperature, K

[v.sub.c] = condensed phase specific volume, [m.sup.3]/mol

[v.sub.ms] = saturated air volume, [m.sup.3]/mol

[v.sub.s] = saturated air volume, [m.sup.3]/kg

[x.sub.as] = mole fraction of air

[x.sub.ws] = mole fraction of water vapor

[Z.sub.ms] = compressibility factor

[kappa] = isothermal compressibility, [Pa.sup.-1]

[v.sub.h] = virial contribution to enthalpy, kJ/kg

[v.sub.s] = virial contribution to entropy, kJ/kg*K

REFERENCES

ASHRAE. 2002. Psychrometric Analysis CD. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

ASHRAE. 2005. 2005 ASHRAE Handbook--Fundamentals, Chapter 6. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Beattie, J.A. 1949. The computation of the thermodynamic properties of real gases and mixtures of real gases. Chem. Rev. 44:141-91.

Gatley, D.P. 2005. Understanding Psychrometrics, 2d ed. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Hyland, R.W. 1975. A correlation for the second interaction virial coefficients and enhancement factors for moist air. Journal of Research NBS-A, Physics and Chemistry 79A(4):551-60.

Hyland, R.W., and A. Wexler. 1973a. The enhancement of water vapor in carbon dioxide-free air at 30, 40, and 50[degrees]C. Journal of Research NBS-A, Physics and Chemistry 77A(1):115-31.

Hyland, R.W., and A. Wexler. 1973b. The second interaction (cross) virial coefficient for moist air. Journal of Research NBS-A, Physics and Chemistry 77A(1):133-47.

Hyland, R.W., and A. Wexler. 1980. Thermodynamic properties of dry air, moist air and water. ASHRAE research project report 216-RP. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Hyland, R.W., and A. Wexler. 1983a. Formulations for the thermodynamic properties of the saturated phases of H2O from 173.15 K to 473.15 K. ASHRAE Transactions 89(2A):500-19.

Hyland, R.W., and A. Wexler. 1983b. Formulations of the thermodynamic properties of dry air from 173.15 K to 473.15 K, and of saturated moist air from 173.15 K to 372.15 K, at pressures to 5 MPa. ASHRAE Transactions 89(2A):520-35.

Nelson, H.F., H.J. Sauer, and X. Huang. 2001a. Formulations for the thermodynamic properties of moist air at high temperatures. ASHRAE research project report 1060-RP. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Nelson, H.F., H.J. Sauer, and X. Huang. 2001b. High temperature properties of moist air. ASHRAE Transactions 107(2):780-91.

Sauer, H.J., H.F. Nelson, and X. Huang. 2001. The search for high-temperature experimental psychrometric data. ASHRAE Transactions 107(2):768-79.

Jose A. Perez Galindo, PhD

Member ASHRAE

Luis A. Payan Rodriguez

Ignacio R. Martin Dominguez, PhD

Jose A. Perez Galindo and Luis A. Payan Rodriguez are professors in the Department of Mechanical Engineering at the Instituto Tecnologico de Durango, Durango, Mexico. Ignacio R. Martin Dominguez is research scientist at the Centro de Investigacion en Materiales Avanzados, Chihuahua, Mexico.

All engineering calculations required to design an HVAC system start with estimations of the air-water vapor mixture properties that are the basis for heat and mass balances. Some time ago, engineers relied on hand calculations, and tables were their main tool for properties estimation. Tables were prepared from very precise calculations, such as those of Hyland and Wexler (1980), which are iterative and lengthy.

In modern engineering, except for rough initial estimates, most calculations are made with the aid of computers, and equations are needed to calculate properties. These equations may be as simple as the ideal gas model or as complicated as virial equations of state.

Although it would be expected that values obtained from a computer should have very high precision, this is not necessarily so. This fact may be easily verified, as an example, for the Psychrometric Analysis CD (ASHRAE 2002) results, which are obtained from the ideal gas formulation. The reason for this is probably that the calculations must be made fast enough to display results in real time as the cursor moves around the psychrometric charts, which would preclude the use of complicated models.

