AIR.HTM --- Part of Manual for Driver Parameter Calculator --- by Claus Futtrup.
Created 9. September 1996, last revised 16. February 2004. Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

1. Properties of Air
2. Air Density
3. Speed of Sound
4. Sample Calculations for rho and c
• 0 degC
• 20 degC
• 40 degC
• Summary
6. Acknowledgments
7. Litterature

## Properties of Air

As can be seen from MEASURE.HTM the loudspeaker data found by measurements + calculations depends on two factors:

```        rho   : air density

c     : speed of sound in air
```

Since these factors depends on air pressure, air temperature and more, and are so essential to the accuracy of Vas versus Cms, we will treat them here.

Notice that this document concludes that humidity level does not have an influence on Vas versus Cms because changes in rho and c are balanced out.

## Air Density

The air density is defined (as any other density) as "mass per volume," ie. rho = m/V. This expression can be isolated from the gas equation, which assumes that the gas behaves like an ideal gas:

```        pV = nRT                                                     (1)
```

which should be familiar to the reader. Since n = m/M, where M is the molar mass of the molecules in air we get:

```             mRT                             pM
pV = ---      <===>     rho = m/V = ----                     (2)
M                              RT

p is the gas pressure, in Pa (Pascal)
M is the molar mass of air, in kg/mol or alternatively g/mol
T is the gas temperature, in K (Kelvin)
R is the gas constant, R = 8.314510 J/(mol K) or alternatively Nm/(mol K)
m is the mass of the gas
n is the number of molecules, measured in mol
V is the volume of the gas
rho is the mass density of the gas
```

You can measure the temperature on a thermometer and convert to Kelvin:

```        Celcius to Kelvin    : K = degC + 273.15
Fahrenheit to Kelvin : K = degF*5/9 + 255.37
Reamur to Kelvin     : K = 1.25*degR + 273.15
Rankine to Kelvin    : K = R/1.8 = R*5/9
```

Note that Rankine is not specified on a degree scale, which is because it is measuring absolute temperature similarly to Kelvin (which is not a degree scale either).

Note that in some books the shifting value is 273.16 and not 273.15. I have concluded that the variations are due to different temperature scales. I have chosen the 273.15 value.

The Reamur degree-scale is not supported by Driver Parameter Calculator. Quite frankly I do not think anybody is using it anymore. Give me a hint if this is not the case and I will support it.

Calculations at room temperature, as examples:

```         20 degC = 293.15 K
70 degF = 294.25 K
16 degR = 293.15 K
530 R = 294.44 K
```

Similarly you can measure the pressure on a barometer (and convert to Pascal). Here the following relations are valid:

```        101325 Pa = 101.325 kPa = 1.01325 hPa
= 1.01325 bar = 1013.25 mbar
= 1 atm
= 14.69595 psi
= 760 torr = 760 mm Hg = 0.76 m Hg = 29.92126 in Hg
= 407.189 mm H2O
```

101325 Pa -- the standard pressure, in Pascal. Then comes kilo-Pascal and hecto-Pascal (which is similar to bar). The standard pressure is also 1 atmosphere. You can also convert from pounds per square-inch or from torricelli or height of a bar of mercury = quicksilver (meter Hg), or height of a similar bar of water (millimeter H20). You most likely can find other measures for pressure, like the gravitational ones.

To convert from another unit to Pascal you find the appropriate figure on the right side and divide the Pascal figure by that number. You then read the current pressure and multiply with that figure.

Things gets a little harder when it comes to the molar mass because you need to know the distribution of gases in the air. For dry air it approximately is (you can calculate M from a periodic table):

```gas   |   M   |  %  |
------|-------|-----|
N_2   | 28.01 | 78  |   Nitrogen        (M = 28.0134, 78.084% in air)
O_2   | 32.00 | 21  |   Oxygen          (M = 31.9988, 20.947% in air)
Ar    | 39.95 |  1  |   Argon           (M = 39.948, 0.9300% in air)
```

