AIRMODEL.HTM --- Part of Manual for Driver Parameter Calculator --- by
Claus Futtrup.

Created 7. March 2001, last revised 4. February 2004.
Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

Table of Contents:

- Introduction
- Partial pressure of Water Vapor
- Comments
- Old model
- Source / litterature
- Alternative model
- Acknowledgments

In AIR.HTM, section Air Density, right above equation 5, a table of partial pressure for saturated water vapor is given. This table ranges from 0 degC to 59 degC, which is insufficient for studies in car-fi, where a temperature range of -40 degC to +105 degC (worst cases) are to be considered.

For this reason DPC has been implemented with an advanced model of air. Within the temperature range of the earlier table of Pm there is no significant changes. At the extremes of the new temperature range humidity level can play a very significant role, and actually DPC does not support high relative humidity levels at high temperatures (and they are very difficult to achieve in real life). The new model is valid from -100 degC to +200 degC. The temperature range of interest is well within the limits of the model.

Earlier, due to limits in the model, the temperatures treated in DPC were limited to be between -100 degC and +99 degC. This is also the temperature limits given by Hyland and Wexler for humid air.

From release 0.5.2, the model instead takes care that the described situation is a stable situation. This means that the figures must not describe a situation where water boils. To prevent such situations, the pressure is increased automatically when required.

The model described in the source (see below) is split into 2 sections, one which covers temperatures above freezing, and another which covers temperatures below and at freezing.

Equation 1 (eq. 18 in the source) covers temperatures from -100 degC to 0 degC:

_5_ \ ln(p_ws) => m_i * T^(i-1) + m_6 * ln(T) [Pa] (1) /___ i=0 = C1/T + C2 + C3*T + C4*T^2 + C5*T^3 + C6*T^4 + C7*ln(T)

Equation 2 (eq. 17 in the source) covers temperatures from 0 degC to 200 degC:

_3_ \ ln(p_ws) => g_i * T^i + g_4 * ln(T) [Pa] (2) /___ i=-1 = C8/T + C9 + C10*T + C11*T^2 + C12*T^3 + C13*ln(T) Pm = p_ws = e^(ln(p_ws))

Pm is the partial pressure of ice/liquid water at saturation.

C1 = -5.674 535 9E+3 (C1 was erroneously given as -5.674 35 9E+3) C2 = 6.392 524 7 (tracked by using the original article) C3 = -9.677 843 0E-3 (C3 was erroneously given as -9.677 843 0E3,) C4 = 6.221 570 1E-7 (but luckily this error was easy to track) C5 = 2.074 782 5E-9 C6 = -9.484 024 0E-13 C7 = 4.163 501 9 C8 = -5.800 220 6E+3 C9 = 1.391 499 3 C10 = -4.864 023 9E-2 C11 = 4.176 476 8E-5 C12 = -1.445 209 3E-8 C13 = 6.545 967 3

I have chosen to use equation 18 at 0 degC, which gives the most correct value at 0 degC. The two equations does not meet 100% at the same value at 0 degC, and the situation of water vapor at 0 degC would be an extrapolation of measured values. The H2O molecules in the air are in the form of (hexagonal) ice at 0 degC and below.

Notice : There is a discontinuity in the implementation in DPC from T = 1E-14 degC (Pm = 611.212867444683) and T = 1E-15 degC (Pm = 611.53570887130). The difference is only 97.0 ppm.

These equations are very accurate from -100 to +200 degC. Accuracy is 0.5% at -100 degC, 0.012% at 0 degC, 0.015% at 100 degC and 0.037% at 200 degC.

The derivation works with the TTS (TTS = Thermodynamic Temperature Scale) above 0 degC, and the IPTS-68 temperature scale below 0 degC (as an approximaton to the TTS). On the TTS water boils at 99.97 degC (at 101325 Pa). A more recent temperature scale is IPTS-90 (International Practical Temperature Scale of 1990). The maximum difference between TTS and IPTS-68 is at +150 degC. and 1E-15 (Pm = 611.53570887130). The difference is only 97.0 ppm.

I assume that IPTS-90 is the same temperature scale as ITS-90. What such a temperature scale does, at least in this case, is to describe measuring methods in details as well as a high number of fix points. For example, between 13.8033 Kelvin and 961.78 degC (= 1234.93 Kelvin), a platinum-resistance thermometer is used. Fixpoints are eg. 0.01 degC for H2O (triple point). At higher temperatures the melting point of various metals (at 1 atm air pressure) is used as fix points.

The difference between ITS-90 and IPTS-68 is never more than 0.4 Kelvin, and at 100 degC it is only 0.025 Kelvin.

I am not considering this to be a problem - use of a different temperature scale will open the same discussion as if DPC had not been using this new equation set. The temperature scales differ in an way that is not significant in the valid temperature range, and should under normal circumstances not alter a conclusion based on the data from DPC. Such a situation would be very subtle.

Notice that Pm (= p_ws) is described entirely as a function of the temperature. A previously written article by Hyland and Wexler prepared the work, and it claims the models to be valid (from vacuum) up to a pressure of 50 bar (5 MPa).

R. W. Hyland and A. Wexler, Formulations for 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, vol 89, 1983a, pages 520-535.

Another author, Lester Haar et al. (1983) has described air for more extreme temperatures and pressures:

Haar, L., Gallagher, J., and Kell, G.S. 1983 "NBS/NRC Steam Tables" New York: Hemispheric Publishing Corp. (in press) NBS : National Bureau of Standards

The model implemented into DPC is in conformity with the previous table. Only at 0 degC it will be possible to observe very minor differences (like 0.03%) in the values of the calculated sound speed and air density. Otherwise the model works all the way up to 60 degC in conformity with the table. Below 0 degC the new model is much more accurate, because the old model ignored humidity levels (ice vapor in the air) below 0 degC. This does not change much, though. Above 60 degC the new model provides significant improvements.

