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Thermodynamic properties for saturated air, an engineering correlation.



INTRODUCTION

All engineering calculations required to design an HVAC (Heating Ventilation Air Conditioning) In the home or small office with a handful of computers, HVAC is more for human comfort than the machines. In large datacenters, a humidity-free room with a steady, cool temperature is essential for the trouble-free  system start with estimations of the air-water vapor vapor /va·por/ (va´por) pl. vapo´res, vapors   [L.]
1. steam, gas, or exhalation.

2. an atmospheric dispersion of a substance that in its normal state is liquid or solid.
 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 estimation

In mathematics, use of a function or formula to derive a solution or make a prediction. Unlike approximation, it has precise connotations. In statistics, for example, it connotes the careful selection and testing of a function called an estimator.
. Tables were prepared from very precise calculations, such as those of Hyland Hyland can refer to any of the following:
  • Brian Hyland, American musician
  • Diana Hyland, American actress
  • Drew Hyland, American academic and philosopher
  • Frances Hyland, Canadian actress
  • Harry Hyland, Canadian ice hockey player
 and Wexler (1980), which are iterative it·er·a·tive  
adj.
1. Characterized by or involving repetition, recurrence, reiteration, or repetitiousness.

2. Grammar Frequentative.

Noun 1.
 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 Virial equation

An equation of state of gases that has additional terms beyond that for an ideal gas, which account for the interactions between the molecules.
 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 ver·i·fy  
tr.v. ver·i·fied, ver·i·fy·ing, ver·i·fies
1. To prove the truth of by presentation of evidence or testimony; substantiate.

2.
, as an example, for the Psychrometric Analysis CD (ASHRAE ASHRAE American Society of Heating, Refrigerating & Air Conditioning Engineers  2002) results, which are obtained from the ideal gas formulation formulation /for·mu·la·tion/ (for?mu-la´shun) the act or product of formulating.

American Law Institute Formulation
. The reason for this is probably that the calculations must be made fast enough to display results in real time as the cursor (1) The symbol used to point to some element on screen. On Windows, Mac and other graphics-based screens, it is also called a "pointer," and it changes shape as it is moved with the mouse into different areas of the application.  moves around the psychrometric charts, which would preclude pre·clude  
tr.v. pre·clud·ed, pre·clud·ing, pre·cludes
1. To make impossible, as by action taken in advance; prevent. See Synonyms at prevent.

2.
 the use of complicated models.

Even though computer time is increasingly cheaper and computers are faster, there are still instances in which computing computing - computer  time is a premium commodity. This is true of real-time 1. real-time - Describes an application which requires a program to respond to stimuli within some small upper limit of response time (typically milli- or microseconds). Process control at a chemical plant is the classic example.  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, river: see Süre.  (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 enthalpy (ĕn`thălpē), measure of the heat content of a chemical or physical system; it is a quantity derived from the heat and work relations studied in thermodynamics.  and entropy entropy (ĕn`trəpē), quantity specifying the amount of disorder or randomness in a system bearing energy or information. Originally defined in thermodynamics in terms of heat and temperature, entropy indicates the degree to which a given  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 Beattie is a surname, and may refer to:
  • A. L. Beattie, pioneering Chief Mechanical Engineer of the New Zealand Railways Department.
  • Ann Beattie (1947-), American writer
  • Craig Beattie, an Scottish footballer.
 1949) for enthalpy and entropy, and the so-called so-called
adj.
1. Commonly called: "new buildings ... in so-called modern style" Graham Greene.

2.
 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 molar /mo·lar/ (mo´lar)
1. pertaining to a mole of a substance.

2. a measure of the concentration of a solute, expressed as the number of moles of solute per liter of solution. Symbol M, , or mol/L.
 fractions at saturation saturation, of an organic compound
saturation, of an organic compound, condition occurring when its molecules contain no double or triple bonds and thus cannot undergo addition reactions.
 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 Kappa

Used in regression analysis, Kappa represents the ratio of the dollar price change in the price of an option to a 1% change in the expected price volatility.

Notes:
Remember, the price of the option increases simultaneously with the volatility.
][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 i·so·ther·mal
adj.
Of, relating to, or indicating equal or constant temperatures.



isothermal, isothermic

having the same temperature.
 compressibility com·press·i·ble  
adj.
That can be compressed: compressible packing materials; a compressible box.



com·press
 . ([Pa.sup.-1]), saturation pressure [P.sub.W] (Pa), condensed con·dense  
v. con·densed, con·dens·ing, con·dens·es

v.tr.
1. To reduce the volume or compass of.

2. To make more concise; abridge or shorten.

3. Physics
a.
 phase specific volume [v.sub.c] ([m.sup.3), Henry's law Henry's law, chemical law stating that the amount of a gas that dissolves in a liquid is proportional to the partial pressure of the gas over the liquid, provided no chemical reaction takes place between the liquid and the gas.  constant, [kappa]([Pa.sup.-1]), and the virial coefficients Virial coefficients appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density.  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 humidity, moisture content of the atmosphere, a primary element of climate. Humidity measurements include absolute humidity, the mass of water vapor per unit volume of natural air; relative humidity (usually meant when the term 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 polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a  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 The compressibility factor (Z) is used to alter the ideal gas equation to account for the real gas behaviour.[1] The compressibility factor is usually obtained from the compressibility chart. , are functions of temperature and pressure.