This makes up about 100% (and M becomes 28.9673 g/mol), but actually there are more gases in the air, they just do not have any considerable influence, the content of these gases should be measured in ppm (parts per million), not percent:

```gas  |     M   |
-----|---------|
CO_2 |  44.010 | typically around 0.0340% = 340 ppm     (M = 44.0098)
Ne   |  20.179 | typically around 0.0018% = 18 ppm
He   |   4.003 | typically around 0.00052% = 5.2 ppm
Kr   |  83.80  | typically around 0.000114% = 1.14 ppm
H_2  |   2.016 | typically around 0.00005% = 0.5 ppm
Xe   | 131.29  | typically around 0.0000086 = 0.086 ppm
O_3  |  47.997 | typically around 0.000004% = 0.04 ppm (varies)
H2O  |  18.020 |
SO_2 |  64.065 |
CH_4 |  16.043 |
NO   |  30.006 |
NO_2 |  46.006 |
NH_3 |  17.031 |
CO   |  28.010 |
```

Some of the gases are reactive and are therefore not stable constituents of air. Others shift toward lower/higher values, eg. the CO_2 level is continuously increasing. The last seen estimate for CO_2 was at 370 ppm, which is the highest level in 420.000 years, and at the end of the 1880'ies the level was measured/estimated at 280 ppm. In the year 2050 it is expected to be between 450 and 650 ppm.

To find a value for M that everybody can agree about has proven to be quite difficult. We are talking average values measured over time and at different places. My book (2. reference) says 28.970, which does not match the gas distribution from the same book (see tables above). Other sources agrees that M is as low as 28.964. The difference between the highest and the lowest value is only 0.02% anyway.

If we include the other stable gasses and take advantage of the higher precision the more accurate calculation of the molar mass becomes : M = 28.96375 for 99.998% of the air - but we want 100% so by simple scaling we totally get M = 28.96375 / 0.99998 = 28.96433. The initial data does not support this many digits, so from now on the last 2 digits will be cut off and M = 28.965 +/- 0.002 is assumed to be a reasonable value for the molar mass of air as an average value.

Though the mixture of air may change for different places on earth, I think we will be surprised how little influence it has on the molar mass. Whether your are in asia, in the jungle or on the ocean. Even if you live in a town and the amount of CO_2 could be much higher when compared to, say the Himalaya Mountains, then CO_2 is only found in small concentations. The typical values of gas concentrations versus location does not vary much because of the mixing of air in the troposphere (from ground level up to a height of 12 km). You would have to live close to a significant source of CO_2 gases to see any real and permanent change of the average molar mass.

Local variations in the mixture of gases can be large but they are normally quickly distributed and vary from day to day. For meteorological studies only the 3 most significant gases are used, as far as I know. In a room with a lot of welding the inert gas Argon may be found in slightly higher concentrations too.

I believe the above figure for M is as precise as it gets for dry air. For many applications it will suffice to set M = 29 g/mol for the air, whether humid or not.

Something which will have an influence is water vapour from the humidity. But at low temperatures the influence is low, since cold air cannot carry much humidity before the relative humidity is close to 100% and then the influence is low anyway.

With the above in mind, the approximation for rho and c are usually considered sufficient for a good precision. Below we will include humidity in the calculations and add further to the precision of measuring/calculating driver data.

To measure the relative humidity you have several opportunities, of which the easiest is to use a hygrometer (you can buy one wherever you buy barometers). Hygrometers will display the relative humidity. Nowadays electronic hygrometers are inexpensive.

Alternatively you can use a thermometer, you read the temperature T_dry. Then moisturize a sock and put it around the thermometer. Then ventilate heavily (with a fan) on the system and reread the temperature T_wet (this method will not give you accurate results). Calculate the drop in temperature (T_drop = T_dry - T_wet) and use the following approximative equation (correct within 1% from 2 degC to 36 degC) to calculate the relative humidity:

```  phi = 100 - (445/(T_dry + 23) - 1.5) * (T_drop - (T_drop^2)/80)    (3)
```

T_dry and T_drop are in degrees Celcius, and phi is the relative humidity in percent.

Since the molar mass of dry air weights more than water, the air actually gets lighter, as it becomes more humid. This is easy to understand when the pressure is assumed to be the same (you measure it), and therefore the total amount of molecules will be the same, only each molecules weighs something different---and since the light water molecules substitute heavier dry-air molecules, totally the air becomes lighter.