Even though computer time is increasingly cheaper and computers are faster, there are still instances in which computing time is a premium commodity. This is true of real-time system simulation and control applications. For these cases, simple and accurate equations are needed.

This work started in response to readers' comments regarding precision of the ideal gas model in Chapter 6 of the ASHRAE Handbook--Fundamentals (ASHRAE 2005). In particular, the comments were about errors in the [W.sub.s] and [h.sub.ms] predictions, but they led to identification of causes and possible solutions and finally to development of the model presented in this work.

IDEAL AND REAL GAS MODELS

Model Description

Two models set the range of precision and complexity of calculations. On one hand, there is the ideal gas model for air-water vapor mixtures (ASHRAE 2005) and on the other the real gas model presented by Hyland and Wexler (1983a, 1983b) and Nelson and Sauer (Sauer et al. 2001; Nelson et al. 2001b).

Because of its simplicity and first principle foundation, results of the ideal gas model set the minimum for both acceptable precision and complexity of calculations. In this model, the specific volume is calculated from the ideal gas PVT relation and enthalpy and entropy from ideal mixture laws.

The real gas model provides a standard for comparison because it yields results of very high precision (as shown in the Hyland and Wexler work) but with considerable complexity and calculation effort. This real gas model is based on the use of a virial equation of state for specific volume, Beattie's formulation (Beattie 1949) for enthalpy and entropy, and the so-called enhancement factor equation for the interaction of air-water vapor molecules (Hyland and Wexler 1973a,1973b; Hyland 1975).

To show the gas real model complexity and as background for model development, the Hyland and Wexler (1980) model will be briefly described. The air and water vapor molar fractions at saturation are related to the enhancement factor through the following equations:

[x.sub.as] = [[P - f[P.sub.w]]/P][x.sub.ws] = [[f[P.sub.w]]/P] (1)

ln(f)=[[(1+[kappa][P.sub.w])(P-[P.sub.w])-[kappa](P-[P.sub.w.sup.2])/2]/[RT]][v.sub.c] + ln[gamma](1 - k[x.sub.as]P) + ([[x.sub..sup.2]P]/[RT])[B.sub.ww] - ([2[x.sub.[as].sup.2]P]/[RT])[B.sub.aw] - ([P - [P.sub.w] - [x.sub.[as].sup.2]P]/[RT])[B.sub.ww] + [[3[x.sub.[as].sup.3][P.sup.2]]/[(RT).sup.2]][C.sub.aaa] + [[3[x.sub.[as].sup.2](1 - 2[x.sub.as])[P.sup.2]]/[2[(RT).sup.2]]][C.sub.aaw] - [[3[x.sub.[as].sup.2](1 - [x.sub.as])[P.sup.2]]/([RT.sup.2])][C.sub.aww] - [[(1 - 2[x.sub.as])[(1 - [x.sub.as]).sup.2][P.sup.2] - [P.sub.w.sup.2]]/[2[(RT).sup.2]]][C.sub.www] - [[[x.sub.[as].sup.2](1 - 3[x.sub.as])[(1 - [x.sub.as]).sup.2][P.sup.2]]/[(RT).sup.2]][B.sub.aa][B.sub.ww] - ([3[x.sub.[as].sup.4][P.sup.2]]/[2[(RT).sup.2]])[B.sub.[aa].sup.2] - [[2[x.sub.[as].sup.2](1 - [x.sub.as])(1 - 3[x.sub.as])[P.sup.2]]/[(RT).sup.2]][B.sub.[aw].sup.2] - [[[P.sub.w.sup.2] - (1 + 3[x.sub.as])[(1 - [x.sub.as]).sup.3][P.sup.2]]/[2[(RT).sup.2]]][B.sub.[ww].sup.2] (2)

The isothermal compressibility . ([Pa.sup.-1]), saturation pressure [P.sub.W] (Pa), condensed phase specific volume [v.sub.c] ([m.sup.3), Henry's law constant, [kappa]([Pa.sup.-1]), and the virial coefficients are functions of temperature only, and equations for their calculation are given in Hyland and Wexler (1980).