Enthalpy and entropy are calculated from:

[h.sub.ms] = [x.sub.as]([5.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  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 See parenthesis.

parentheses - See left parenthesis, right parenthesis.
 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 spreadsheet

Computer software that allows the user to enter columns and rows of numbers in a ledgerlike format. Any cell of the ledger may contain either data or a formula that describes the value that should be inserted therein based on the values in other cells.
 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 (programming) debugging - The process of attempting to determine the cause of the symptoms of malfunctions in a program or other system. These symptoms may be detected during testing or use by real users.  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 saturated /sat·u·rat·ed/ (sach´ah-rat?ed)
1. denoting a chemical compound that has only single bonds and no double or triple bonds between atoms.

2. unable to hold in solution any more of a given substance.
 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 ref·er·ence values
pl.n.
A set of laboratory test values obtained from an individual or from a group in a defined state of health.
 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 atmospheric pressure
 or barometric pressure

Force per unit area exerted by the air above the surface of the Earth. Standard sea-level pressure, by definition, equals 1 atmosphere (atm), or 29.92 in. (760 mm) of mercury, 14.70 lbs per square in., or 101.
 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 lee·way  
n.
1. The drift of a ship or an aircraft to leeward of the course being steered.

2. A margin of freedom or variation, as of activity, time, or expenditure; latitude. See Synonyms at room.
 for extrapolation (mathematics, algorithm) extrapolation - A mathematical procedure which estimates values of a function for certain desired inputs given values for known inputs.

If the desired input is outside the range of the known values this is called extrapolation, if it is inside then
, 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 PSI - Portable Scheme Interpreter ) 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 Vir´i`al

n. 1. (Physics) A certain function relating to a system of forces and their points of application, - first used by Clausius in the investigation of problems in molecular physics.
 contribution to properties and were introduced in an attempt to eliminate the "near zero" indetermination in·de·ter·mi·nate  
adj.
1.
a. Not precisely determined, determinable, or established: a person of indeterminate age.

b.
 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 ful·fill also ful·fil  
tr.v. ful·filled, ful·fill·ing, ful·fills also ful·fils
1. To bring into actuality; effect: fulfilled their promises.

2.
 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 infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e.  as the function tends to zero) and the second is that errors are lower elsewhere. The latter, when analyzed an·a·lyze  
tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es
1. To examine methodically by separating into parts and studying their interrelations.

2. Chemistry To make a chemical analysis of.

3.
 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 This article or section is written like a personal reflection or and may require .
Please [ improve this article] by rewriting this article or section in an .
 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 pre·scribe  
v. pre·scribed, pre·scrib·ing, pre·scribes

v.tr.
1. To set down as a rule or guide; enjoin. See Synonyms at dictate.

2. To order the use of (a medicine or other treatment).
 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 moist

having a moderate moisture content, slightly wet to the touch.


moist dermatitis
see moist dermatitis of rabbits.

moist grain storage
grain stored at about 30% moisture in airtight silos.
 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 Substitution
Arsinoë

put her own son in place of Orestes; her son was killed and Orestes was saved. [Gk. Myth.: Zimmerman, 32]

Barabbas

robber freed in Christ’s stead. [N.T.: Matthew 27:15–18; Swed. Lit.
 of the virial coefficient contribution to property values in favor of upon the side of; favorable to; for the advantage of.

See also: favor
 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 nomenclature /no·men·cla·ture/ (no´men-kla?cher) a classified system of names, as of anatomical structures, organisms, etc.

binomial nomenclature
 

[a.sub.i] = correlation coefficients Correlation Coefficient

A measure that determines the degree to which two variable's movements are associated.

The correlation coefficient is calculated as:
 

[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 coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

2.
 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 The introduction to this article provides insufficient context for those unfamiliar with the subject matter.
Please help [ improve the introduction] to meet Wikipedia's layout standards. You can discuss the issue on the talk page.
 

[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 vapor pressure, pressure exerted by a vapor that is in equilibrium with its liquid. A liquid standing in a sealed beaker is actually a dynamic system: some molecules of the liquid are evaporating to form vapor and some molecules of vapor are condensing to form liquid. , Pa

R = universal gas constant universal gas constant: see gas laws. , J/mol*K

[R.sub.2] = coefficient of correlation coefficient of correlation
n. pl. coefficients of correlation
See correlation coefficient.

Noun 1. 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 mole fraction
n.
The ratio of the moles of one component of a system to the total moles of all components present.
 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 re·frig·er·ate  
tr.v. re·frig·er·at·ed, re·frig·er·at·ing, re·frig·er·ates
1. To cool or chill (a substance).

2. To preserve (food) by chilling.
 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 Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking.  of the thermodynamic properties Here is a partial list of thermodynamic properties of fluids:
  • temperature [K]
  • density [kg/m3]
 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
This article is about the Mexican city. For other uses of the same name, see Durango (disambiguation).


The city of Durango or Victoria de Durango
, Mexico. Ignacio R. Martin Dominguez is research scientist at the Centro de Investigacion en Materiales Avanzados, Chihuahua Chihuahua, state, Mexico
Chihuahua (chēwä`wä), state (1990 pop. 2,441,873), 94,831 sq mi (245,612 sq km), N Mexico, on the border of N.Mex. and Texas. The city of Chihuahua is the capital.
, Mexico.
COPYRIGHT 2007 American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
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Author:Galindo, Jose A. Perez; Rodriguez, Luis A. Payan; Dominguez, Ignacio R. Martin
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1MEX
Date:Jul 1, 2007
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