Water Vapor - Other Formulas

My programs implement a number of formulas which claim to compute saturated water pressure at a given temperature. However, from time to time I run across new formulas - and occasionally, flat out wrong formulas. Since I don't want to be continuously updating my program, I will simply add these "new" formulas here.


The Engineering ToolBox

The Engineering ToolBox is a source of fairly reliable data. As a result, I was shocked (and disappointed) to find that their Saturation Pressure of Water Vapor formula is totally different than all the other formulas I've collected. In general, the results agree fairly well with GoffGratch up to about 0°C, then it begins to strongly diverge. For instance, (based on this table) Only GISS-AR4 (NASA) is worse.

Unfortunately, they do not reference their source for this equation - it is merely presented as "a statement of fact".

A Google search for that equation returned numerous hits. Unfortunately, all the examples I checked are obviously just quoting The Engineering ToolBox - either an explicit reference or the identical text and formatting.

In addition, on their Dry Bulb, Wet Bulb and Dew Point Temperatures page they say

which is, of course, wrong! It should be -273.15°C. (Yes - it does make a difference! Particularly in a reference book.)


NASA Tech Note D-8401

Equations for the Determination of Humidity from Dewpoint and Psychrometric Data - NASA Tech Note D-8401, Parish and Putnam, January 1977 - provides yet another set of equations with derivations. They claim that the following is the Magnus formula. which has the general form None of the other formulas I have seen to date have that form. The MagnusTetens formula is To be clear, several formulas contain Ta, but a is usually positive and there are always a lot more terms.

Their derivations of the constants - a, b, c - are provided in Appendix A. They always use 273 to convert between K and C (should be 273.15) and they use 2 vapor pressure calibration points - both are at 0°C (32°F).

Appendix A, Example 2 computes relative humidity from a dew point below freezing. Their value is 17.3%, my program, using MurphyKoop, computes 19.2% over water and 17.1% over ice. Example 3 uses 2 temperatures above freezing - they compute 84.6%, my program finds the same for most of the provided algorithms.

In addition, their definition of relative humidity (based on that giving in the Smithsonian Meteorological Tables, 1971) is a bit strange.

I have not seen any other papers specify that the denominator always be with respect to the saturation pressure over liquid water.


New Magnus Coefficients

Improved Magnus Form Approximation of Saturation Vapor Pressure, Alduchov (Ru) and Eskridge (US), April 1996, Journal of Applied Meteorology

This paper argues that the standard Magnus coefficients should be replaced so that the Magnus equation can be used instead of the much more complex equations of Wexler (1976), Sonntag (1990), and Goff and Gratch (1946).

As an aside, these dates do not match those provided by Vömel - Wexler (1983), and Sonntag (1994) - and the MurphyKoop (2005) formulation was published after this paper.

Part of the justification of this approach is that atmospheric temperature is reported to only 0.1°C accuracy world wide and only 0.5°C in the US. That, plus the general inaccuracies of the reported humidity proxies (such as dew point) means that the true accuracy of any computation is low enough that several formulations produce calculated saturation pressures that are "the same" within some margin of error.

This is the general form of the Magnus formula.

Alduchov provides 6 different sets of Magnus coefficients by others, and one "better formulation" suggested by himself. The following table only shows 3 - MagnusTetens from Vömel, one of the six "other" formulations, and the new "suggested" Alduchov formulation.


Full equations

The Alduchov paper discusses 4 "complex" equations by 3 authors - its purpose was to replace these "difficult to compute" formulations with a simpler Magnus form. (With modern computers, this goal is questionable.)


Wexler

New Equations for Computing Vapor Pressure and Enhancement Factor, Arden L.Buck, 1981.

This paper contains a good overview of the progress toward developing equations to compute saturation vapor pressure. It includes a modification to the Magnus equation by Bögel and 2 equations by Wexler that are not included in Vömel.


washington.edu skewt.js

While developing a skew-T charting application, I found washington.edu skewt.js. This is their saturation vapor pressure subroutine (no source given) This is almost identical to the Goff Gratch equation (only one constant is different), but with the terms multiplied out (which presumably makes the computation faster) which, unfortunately, obscures the use of the triple point temperature. The following computations were performed using a spreadsheet. A few test computations show the same results as Goff Gratch to 2 decimal places - 4 significant figures.


MODTRAN

MODTRAN is a binned-spectrum transmittance program used by the US Air Force. The code for version 3 (dated April 1985, the current version is 6, but I don't have the code) uses They claim that the formula "is accurate to better than 1 percent from -50 to +50 deg C."

However, since this formula computes density (gm/m3), I need to use the gas law to get the pressure.