An iterative solution of Equations 1 and 2 to obtain the enhancement factor and, hence, the saturation air and water vapor molar fractions and the humidity ratio is the first step in the real gas model. From these equations we may also conclude that, albeit complex and implicit, the enhancement factor is a function of temperature and pressure. This was recognized by Nelson et al. (2001a), who used sixth-degree polynomials for f as a function of T in their routine calculations of air-water vapor mixture properties at high temperature. They employed one polynomial for each investigated pressure.

The following virial equation of state is used for the mixture volume:

[v.sub.ms] = [[RT]/P][1 + [[B.sub.ms]/[v.sub.ms]] + [C.sub.ms]/[v.sub.[ms].sup.2]] = [[RT]/P][Z.sub.ms] (3)

In this equation, the mixture virial coefficients are related to the saturation air and water vapor molar fractions and to the components' virial coefficients through the following:

[B.sub.ms] = [x.sub.[as].sup.2][B.sub.aa] + 2[x.sub.as][x.sub.ws][B.sub.aw] + [x.sub.[ws].sup.2][B.sub.ww] [C.sub.ms] = [x.sub.[as].sup.3][C.sub.aaa] + 3[x.sub.[as].sup.2][x.sub.ws][C.sub.aaw] + 3[x.sub.as][x.sub.[ws].sup.2][C.sub.aww] + [x.sub.[ws].sup.3][C.sub.www] (4)

Equations 3 and 4 constitute the second iterative loop of this model and solution results, in particular the compressibility factor, are functions of temperature and pressure.

Enthalpy and entropy are calculated from:

[h.sub.ms] = [x.sub.as]([5.summation over (i = 0)][a.sub.i][T.sup.i] + [H.sup.a]) + [x.sub.ws]([5.summation over (i = 0)][b.sub.i][T.sup.i] + [H.sub.w]) + RT[([B.sub.ms] - T[d[B.sub.ms]]/[dT])[1/[v.sub.ms]] + ([C.sub.ms] - [1/2]T[d[C.sub.ms]]/[dT])1/[v.sub.[ms].sup.2]] [s.sub.ms] = [x.sub.as]([4.summation over (i = 0)][c.sub.i][T.sup.i] + [c.sub.5]lnT + [s'.sub.w]) (5)

- Rln(P/[101,325]) + [x.sub.as]Rln([Z.sub.ms]/[x.sub.as]) + [x.sub.ws]Rln([Z.sub.ms]/[x.sub.ws]) - R[([B.sub.ms] + T[d[B.sub.ms]]/[dT])[1/[v.sub.ms]] + ([C.sub.ms] + [1/2]T[d[C.sub.ms]]/[dT])1/[v.sub.[ms].sup.2]] (6)

Expressions in the first two parentheses of Equations 5 and 6 are curve fits of the ideal gas properties (enthalpy and entropy, respectively) plus reference state values (primed quantities) for the mixture components. Hyland and Wexler (1980) give the required constants.

Real Gas Model Predictions

The first step to evaluate the ideal gas (or any other) model was to program the real gas model equations. A spreadsheet was implemented to generate comparison databases, and predictions of this program at 0.1 MPa (14.5 psia) were evaluated with respect to those of ASHRAE RP-216 (Hyland and Wexler 1980). This report proved to be a very useful debugging aid since their tabulations include not only the virial coefficients and all properties but values of the various terms of each equation used in calculations. They are available at 10[degrees]C (5.55[degrees]F) intervals for temperatures of -100[degrees]C to 90[degrees]C (-148[degrees]F to 194[degrees]F) for 0.1 MPa (14.5 psia), -100[degrees]C to 150[degrees]C (-148[degrees]F to 302[degrees]F) for 0.5 MPa (72.5 psia), -100[degrees]C to 170[degrees]C (-148[degrees]F to 338[degrees]F) for 1.0 MPa (145.03 psia), and -100[degrees]C to 200[degrees]C (-148[degrees]F to 392[degrees]F) for 5.0 MPa (725.18 psia).