From thermodynamics we have the following equation:

```               (ML - MH)*phi*Pm
rho = rho_0 - ------------------                                     (4)
RL * T

rho_0 is the density of dry air
MH = 18.020 is the molar mass for water
ML = 28.965 is the molar mass for dry air (see above)
RL = R = 8.314510, R is the gas constant 8.314510 J/(mol K)
T is the temperature
phi is the relative humidity
Pm is the partial pressure for water vapour at similar temperature, but
at 100% humidity, given in the table below
```

RL only counts for dry air, but it is usually a good figure for humid air if the water vapour is in thermal balance with the dry air. Besides being in the air as steam, H2O can also be in the air in fluid or solid form because the critical temperature for H2O is higher than for the gases of dry air.

Pm can be calculated from the ideal gas equation (though air is not an ideal gas, the results should be valid in the temperature range where you will be measuring, say 5 degC to 60 degC = 41 degF to 140 degF), it is tabulated below (the partial pressure is given in Pascal):

``` T  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9  |
----|-----------------------------------------------------------|
0 |  611|  657|  705|  757|  813|  872|  935| 1001| 1072| 1147|
10 | 1227| 1312| 1401| 1497| 1597| 1704| 1817| 1936| 2062| 2196|
20 | 2337| 2485| 2642| 2808| 2982| 3166| 3360| 3564| 3778| 4004|
30 | 4242| 4491| 4753| 5029| 5318| 5622| 5940| 6274| 6624| 6991|
40 | 7370| 7780| 8200| 8640| 9100| 9580|10090|10610|11160|11740|
50 |12330|12960|13610|14290|15000|15740|16510|17310|18140|19010|
```

For eg. car-fi this table may not cover a sufficient range of temperatures. For this application an approximative (curve-fitted) model should be applied instead. Such an advanced model is implemented into DPC, see AIRMODEL.HTM for further information.

From equation (2) and (4) we can make a single equation for rho:

```       p * ML - (ML - MH)*phi*Pm
rho = ---------------------------                                    (5)
R * T
```

This equation (5) is the important one for our future calculations.

## Speed of Sound

Speed of sound can be calculated as:

```        c = sqrt(gamma*p/rho) = sqrt(gamma*RT/M)                     (6)
```

Since R, T and M have been discussed above the latter mentioned equation (6) at a glance looks better, but the first equation (6) can be used as well. Specifying the air density (rho) we just calculated is actually the easiest way to deal with humid air.

for each gas in the air, gamma is given as:

```                 cp     cv + R
gamma = ---- = --------                                      (7)
cv       cv
```

where cp is the heat capacity at constant pressure, and cv is the heat capacity at constant volume. This factor is basically given by the structure of the molecule (number of atoms + how well it fits the ideal gas model). From kinetic gas theory cv = f/2 * R, where f is the number of freedoms. For triatomic gases the freedom of the third atom is given by the molecule structure:

```        For monoatomic gas : cv = 3/2 * R

For diatomic gas : cv = 5/2 * R
```

This gives for ideal monoatomic gas : gamma = (cv + R) / cv = 5 / 3 = 1.667 For an ideal diatomic gas we get : gamma = 7 / 5 = 1.4

Here is gamma for some gasses, at 20 degC (cv is dependent on temperature):

```gas   | gamma |
------|-------|
N_2   |  1.40 |
O_2   |  1.40 |
Ar    |  1.65 | (for an ideal monoatomic gas, gamma would equal 1.667)
CO_2  |  1.29 |
CO    |  1.40 |
He    |  1.63 |
SO_2  |  1.27 |
CH_4  |  1.31 |
NO    |  1.39 |
NO_2  |  1.17 |
NH_3  |  1.31 |
H_2   |  1.40 |
H_2O  |  1.30 |
```

For dry air gamma = 1.40 is a good estimate, the temperature may even differ from 20 degC, the assumption is valid within 0.3% from about -10 degC to about 50 degC (from 14 degF to 122 degF). I do not know who would measure (or play music on hi-fi loudspeakers) at temperatures outside this range. I assume that the influence on humid air is relatively small too (because c is calculated as a square-root), but having measured it as described above you may as well include it here too, simply use rho instead of rho_0.