Where the molecular weight of water (molWt) used in the program is 18.015 gm/mol.

Via the engineeringtoolbox, the gas constant is

And the decimal point needs to be adjusted.

Assuming T is in kelvin (it works), this table compares the adjusted MODTRAN formula with the MurphyKoop formulation as computed using my calculator.

So, their formula is pretty good, but it does not exactly match any of the other formulations (models) in the calculator.

They do not provide a formula for finding the relative humidity over ice.


More

As a part of my normal research, I performed a google search for There were a few hits. I checked a number of additional references and they all reference that equation to LOTRAN.


Vaisala

In Humidity Conversion Formulas, Vaisala suggests using the IAPWS formula over liquid water and the following over ice However, the voemel reference says


Lowe

Paul R. Lowe wrote at least 2 papers on attempting to find faster algorithms for computing the water vapor saturation pressure. Basically, the accepted standard at the time was Goff-Gratch 1946 which requires multiple logs and powers. Several other formulations also existed then, but they all required at least one exponential. Even today, all the other algorithms I have seen require at least one exponential.

In The Computation of Saturation Vapor Pressure (March 1974), Lowe argues for finding an algorithm with acceptable accuracy and a fast computation speed. The paper presents two 6th order polynomials for computing saturated water vapor over ice and liquid water that, he claims, run faster and are more accurate than Tetens.

In An Approximating Polynomial for the Computation of Saturated Vapor Pressure (1976), he repeats the polynomials and provides additional data supporting his claims.

These are just the coefficients for temperature in C and over water. The papers also contain coefficients for the temperature in kelvin over water and for over ice.


Accuracy

In 1976, to determine the accuracy of his polynomial, Lowe selected Goff-Gratch 1946 as the gold standard. According to New Equations for Computing Vapor Pressure and Enhancement Factor (Buck 1981), Wexler 1976 is a more accurate formulation.

In Table 1, Lowe 1976 claims that Goff-Gratch computes the saturation pressure as 6.1078 mb at 0°C, but, according to my calculator, that is the pressure computed at 0.01°C (the triple point) and the Goff-Gratch value at 0.00°C is actually 6.10336 mb. Via a0 in the coefficients above, Lowe's formula sets the saturation pressure to 6.107799961 mb at 0.00°C. (The current standard is 6.11657 ± 0.0001 mbar at 273.16 K (0.01°C) - the triple point.)

I checked a number of Goff-Gratch and Tetens values in his table with those computed by my calculator - they do not match. One of us has an obvious math error.

Compared to voemel, there is only a slight difference in one of the Lowe 1976 Tetens formula coefficients, but a fairly significant difference in the values shown in Table 1.

However, the computed and reported values differ just a bit too much This is bad! Using 20°C as an example In my opinion, the table values and those computed via my calculator should have been identical ... and they are not !!

Whether or not these errors are significant depends on your application. But, in a peer-reviewed paper about algorithm accuracy, I expect better. On the other hand, this is an old paper and some of the standards and/or accepted calibration points have changed since then. As a result, his coefficients may need to be tweaked anyway.

At any rate, the Lowe paper is important since it is the main reference on the wikipedia Vapour pressure of water page and since these are the only formulations I've seen that use simple polynomials.

At some point, I want to compare the computation speeds and accuracies of all the formulations I have access to, in a manner similar to Lowe, but using MurphyKoop or HylandWexler as the accuracy standard.


Additional Formulas

Lowe 1976 mentions 3 other formulas I have not seen previously

A number of formulations, including Goff-Gratch, identify 373.16 K as the steam point. However, before 1990 the steam point was actually 373.15 K, after 1990, it became 373.124 K.

The Richards 1971 formula is identical to the one used in Atmospheric Chemistry and Physics by Seinfeld and Pandis with the exception that they used 373.15 K and not 373.16 K.


CRC

The CRC Handbook of Chemistry and Physics (85th ed. 2004) has tables of vapor pressures over ice from 0.01°C to -80°C (page 6-9) and over water from 0°C to 370°C (pages 6-10 & 6-11). (The number of displayed pages is not predictable - if it is hidden, scroll up a few pages and then you can scroll down and see page 6-11.)

There is no formula for finding the values - just tables.

The values in the "over water" table are from the NBS/NRC Steam Tables, 1984 and do not agree with any of the formulas used in my main program. I have not checked the values against those produced by formulas on this page.

The "over ice" values closely track MurphyKoop and HylandWexler from 0.01°C to -40°C and then they begin to diverge and follow Sonntag from -50°C to -80°C.

The tables do not cover "over water" below freezing.


Author: Robert Clemenzi
URL: http:// mc-computing.com / Science_Facts / Water_Vapor / Other_Formulas.html