Percent error listings of the spreadsheet predictions of the saturated mixture volume [v.sub.ms], enthalpy [h.sub.ms], and entropy [s.sub.ms] at 0.1 MPa (14.5 psia), as compared with those of the ASHRAE report, are given in Table 1. From these results it may be concluded that our real gas model program may be used to generate reference values of acceptable precision, since the maximum and minimum errors are 6.4 x [10.sup.-4%], and -1.09 x [10.sup.-3%]respectively.

Ideal Gas Model Predictions

Ideal gas model errors (100 x [ideal gas prediction - real gas value] / real gas value) are shown in Figures 1, 2, and 3. Figure 1 shows mixture volume error as a function of temperature for 75 and 101.325 kPa (10.87 and 14.695 psia). These results show that the ideal gas model predicts the air-water vapor mixture volume within the desired precision ([+ or -]0.2%).

Table 1. Spreadsheet Prediction Errors Error, % T, K [v.sub.ms] [h.sub.ms] [s.sub.ms] 213.15 4.08E-04 5.55E-05 1.69E-04 223.15 4.71E-04 -7.15E-06 -2.28E-04 233.15 1.18E-04 1.63E-05 -3.20E-04 243.15 6.40E-04 -1.78E-04 3.27E-04 253.15 4.72E-04 -1.74E-04 -2.17E-04 263.15 6.25E-04 -8.91E-04 -1.09E-03 273.15 5.24E-04 -5.18E-04 3.14E-04 283.15 -1.86E-04 3.65E-05 5.94E-05 293.15 3.24E-04 -3.75E-05 -4.41E-05 303.15 5.23E-04 -1.58E-04 -1.95E-04 313.15 -2.37E-05 -9.96E-05 -8.84E-05 323.15 9.23E-05 -1.06E-04 -1.05E-04 333.15 -5.29E-05 -4.25E-05 -5.09E-05 343.15 -1.62E-05 -7.27E-05 -7.47E-05

[FIGURE 1 OMITTED]

As shown in Figure 2, errors in enthalpy are very small for atmospheric pressure and low temperatures and then they become very large as enthalpy approaches zero. As pointed out by Gatley (2005) this is because absolute errors, which are small, are divided by the very small enthalpy values. Finally, as temperature goes up, enthalpy errors become bigger and go beyond the set precision objective. The same problem occurs for 75 kPa (10.87 psia) but with slightly greater errors.

[FIGURE 2 OMITTED]

Figure 3 shows that for entropy, although tendencies are similar, errors are larger and, therefore, we may conclude that this model does not meet our specified precision limit.

ENGINEERING CORRELATIONS MODEL

Correlation Objectives

The main objective was to develop simple yet accurate correlations to calculate the air-water vapor mixture properties v, h, and s. Early in the project we detected that restriction of the range was important and selected to encompass ASHRAE psychrometric charts for normal and low temperatures (No. 1 and 2) at sea level as well as the normal temperature charts for altitudes up to 2250 m (7282 ft) above sea level (No. 5, 6, and 7).

Therefore, the range of temperature for the correlations was -40[degrees]C to 50[degrees]C (-40[degrees]F to 122[degrees]F) and the range of pressure was from 77.059 to 101.325 kPa (11.175 to 14.695 psia). To provide some leeway for extrapolation, the study was performed for temperatures from -60[degrees]C to 70[degrees]C (-76[degrees]F to 158[degrees]F) at 5[degrees]C (2.78[degrees]F) intervals while pressures varied from 75 to 105 kPa (10.87 to 15.22 psia) at 5 kPa (0.73 psi) intervals.

