#! /bin/zsh
#
# this file:
# https://dataswamp.org/~incal/conf/.zsh/air
#
# thanks bildramer at ##math, freenode:
# the simplest way for 2D data is bilinear
# interpolation: given that the next lowest data
# point is (x0, y0) and the next highest is
# (x1, y1) you calculate s = (x - x0)/(x1 - x0) and
# t = (y - y0)/(y1 - y0). then the output is
# (1 - s)((1 - t)f00 + t*f01) + s((1 - t)f10 + t*f11)
#
# Let's model an inflated bicycle tube/tire in the form of
# a torus, as in this figure: [1]
#
# We use the ISO ETRTO notation for bicycle tire sizes.
# For example, a typical 29-er MTB tire is 57-622, where 57 is
# the tire width and 622 the rim BSD (Bead Seat Diameter) in
# mm. See [2]. Let's denote the width w and the BSD b.
#
# Comparing [1] and [2], we understand that
#
# r = w/2
#
# and
#
# R = b/2 + r
#
# R = b/2 + w/2
#
# with the volume of a torus being
#
# V = 2 * pi^2 * R * r^2
#
# V = 2 * pi^2 * (b/2 + w/2) * (w/2)^2
#
# V = pi^2w(b+w^2)/4
#
# Now let's put all that into a function to evermore do
# the math for us:
bike-tire-volume () {
local width=$1
local bsd=$2
local pi=3.1415
local vmm=$(( $pi**2 * $width * ($bsd + $width**2) / 4 ))
local vcm=$(units -t "$vmm mm3" cm3)
local rvcm=$(( rint($vcm) ))
printf "V = %d cm3\n" $rvcm
return $rvcm
}
# Example invocation, our MTB 29-er:
#
# $ bike-tire-volume 57 622
# $ V = 544 cm3
#
# Next step...
#
# We assume that air behaves like an ideal gas at reasonable
# preassure levels with respect to bike tires, and at the
# given temperature. That means we can use the ideal gas law,
# PV = nRT (even tho air isn't an ideal gas, but it's close
# enough for our purposes. [3])
#
# Finally, we change the units into SI and then compute.
#
# SUMMARY
#
# The air, in mols, in a bicycle tire is
#
# n = Ppi^2w(b + w^2)/4RT
#
# where
#
# P is the air pressure (in liters; her inputted in psi)
# w .. the tire width in mm
# b .. the rim BSD (in mm)
# R .. the gas constant, 0.0822 l-atm/mol-K
# T .. the temperature, arbitrarily set to 290.45K (+17.3C) [4]
#
# Now we put everything together...
bike-tire-air () {
local width=$1
local bsd=$2
local psi=$3
bike-tire-volume $width $bsd
local vcm=$?
local p=$(units -t "$psi psi" atm)
local v=$(units -t "$vcm cm3" liters)
local t=290.45
local r=0.0821
local n=$(( ($p * $v) / ($r * $t) ))
printf "%f moles of air gas\n" $n
}
# Here are some interesting results, all using common bicycle
# tire/preassure configurations as indata.
#
# MTB 29-er: 57-622, 65 psi
# $ bike-tire-air 57 622 65
# $ V = 544 cm3
# $ 0.100902 moles of air gas
#
# road bike, tire at maximum psi: 23-622, 145 psi
# $ bike-tire-air 23 622 145
# $ V = 65 cm3
# $ 0.026895 moles of air gas
#
# we compare the road bike to the MTB:
# (/ 0.026895 0.100902) ; 26% of MTB tire air in the road bike tire
#
# road bike, a realistic psi for a ~80 kg rider: 23-622, 115 psi
# $ bike-tire-air 23 622 115
# V = 65 cm3
# 0.021330 moles of air gas
# (/ 0.021330 0.100902) ; 21% v MTB
#
# road bike: is it worth upgrading from 23 mm to 25 mm?
# $ bike-tire-air 23 622 115
# V = 65 cm3
# 0.021330 moles of air gas
# $ bike-tire-air 25 622 100
# V = 77 cm3
# 0.021972 moles of air gas
# (- 100 (* 100 (/ 0.021330 0.021972))) ; only +2.9%! :O
# so two new, expensive tires equals a mere (* 2 2.9) ; +5.8%
#
# some budget tires I have on a fixie
# $ bike-tire-air 28 622 85
# V = 97 cm3
# 0.023528 moles of air gas
# (/ 0.023528 0.100902) ; 23% v. MTB
# Also, that is only a +2% increase compared to the
# 23 mm road tire which is supposedly 5 mm less wide. But this
# may not be shocking tho given the steep drop from 115
# to 85 psi in air preassure.
#
# [1] https://www.mathsisfun.com/geometry/torus.html
#
# [2] https://dataswamp.org/~incal/bike/tire-size.png
#
# [3] In theory, air isn't an ideal gas. To compute
# a meticulous result one would have to include Z, the air
# compressibility factor: PV = nRTZ (for an ideal gas,
# Z is 1 and thus falls from the equation). However, even
# with air, at temperature T = 300 K (+27C) and preassure
# P between 5-10 bar (73-145 psi), Z is between
# 0.9974-0.9987. So for the purpose of this document air
# is practically ideal.
#
#
#
#
# [4] 290.45K or +17.3C is the average temperature in Liège in
# July, the hottest month of the year.
#