If rho changes, then M changes similarly, according to the ideal gas equation (2) : P * M = rho * R * T. Here P, R and T are held constant. If M is changed because it now consist of a new mixture of gases we get, say the air is now 2% water vapour (H20) and 98% dry air:

```        Mnew = 0.98 * 28.965 + 0.02 * 18.020 = 28.7461 g/mol         (8)
```

Then rho will change proportionally:

```         rho       Mnew
------- = ------                                             (9)
rho_0      M
```

Remixing gamma as well you get that gamma changes from 1.40 to 1.398, which is a change of about 0.14%, or basically no changes at all, and outside the precision range for the tabulated gamma values. My second reference gives cp and cv explicitly for the air, from which I get gamma = 1.401 at 0 degC (32 degF) and standard pressure 101325 Pa. M has changed 0.76%. The changes work in opposite directions and we get that the expression of c changes 0.31%, which in my opinion may be neglected.

Note, though that c usually is squared in the Thiele-Small parameter for Cms, whereby the error in the transformation from Vas to Cms diminishes. This means that when calculating new values for air density and sound speed it is sufficient to base it on temperature and pressure because including humidity will not increase the precision of Cms. This can be seen from the first equation for the speed of sound (6), where c is inverse proportional to the square-root of rho.

The sound speed c is used in several other equations, though, like in the calculations of the efficiency of the driver, where no is proportional to 1 / c^3. A change of 0.31% in c will give you 1% error in the reference efficiency, no.

## Sample Calculations for rho and c

The following temperatures will be calculated, always assuming standard pressure p = 101325 Pa, for dry air the molar mass is assumed to be ML = 0.028965 kg/mol and for water vapour MH = 0.018020 kg/mol:

• 0 degC (32 degF)
• 20 degC (68 degF)
• 40 degC (104 degF)

at 0 and 100% relative humidity (phi = 0 and phi = 1).

### 0 degC

```T   = 273.15 K measured on a thermometer
Pm  = 611 Pa from the above table
```

rho_0 = 1.2923 kg/m3

Substituting 0 with 1 in the above equation (5), for 100% humidity, (we have precalculated ML - MH once and for all) we get:

```       p * ML - (ML - MH)*Pm*phi
rho = ---------------------------
R * T

101325 * 0.028965 - 0.010945 * 611 * 1
= --------------------------------------                        (10)
8.314510 * 273.15

= 1.2893 kg/m3
```

The speed of sound for the desert (dry air) and for the rain forrest (humid air) becomes:

```   c0 = sqrt(1.40*101325/1.2923) = 331.32 m/s      (dry)
c  = sqrt(1.40*101325/1.2893) = 331.70 m/s      (humid)
```

At 0 degC the changes are really insignificant.

Furthermore, please note that the total change in rho*c^2 = 0% and that expression is what is actually used in the Thiele/Small equations for suspension compliance of loudspeakers. The total change in c^3 = 0.3%.

### 20 degC

```T = 293.15 K measured on a thermometer
Pm = 2337 Pa from the above table
```

Insertion in the equations gives:

``` * rho_0 = 1.2041       * c0 = 343.23
* rho   = 1.1936       * c  = 344.74
```

The difference on rho are now up to 0.9% and should be taken into considerations. The difference on c is .4% in the opposite direction so that if you calculate rho*c^2 the total error will be zero. If you calculate c^3 the total error becomes 1.3%.

### 40 degC

```T = 313.15 K measured on a thermometer
Pm = 7370 Pa from the above table
```

Insertion in the equations gives:

``` * rho_0 = 1.1273       * c0 = 354.74
* rho   = 1.0963       * c  = 359.71
```

The difference on rho is now 2.8% and as we can see the humidity becomes more important for higher temperatures. The change in c^3 = 4%.

### Summary

Considering humidity will in other words change the numbers for rho and c slightly, but totally it has no importance when calculating the loudspeaker suspension compliance, Cms, from its volume equivalent, Vas. Driver Parameter Calculator will continue to support humidity because you may find the calculations of c or rho useful individually in other areas where they are not treatet together, like if you want to know the time-distance in a measuring setup from a loudspeaker driver to a microphone.

