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.
Note:
As kbk pointed out, some or all of these formulae will have to be rearranged so that they can be evaluated in a fast and numerically stable way. For now, we are just gathering them in one place.
# Auxiliary procedures:
#
# Check the range of a parameter
#
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"
}
}
# Efficiently evaluate a polynomial -
# coefs is a list with the highest power first
proc poleval {coefs x} {
set y 0
foreach c $coefs {
set y [expr {$y * $x + $c}]
}
}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:
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 {Salinity Temperature Pressure} {
#
# compute the density of seawater in kg/m3 as a function of:
#
# Salinity -> salinity (in psu, practical salinity units)
# Temperature -> temperature (in ℃, degrees Celsius)
# Pressure -> pressure (in bar)
#
check Temperature $Temperature {-2 40}
check Salinity $Salinity {0 40}
check Pressure $Pressure {0 12000}
set rhow [expr {
999.842594
+ 0.06793952 * $Temperature
- 0.00909529 * pow($Temperature,2)
+ 0.0001001685 * pow($Temperature,3)
- 1.120083e-06 * pow($Temperature,4)
+ 6.536332e-09 * pow($Temperature,5)
}]
set A [expr {
0.824493
- 0.0040899 * $Temperature
+ 7.6438e-05 * pow($Temperature,2)
- 8.2467e-07 * pow($Temperature,3)
+ 5.3875e-09 * pow($Temperature,4)
}]
set B [expr {
-0.00572466
+ 0.00010227 * $Temperature
- 1.6546e-06 * pow($Temperature,2)
}]
set C 0.00048314
set rho0 [expr {
$rhow
+ $A * $Salinity
+ $B * pow($Salinity, 3.0/2)
+ $C * pow($Salinity, 2)
}]
set Ksbmw [expr {
19652.21
+ 148.4206 * $Temperature
- 2.327105 * pow($Temperature,2)
+ 0.01360477 * pow($Temperature,3)
- 5.155288e-05 * pow($Temperature,4)
}]
set Ksbm0 [expr {
$Ksbmw
+ $Salinity * (
54.6746
- 0.603459 * $Temperature
+ 0.0109987 * pow($Temperature,2)
- 6.167e-05 * pow($Temperature,3))
+ pow($Salinity,3.0/2) * (
0.07944
+ 0.016483 * $Temperature
- 0.00053009 * pow($Temperature,2))
}]
set Ksbm [expr {
$Ksbm0
+ $Pressure * (
3.239908
+ 0.00143713 * $Temperature
+ 0.000116092 * pow($Temperature,2)
- 5.77905e-07 * pow($Temperature,3))
+ $Pressure * $Salinity * (
0.0022838
- 1.0981e-05 * $Temperature
- 1.6078e-06 * pow($Temperature,2))
+ $Pressure * pow($Salinity,3.0/2) * 0.000191075
+ $Pressure * $Pressure * (
8.50935e-05
- 6.12293e-06 * $Temperature
+ 5.2787e-08 * pow($Temperature,2))
+ pow($Pressure,2) * $Salinity * (
-9.9348e-07
+ 2.0816e-08 * $Temperature
+ 9.1697e-10 * pow($Temperature,2))
}]
return [expr {$rho0/double(1 - double($Pressure)/$Ksbm)}]
}or using poleval
proc rho {Salinity Temperature Pressure} {
set rhow [poleval {+6.536332e-09 -1.120083e-06 +0.0001001685 -0.00909529 +0.06793952 999.842594} $Temperature]
set A [poleval {+5.3875e-09 -8.2467e-07 +7.6438e-05 -0.0040899 0.824493} $Temperature]
set B [poleval {-1.6546e-06 +0.00010227 -0.00572466} $Temperature]
set C 0.00048314
set rho0 [expr {$rhow + $A * $Salinity + $B * pow($Salinity,3.0/2) + $C * pow($Salinity,2)}]
set Ksbmw [poleval {-5.155288e-05 +0.01360477 -2.327105 +148.4206 +19652.21} $Temperature]
set Ksbm0 [expr {
$Ksbmw
+ $Salinity * (
54.6746
- 0.603459 * $Temperature
+ 0.0109987 * pow($Temperature,2)
- 6.167e-05 * pow($Temperature,3))
+ pow($Salinity,3.0/2) * (
0.07944
+ 0.016483 * $Temperature
- 0.00053009 * pow($Temperature,2))
}]
set Ksbm [expr {
$Ksbm0
+ $Pressure * (
3.239908
+ 0.00143713 * $Temperature
+ 0.000116092 * pow($Temperature,2)
- 5.77905e-07 * pow($Temperature,3))
+ $Pressure * $Salinity * (
0.0022838
- 1.0981e-05 * $Temperature
- 1.6078e-06 * pow($Temperature,2))
+ $Pressure * pow($Salinity,3.0/2) * 0.000191075
+ $Pressure * $Pressure * (
8.50935e-05
- 6.12293e-06 * $Temperature
+ 5.2787e-08 * pow($Temperature,2))
+ pow($Pressure,2) * $Salinity * (
-9.9348e-07
+ 2.0816e-08 * $Temperature
+ 9.1697e-10 * pow($Temperature,2))
}]
return [expr {$rho0/double(1 - double($Pressure)/$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 ℃ 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. And: we should stick to SI units, so one result can be fed into the next formula without conversion.
Sarnold - Add something like accuracy? Take a look at math::bigfloat if you need to manage accuracy across computations. I must admit it may obfuscate the source code for such a package. (When I designed it, I remembered my physicist background...)
As a small contribution to this project, I was thinking that it could be useful to have a "formula" proc that created a proc that applied the formula, converting units if necessary. Here's my whack at a "formula" proc:
package require units
# Author: Eric Kemp-Benedict, 2007
# Use it as you like!
# Usage:
# formula nameOfFormula "Description" formula units V1 U1 V2 U2 ...
proc formula {name args} {
set fargs $args
set procbody1 "
set name $name
set descr {[lindex $fargs 0]}
set formula {[lindex $fargs 1]}
set funits {[lindex $fargs 2]}
foreach {v u} [list [lrange $fargs 3 end]] {
set units(\$v) \$u
}"
set procbody2 {
if {$args eq "info"} {
set infostr "$descr\n\nVariables:\n"
foreach v [lsort [array names units]] {
set infostr "$infostr $v ($units($v))\n"
}
return $infostr
}
foreach {v = val} $args {
if {[lsearch [array names units] $v] == -1} {
error "$name: Variable $v is not valid; Use \"$name info\" for more information"
return 0.0
}
set $v [::units::convert $val $units($v)]
}
return "[expr $formula] $funits"
}
set procbody "$procbody1\n$procbody2"
proc $name args $procbody
}
##
## Example
##
formula CharlesVol "Apply Charles' Law to calculate volume" \
{1.0 * $V1 * $T2/$T1} L \
V1 L T1 K T2 KOnce the example is created, you can do:
(Formula) 97 % CharlesVol info Apply Charles' Law to calculate volume Variables: T1 (K) T2 (K) V1 (L)
to get a summary of the formula, and something like
(Formula) 98 % CharlesVol V1 = 3.2m^3 T1 = 300K T2 = 298K 3178.66666667 L
to apply it.
AM (5 july 2007) Now, that looks very nice, especially the "info" subcommand!