Version 4 of Physical empirical formulae

Updated 2006-05-05 12:03:55 by TR

Arjen Markus (4 may 2006) It occurred to me that we do not yet have a decent module in Tcllib yet that collects the various physical empirical formulae. So, right now, if I want to compute the water content of air, I have to hunt the Internet for the right formula. Similarly, the density of water as a function of temperature and salinity.

I am probably not the only one with that little problem, so I propose to collect such formulae here (with a reference) so that later we can simply wrap them up in a package/module for Tcllib.

 # Auxiliary procedure
 proc check {descr var range} {
   foreach {min max} $range break
   if { $var < $min || $var > $max } { 
       return -code error "Range error: $descr should be between $min and $max"
   }
 }

A formula for the humidity content of air:

   check "dew-point temperature" $td {-20 100}
   set vapordens [expr {5.018+0.32321*$td+8.1847e-3*$td*$td+3.1243e-4*$td*$td*$td}]

Symbols:

  • td the dewpoint temperature in degrees C - between -20 and +100 degrees
  • vapordens the water vapour density in g water/m3

Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/relhum.html#c3

TR - Here comes a formula for the density of seawater as a function of salinity, temperature, and pressure:

 proc rho {S T P} {
        #
      # compute the density of seawater in kg/m3
      # as a function of:
      #
      # S -> salinity (in psu, practical salinity units)
        # T > Temperature (in °C, degrees Celsius)
        # P -> pressure (in bar) 
      #
      check Temperature $T {-2 40}
      check Salinity $S {0 40}
      check Pressure $P {0 12000}
        set rhow [expr {999.842594 + 0.06793952 * $T - 0.00909529 
                * pow($T,2) + 0.0001001685 * pow($T,3) - 1.120083e-06 
                * pow($T,4) + 6.536332e-09 * pow($T,5)}]
        set A [expr {0.824493 - 0.0040899 * $T + 7.6438e-05 * pow($T,2)
                - 8.2467e-07 * pow($T,3) + 5.3875e-09 * pow($T,4)}]
        set B [expr {-0.00572466 + 0.00010227 * $T - 1.6546e-06 * pow($T,2)}]
        set C 0.00048314
        set rho0 [expr {$rhow + $A * $S + $B * pow($S,3.0/2) + $C * pow($S,2)}]
        set Ksbmw [expr {19652.21 + 148.4206 * $T - 2.327105 * pow($T,2)
                + 0.01360477 * pow($T,3) - 5.155288e-05 * pow($T,4)}]
        set Ksbm0 [expr {$Ksbmw + $S * (54.6746 - 0.603459 * $T + 0.0109987 * 
                pow($T,2) - 6.167e-05 * pow($T,3)) + pow($S,3.0/2)
                * (0.07944 + 0.016483 * $T - 0.00053009 * pow($T,2))}]
        set Ksbm [expr {$Ksbm0 + $P * (3.239908 + 0.00143713 * $T + 0.000116092
                * pow($T,2) - 5.77905e-07 * pow($T,3)) + $P * $S
                * (0.0022838 - 1.0981e-05 * $T - 1.6078e-06 * pow($T,2))
                + $P * pow($S,3.0/2) * 0.000191075 + $P * $P
                * (8.50935e-05 - 6.12293e-06 * $T + 5.2787e-08 * pow($T,2))
                + pow($P,2) * $S
                * (-9.9348e-07 + 2.0816e-08 * $T + 9.1697e-10 * pow($T,2))}]
        return [expr {$rho0/double(1 - double($P)/$Ksbm)}]

}

References:

Note: I am not totally sure about the ranges of the three parameters, so I added some reasonable defaults

Example: rho 35 10 0 = 1026.95 (quite standard seawater with a salinity of 35 psu, temperature of 10 °C and pressure of 0 bar (surface water))

IDG May 04 2006: All empirical formulae need to have their range of validity stated. Note what happens to the above as td -> infinity :-)

AM Ah! Good thinking! I have added the valid range.

TR - Perhaps we should also add something like accuracy. It is often not good to give a result like 3.24345643 when the formula only has significance for 3.24

[ Category Physics ]