By the way. If you know the relative humidity out-door, and you know the in-door and out-door temperatures, then you can calculate a new relative humidity. We need:

```        Out-door temperature, T_out
In-door temperature, T_in
The relative humidity outdoor, H_out
```

Say the relative humidity outdoor is 90%, and the temperature is 9 degC (48.2 degF), but your indoor temperature is 20 degC (68 degF), then from the table of Pm we find Pm_out = 1147 and Pm_in = 2337.

```                       Pm_out
H_in = H_out * ------ = 90 * 1147/2337 = 44%                (11)
Pm_in
```

This is a linear equation, which works in the same temperature range as given for the table of partial pressures.

This way you suddenly get a completely new figure for the relative humidity. The partial pressure is kept unchanged, and since the temperature is new you will get a different air-density.

Assuming a constant Gamma is a basis for small errors, perhaps up to 0.3% and assuming that dry air as well as water vapour is an ideal gas at normal temperatures, which is the basis for the table of partial pressures, may provide a basis for similar errors.

What is the weather around the globe? (for designers of speakers, that are exported, this might be of interest). What is needed is the average temperature and pressure at particular locations, and how are they distributed (mean-value + variance?). Perhaps the averages could be weighted in relation to the number of people (who have money to buy speakers) living in the areas. The following internet URL may provide some answers: http://www.ncdc.noaa.gov/

Outdoor data is one issue, but do not forget, that all figures should be modified to in-door figures. Living in hot and humid areas (India?), you must take air-conditioning into considerations as well.

In Denmark, airconditioning is not normal and the temperature is normally approx. 20 degC, with humidity around 40% most of the year being a good approximation. Humidity out-door can be much higher.

Following the danish organisation BST, an organisation for health on work places, it is recommended to keep the relative humidity at 35-65 % for comfortable levels. High relative humidity can increase the number of sick days, and normally the relative humidity is kept at the low end.

On a measuring instrument at Dynaudio, which displays temperature, pressure and relative humidity, the instrument says DRY until you reach 40% relative humidity, then the instrument says COMFORT. The instrument costs more around 100 USD. In a given room situation the relative humidity could be almost 100% (or at least that is how it feels) but the instrument would show max. 45% RH. The reason was simply wrong placement of the instrument. Take care. If the instrument had been able to go above 70% RH then it would probably specify the environment as WET.

A value of 40% RH is probably a good approximation. If 30% RH is chosen, which is the standard setup in DPC, then the values for c and rho are very close to the standard values used in various books.

Loudspeaker and Headphone Handbook, ed. John Borwick (ISBN 0 240 51371 1, 2. edition, p. 439) suggests that "measurings should therefore be carried out at normal room temperature, say 25 +- 5 degrees C". I guess this is not necessary anymore, but still---a loudspeaker driver will change behaviour depending on temperature, and these changes is inherent in the materials. Therefore you should pick a day with a reasonably normal temperature.

The model of air with a table of partial pressures has been exchanged for an approximative (curve-fitted) model, because such a model is slightly more accurate and useful for a wider range of temperatures. Similarly an approximative model for gamma could be made available within Driver Parameter Calculator, but has not been reworked yet.

## Acknowledgments

Since the creation of this document in September 1996 to the beginning of January 2000 I did not know whether this document was right or wrong regarding to humidity. I finally succeeded to have someone look at the content, and several people replied with suggestions for improvement:

Timo Christ, an EE student from University of Bremen.

Ron Ennenga, a ME student from University of Minnesota.

Ray Hopkins from Cloud and Aerosol Sciences Lab, University of Missouri-Rolla.

Claus Petersen from the Research & Development department of the Danish Meteorological Institute, DMI, see http://www.dmi.dk.

## Litterature

This document is based on the following litterature:

```     1. Maskinteknisk Termodynamik, haefte 6
Andreas Andersen & Svend Ishoy Rasmussen
DIAM nr. 571, Juli 1988
```

comment: I have "slaughtered" several bugs in this book.

```     2. DATABOG fysik kemi
Erik Strandgaard Andersen,
Paul Jespersgaard
Ove Gronbaek Ostergaard
9. edition
F & K forlaget
ISBN 87-87229-57-9
```

comment: used for the table of Pm and the ingenious method for measuring relative humidity with a wet sock. This method is actually the "professional" method applied in eg. climate chambers for control of humidity in the chamber.