The boiling point of water is given when the partial pressure for water, Pm (see above equations), is higher than the normal pressure of the air. Whenever such a situation occurs in the setup of the program, DPC will change the air pressure to the partial pressure of the water vapor.

For example, if you are in the mountains, altitude 3 km, where (in this case) the normal pressure could be 0.7 bar, then water will boil at ca. 90 degC. If you specify this situation in DPC and 100 % RH, then you are at the limits. Lowering the pressure any further is not allowed (in fact DPC will increase the pressure to 702 mbar).

In this example the entire air pressure is from water vapor - all other constituents have been pushed out of the area. You now have a room full of steam (only).

Regarding boiling, a special situation appears at the critical pressure, p_crit = 221.2 bar, where the boiling point is 374.2 degC. Above this, the situation with boiling water will no longer appear, because H2O does not exist in a condensed form. Then H2O only exist in the air phase. Since the required temperature is above 200 degC, it is not included in the frame of this model.

By the way, water exist as steam in the air from 374.2 degC and up to around a temperature above 520 degC, which is another special situation called overheated steam. Above this temperature, the H2O behaves like a normal air constituent.

The old model, utilizing a table of partial pressures also used a different equation for calculating rho:

(ML - MH)*phi*Pm rho = rho_0 - ------------------ RL * T

This equation does a bad job at higher temperatures. Within the previous given temperature limits of 0 to 59 degC this was working fine. The difference between the new and the old model is less than 1% on the value of rho (and 0.5% on the value of c).

The model used in DPC is from the following source:

Authors : R. W. Hyland and A. Wexler Title : Formulations for the Thermodynamic Properties of the Saturated Phases of H2O from 173.15 K to 473.15 K Year : 1983b Pages : 500-519 Volume : 89 Number : 2A Title : ASHRAE Transactions Print : New York, N. Y. ISSN : 0001-2505

Retrieving the article made it possible to correct a few errors, utilizing the original equations on pages 506-507.

An alternative source (with a few errors - see above equations) is:

1997 ASHRAE HANDBOOK, FUNDAMENTALS, SI Edition, ISBN 1-883413-45-1, Chapter 6 : Psychrometrics (PART : Theory), page 6.2, section "Thermodynamic Properties of Water at Saturation".

ASHRAE : American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (http://www.ashrae.org).

Since the edition of the handbook is revised in 1997 I believe the quality of the article is impeccable and of very high quality.

The quality of the handbook is less good, though. When checking the tables I have found an error at 0* degC (extrapolated from equation 17) in the value of P_w (= P_ws), and the table of Ws (the humidity ratio, given in the table on page 6.3, adjacent to the quoted equations) gives errors too.

This table is a cut from the ASHRAE Fundamentals Handbook:

degC |10^3 Pa p -----|--------- -60 | 0,00108 -50 | 0,00394 -40 | 0,01285 -30 | 0,03802 -20 | 0,10326 -10 | 0,25990 0 | 0,61115 0 | 0,61112 (this is probably an error, it should read 0,61121) 10 | 1,2280 20 | 2,3388 30 | 4,2460 40 | 7,3835 50 | 12,3499 60 | 19,944 70 | 31,198 80 | 47,412 90 | 70,180 100 | 101,419 110 | 143,384 120 | 198,685 130 | 270,298 140 | 361,565 150 | 476,198 160 | 618,275 170 | 792,235 180 | 1002,871 190 | 1255,324 200 | 1555,074

There is an "alternative" description of air properties. This model cannot be used in the same application in DPC, but it describes different features / properties of air. This model has been described in the Journal of the Audio Engineering Society.

JAES Vol 42, nr. 11 November 1994 page 927-933. ENGINEERING REPORTS : Acoustical Properties of Air versus Temperature and Pressure, Gavin R. Putland.

This is another view on the air properties, but humidity is not considered, and is taken to provide small errors on the calculation of air properties (at room temperature). This is probably true for the considered properties (I am not so sure when considering car-fi and extreme temperatures). I believe that the voltage sensitivity, USPL, is of great importance, and that small variations, eg. of 0.2 dB (possible changes even though the temperature is kept at 20 degC), should be taken into considerations - because it is possible, and because it could be highly relevant for certain working areas for acoustics, eg. calibration of microphones, or for reduction of a tolerance in measurements (like in DPC), or whenever it is relevant to find an accurate sound pressure level, relative to 20 uPa or absolute, eg. when communicating with other people.

Gavin Richard Putland has written a PhD thesis, which BTW can be downloaded from the internet in a .PDF format. The title is "Modeling of Horns and Enclosures for Loudspeakers." In chapter 9, Acoustical properties of air vs. temperature and pressure. The thesis refers to his article in JAES and describes the chapter in the thesis to be basically identical - just a bit of rewriting, and probably some corrections and updates are included.

He concludes that the sound absorbtion is the only parameter where humidity cannot be neglected. His analysis is only valid for small acoustical devices.

The thesis is written from Department of Electrical and Computer Engineering, University of Queensland. Submitted December 23. 1994. Revised November 1995, Accepted 6. February 1996.

Ron Ennenga helped me get on the track of the articles by Hyland and Wexler. Mutual correspondence helped me to develop the new model of air in DPC.