For precision, our goal was to obtain results with errors below ([+ or -]0.2% of the exact model predictions.

Model Description

Our first model was based on the use of polynomial correlations of the enhancement factor to obtain better predictions of the air and water vapor molar fractions. Hence, the ideal gas model would then provide better predictions of the properties. Although this simple option did indeed raise the precision of the enthalpy and entropy predictions, the errors as h and s approach zero were not eliminated and the mixture volume prediction errors were larger than those of the ideal gas model for the higher investigated temperatures.

On these bases we decided that a better model should include correlations of the enhancement and compressibility factors as functions of pressure and temperature, to be used with Equation 3 and simplified versions of Equations 5 and 6. The proposed model is composed of the following equations.

f = [g.sub.1](T,P)[Z.sub.ms][g.sub.2](T,P) (7)

[v.sub.s] = [[[R.sub.a]T]/(P - f[P.sub.w])][Z.sub.ms] (8)

[x.sub.as] = [[P - f[P.sub.w]]/P][x.sub.ws] = [[f[P.sub.w]]/P] (9)

[h.sub.s] = ([g.sub.3](T) + [H.sub.a]) + [[x.sub.ws]/[x.sub.as]]([g.sub.4](T) + [H.sub.w]) + [v.sub.h] (10)

[s.sub.s] = ([g.sub.5](T) + [s'.sub.a]) + [[x.sub.ws]/[x.sub.as]]([g.sub.6](T) + [s'.sub.w]) - [[R.sub.a]/[x.sub.as]]ln(P/[101,325]) + [[R.sub.a]/[x.sub.as]]ln[gamma]([Z.sub.ms]/[x.sub.as]) + [[x.sub.ws]/[x.sub.as]][R.sub.a]ln[gamma]([Z.sub.ms]/[x.sub.ws]) + [v.sub.s] (11)

The v terms in Equations 10 and 11 represent the virial contribution to properties and were introduced in an attempt to eliminate the "near zero" indetermination noted in the ideal gas model results. It was possible to calculate them as a function of pressure.

[v.sub.h] = [g.sub.7](P)[v.sub.s] = [g.sub.8](P) (12)

To fulfill our objectives, [g.sub.2] to [g.sub.2] should be the simplest possible equations.

Model Functions

Various curve fits for f and Z as a function of temperature led us to conclude that fifth-degree polynomials were very good ([R.sup.2] > 0.9993) and further tests showed that pressure effects were approximately linear. A curve fit with fifth-order dependence in T and first-order in P, which will be named model 1, produced the following equations (with [R.sup.2] values of 0.9843 and 0.9693 for f and Z, respectively).

[g.sub.1] = 2.2770286 - 0.02406584T + 1.8213945X[10.sup.[ - 4]][T.sup.2] - 6.8894708X[10.sup.[ - 7]][T.sup.3] + 1.297668X[10.sup.[ - 9]][T.sup.4] - 9.7078508X[10.sup.[ - 13]][T.sup.5] + 3.9945654X[10.sup.[ - 8]]P (13)

[g.sub.2] = 1.9208388 - 0.018226313T + 1.4278754X[10.sup.[ - 4]][T.sup.2] - 5.5526317X[10.sup.[ - 7]][T.sup.3] + 1.073933X[10.sup.[ - 9]][T.sup.4] - 8.2740746X[10.sup.[ - 13]][T.sup.5] + 3.9755302X[10.sup.[ - 9]]P (14)

To obtain equations as simple as possible for property predictions, regressions with lower-order dependence on T were also investigated. The simplest equation (model 2) is of second order in T and first order in P (with [R.sup.2] values of 0.9686 for f and 0.7766 for Z).

