Flattrable Reactive Programming

I have occasionally seen these “flattr this!” buttons, notably on the blog of Luke Palmer, and occasionally felt it was exactly what I wanted to do. However, inertia always got the better of me – no doubt I was busy being distracted by some lolcat screaming for adoration.

Lately I noticed a flattr button on the blog of Heinrich Apfelmus too, and again felt a desire to give. Heinrich’s blog was the tipping point (pun intended) that finally got me to actually visit flattr.com, and this great quote from Alexandros closed the sale:

This month, thanks to Flattr, I’we shared my world to the world. I earned 0.03 cents - the best 0.03 cents ever!

How can that not make me want to give? (Is Alexandros being sarcastic? He looks happy enough in his profile pic…)

My first flattrs go to Luke and Heinrich; 0.03 cents are coming your way! ;)

Alexandros gets flattred too. And now that I’ve signed up I’ll be on the lookout for more flattr buttons (so don’t feel bad if you weren’t mentioned here!).

As an aside, the common denominator that made me want to flattr Luke and Heinrich is their efforts to explore Functional Reactive Programming. It seems to me that much remains to be discovered in the FRP design space and I hope my flattrs encourage these brave explorers to push the boundaries further… Oh, look! Conal Elliott has a flattr button too!

Dimensional example: leaky container pressure

Herein I demonstrate a calculation performed today in my work. The details are slightly different to protect the innocent but the problem is the same. The purpose of all this is to demonstrate use of the dimensional library. What this blog post unfortunately fails to demonstrate is how the compiler helps catch errors in units and formulae, effectively double-checking any derivations for me – this is the real value of dimensional! In case you have never heard of dimensional before here is the blurb from the project site:

Dimensional is a library providing data types for performing arithmetic with physical quantities and units. Information about the physical dimensions of the quantities/units is embedded in their types and the validity of operations is verified by the type checker at compile time. The boxing and unboxing of numerical values as quantities is done by multiplication and division with units. The library is designed to, as far as is practical, enforce/encourage best practices of unit usage.

This is a literate Haskell program. You can copy and paste this blog post into a .lhs file and it should compile and run (provided you have installed dimensional). First the formalities:

> import Numeric.Units.Dimensional.Prelude
> import Numeric.Units.Dimensional.NonSI
> import qualified Prelude

(In the below I have used parentheses rather than juxtaposition to denote function application for functions of time, e.g. p(t). Does this help or hurt clarity given the subject matter?)

Consider a leaky container of volume v filled with gas to pressure p₀ and launched into space at time t₀ (insert favorite monad tutorial joke here). There is a requirement that at some future time t₁ said container must retain no less than a given amount of gas. The temperature T of the container is held constant by a thermal control system so per the ideal gas law (a good enough approximation) we can characterize the requirement to remnant gas by its pressure p_min. The problem at hand was to determine the maximum allowable leak rate of the container at t₀ to ensure that the pressure at time t₁ is no less than p_min.

Here are the inputs (don’t blame me for the choice of units, this is what I was given):

> v     =      2 *~ liter
> t_0   =      0 *~ hour
> p_0   =    750 *~ mmHg  -- roughly atmospheric pressure
> t_1   = 133650 *~ hour  -- about 15.25 years
> p_min =      5 *~ mmHg
> temp  = fromDegreeCelsiusAbsolute 22

For slow leakage it can be assumed that the leakage rate is proportional to the pressure of the gas inside the container (more generally, the pressure difference between the gas in the container and the surrounding medium). Thus the rate of change of the pressure is described by the first order linear differential equation

> dpdt(t) = negate k * p(t)

with solution

> p(t) = c * exp (negate k * t)

The integration constant c is determined from the initial conditions at t₀:

> c = p_0  -- follows from t_0 = 0 s.

We chose the worst case conditions at t₁ to determine k:

> k = log (c / p_min) / t_1

Now, per the ideal gas law the amount of substance n in the container is described by

> n(t) = p(t) * v / (r * temp)

where

> r = 8.314472 *~ (joule / (mole * kelvin))

is the ideal gas constant. Differentiating n(t) give the rate of change of substance in the container:

> dndt(t) = dpdt(t) * v / (r * temp)

and the maximum allowed rate of change (leakage rate) at time t₀ is produced by

> main = print (dndt t_0)  -- -8.486687441280605e-10 s^-1 mol

That’s it!