[g.sub.1] = 1.0398885 - 0.028774113T + 5.2013515X[10.sup.[ - 7]][T.sup.2] + 3.9753593X[10.sup.[ - 8]]P (15)

[g.sub.2] = 0.97165359 - 2.0355266X[10.sup.[ - 4]]T + 3.6608098X[10.sup.[ - 7]][T.sup.2] + 4.0079476X[10.sup.[ - 9]]P (16)

A constant specific heat model was used to calculate the ideal gas enthalpies [g.sub.3] and [g.sub.4]. The following equations were obtained by least-squares minimization of percent errors of enthalpy values with respect to those of the Hyland and Wexler (1980) ideal gas predictions.

[g.sub.3] = 1.0041923(T - 273.15)[g.sub.4] = 1.8642569(T - 273.15) (17)

Within the range of temperatures used for this estimation,--60[degrees]C to 90[degrees]C (-76[degrees]F to 194[degrees]F), errors in enthalpy were from -0.187% to 0.115% for air and from -0.022% to 0.003% for vapor.

Constants [h'.sub.a] and [h'.sub.w] were calculated to obtain zero enthalpy values at the reference states, 0[degrees]C (32[degrees]F) and 101325 Pa (14.696 psia) for air and the steam triple point. With Equation 17, their values are zero and 2700.7876, respectively.

For the constant specific heat model, the ideal gas entropy functions [g.sub.5]and [g.sub.6] are

[g.sub.5] = 1.0041923ln[gamma](T),[g.sub.6] = 1.8642569ln[gamma](T) (18)

with reference values [s'.sub.a] = -5.63354 and [s'.sub.w] = -3.661889.

The error functions [g.sub.7] and [g.sub.8] depend on values of the enhancement and compressibility factors, but, fortunately, they do not differ much when either model 1 or model 2 is used. This is consistent with previous work (Gatley 2005), which states that the use of an average value of the enhancement factor yields results of acceptable precision.

The largest deviations occur for entropy when model 2 is used for calculation, and Figure 4 shows values of [v.sub.s] as percent of the exact values of entropy, which would be an error if this quantity was not included.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

A comparison of these errors with those of the ideal gas model (Figure 3) resulted in two observations. The first one is that errors near zero are still present (in fact they tend to infinity as the function tends to zero) and the second is that errors are lower elsewhere. The latter, when analyzed together with the values of the functions and their absolute errors, led to a very simple means for error calculation.

Absolute values of [v.sub.s] are shown in Figure 5, where it may be noted that, except for a few points at the large end of the investigated temperatures, a small constant value of [v.sub.s] at each pressure would be a good representation of error. If this fact is analyzed together with the values of ([s.sub.s] - [v.sub.s]), shown in Figure 6, it may be concluded that the effect of adding a small constant is negligible at the higher temperatures.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Therefore, to eliminate the large percent errors in enthalpy and entropy as their values approach zero, the constant (with respect to temperature) to be added was found by least-squares minimization of the calculated percent errors. This procedure was followed at each investigated pressure, and results were curve-fitted as functions of pressure to obtain, for model 1:

[g.sub.7] = - 4.1281106X[10.sup.[ - 18]][P.sup.3] + 1.8191667X[10.sup.[ - 12]][P.sup.2] - 3.0990568X[10.sup.[ - 6]]P + 0.28844128 (19)

[g.sub.8] = 1.3323852X[10.sup.[ - 23]][P.sup.4] - 6.0489891X[10.sup.[ - 18]][P.sup.3] + 1.0524313X[10.sup.[ - 12]][P.sup.2] - 8.8970830X[10.sup.[ - 8]]P + 3.2465229X[10.sup.[ - 3]] (20)

and for model 2:

[g.sub.7] = - 4.1877292X[10.sup.[ - 18]][P.sup.3] + 1.8317367X[10.sup.[ - 12]][P.sup.2] - 3.0980589X[10.sup.[ - 3]]P + 0.28801940 (21)

[g.sub.8] = 1.3900054X[10.sup.[ - 23]][P.sup.4] - 6.5157721X[10.sup.[ - 18]][P.sup.3] + 1.1762918X[10.sup.[ - 12]][P.sup.2] - 1.0177460X[10.sup.[ - 7]]P + 3.6530569X[10.sup.[ - 3]] (22)

It should be mentioned that the order of the [g.sub.7] and [g.sub.8] correlations was the minimum to obtain results within the prescribed precision (in fact, [R.sup.7] was 1.0000000). It should also be noted that in Equations 13 to 22, eight-digit accuracy must be used. This was defined by analyzing the effect of the number of digits on the f and Z predictions with the aid of a commercial curve-fitting program.

CORRELATION RESULTS

An error analysis of model 1 and 2 results, based on predictions for all investigated temperatures and pressures as defined in the objectives, is shown in Table 2. By comparison, it may be concluded that the advantage of model 1 is very small for all the properties so that model 2 would be the preferred choice because it is simpler.

Although enthalpy errors in Table 2 are a little above the prescribed [+ or -]0.2%, they occur only at the highest temperature included in the study (70[degrees]C [158[degrees]F]). If the analysis is restricted to the range of the psychrometric charts (50[degrees]C [122[degrees]F]), the set goal for precision is met.

As shown in Figures 7 to 9 for volume, enthalpy, and entropy, the variables are predicted with the specified precision within the prescribed range with the simpler model.

Table 2. Model Errors (%) Model 1 v h s Average 0.0001 -0.0285 0.0037 Std. Dev. 0.0119 0.0888 0.0376 Max. 0.0642 0.2136 0.1250 Min. -0.0422 -0.1554 -0.0747 Model 2 v h s Average 0.0022 -0.0262 0.0080 Std. Dev. 0.0356 0.0951 0.0566 Max. 0.1735 0.2750 0.1916 Min. -0.0493 -0.1550 -0.0687

It was mentioned that errors calculated by the ideal gas model become very large because the reference values for enthalpy and entropy approach zero within the range of calculation. To effect a better comparison of the present and ideal gas models and since only the difference of values between two states are meaningful for enthalpy and entropy, errors of enthalpy and entropy differences with respect to real gas predictions were calculated. The reference state was any selected pressure and the lowest temperature for ASHRAE psychrometric chart No. 2, -40[degrees]C (-40[degrees]F).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

The enthalpy difference prediction errors as a function of temperature at two pressures for the ideal gas model and the present model are shown in Figure 10. As may be seen there, at the low end of temperature, the present model has slightly higher absolute errors than the ideal gas model for both pressures; however, these errors are within the desired [+ or -]0.2%. As temperature goes up, errors for the present model diminish, while those for the ideal gas model grow beyond the desired value.

[FIGURE 10 OMITTED]

A similar behavior is in evidence for the entropy difference prediction errors shown in Figure 11.

[FIGURE 11 OMITTED]

CONCLUSIONS

The scope of models for saturated moist air properties prediction was addressed and a real gas model described. Based on this model, a spreadsheet program was implemented, tested, and used to evaluate predictions of the ideal gas model. These were found to have very large percent errors in enthalpy and entropy as properties approach zero. For all investigated pressures, errors were found to be above 0.5% for the high end of the range of temperatures of interest for this work, which was defined as that of ASHRAE psychrometric charts 1, 2, 5, 6, and 7 (-40[degrees]C to 50[degrees]C, -40[degrees]F to 122[degrees]F).

An engineering model for calculation of properties of saturated moist air was tailored after the real gas model. The precision objective, set at [+ or -]0.2%, was met with a model that does not require an iterative solution and is based on simple polynomial correlations of the enhancement and compressibility factors. Two tested submodels, which closely resemble the real gas model, were based on substitution of the virial coefficient contribution to property values in favor of a simple correlation. Within the defined range of temperature and pressure, both had acceptable results and, therefore, the simpler one is to be preferred.

NOMENCLATURE

[a.sub.i] = correlation coefficients

[B.sub.aa] = second virial coefficient of dry air, [m.sup.3]/mol

[B.sub.aw] = second virial coefficient of moist air, [m.sup.3]/mol

[b.sub.i] = correlation coefficients

[B.sub.ms] = second virial coefficient of the mixture air-water vapor, [m.sup.3]/mol

[B.sub.ww] = second virial coefficient of water vapor, [m.sup.3]/mol

[C.sub.aaa]= third virial coefficient of dry air, [m.sup.6]/[mol.sup.2]

[C.sub.aaw] = third cross-virial coefficient of moist air, [m.sup.6]/[mol.sup.2]

[C.sub.aww] = third cross-virial coefficient of moist air, [m.sup.6]/[mol.sup.2]

[c.sub.i] = correlation coefficients

[C.sub.ms] = third virial coefficient of the mixture air-water vapor, [m.sup.6]/[mol.sup.2]

[C.sub.www] = third virial coefficient of water vapor, [m.sup.6]/[mol.sup.2]

[d.sub.i] = correlation coefficients

[e.sub.h] = saturated air enthalpy percent error

[e.sub.s] = saturated air entropy percent error

[e.sub.v] = saturated air volume percent error

[e.sub.h] = saturated air enthalpy difference percent error

[e.sub.s] = saturated air entropy difference percent error

f = enhancement factor

[g.sub.i] = correlation functions

[h'.sub.a] = constant for chosen reference state for air, kJ/kg

[h.sub.ms] = saturated air enthalpy, J/mol

[h.sub.s] = saturated air enthalpy, kJ/kg

[h'.sub.w] = constant for chosen reference state for water, kJ/kg

k = Henry's law constant, [Pa.sup.-1]

P = total pressure, Pa

[p.sub.w] = water vapor pressure, Pa

R = universal gas constant, J/mol*K

[R.sub.2] = coefficient of correlation

[R.sub.a] = air gas constant, kJ/kg*K

[s'.sub.a] = constant for chosen reference state for air, kJ/kg.K

[S.sub.ms] = saturated air entropy, J/mol*K

[s.sub.s] = saturated air entropy, kJ/kg*K

[s'.sub.w] = constant for chosen reference state for water, kJ/kg.K

T = temperature, K

[v.sub.c] = condensed phase specific volume, [m.sup.3]/mol

[v.sub.ms] = saturated air volume, [m.sup.3]/mol

[v.sub.s] = saturated air volume, [m.sup.3]/kg

[x.sub.as] = mole fraction of air

[x.sub.ws] = mole fraction of water vapor

[Z.sub.ms] = compressibility factor

[kappa] = isothermal compressibility, [Pa.sup.-1]

[v.sub.h] = virial contribution to enthalpy, kJ/kg

[v.sub.s] = virial contribution to entropy, kJ/kg*K

REFERENCES

ASHRAE. 2002. Psychrometric Analysis CD. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

ASHRAE. 2005. 2005 ASHRAE Handbook--Fundamentals, Chapter 6. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

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Jose A. Perez Galindo, PhD

Member ASHRAE

Luis A. Payan Rodriguez

Ignacio R. Martin Dominguez, PhD

Jose A. Perez Galindo and Luis A. Payan Rodriguez are professors in the Department of Mechanical Engineering at the Instituto Tecnologico de Durango, Durango, Mexico. Ignacio R. Martin Dominguez is research scientist at the Centro de Investigacion en Materiales Avanzados, Chihuahua, Mexico.

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Author: | Galindo, Jose A. Perez; Rodriguez, Luis A. Payan; Dominguez, Ignacio R. Martin |
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Publication: | ASHRAE Transactions |

Article Type: | Report |

Geographic Code: | 1MEX |

Date: | Jul 1, 2007 |

Words: | 4908 |

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