My main vehicle is a 2005 Dodge Ram 2500 with the 5.9 L Cummins diesel.  As of early 2012, diesel fuel is in the $1.25 - $1.30/L range.  I am a chemical engineer.  Time to think about making some of my own fuel.  I will not test the results developmental efforts on my truck as engine replacement costs up to $20,000 have been quoted.  Rather, I will probably source a cheap diesel powered vehicle for testing.

Reference Links

1. Home Made Biodiesel
2. Make Biodiesel
   • Quality Testing
3. Journey to Forever
   • Oil Yields & Characteristics
4. Duda Diesel (equipment & supplies)
5. Iowa State Online Course in Biodiesel

I was thinking about moving along with some alternative energy intiatives here at my home on the hill.  Part of my backyard includes the highest elevation for about a mile in all directions.  Hence, wind power might be a realistic option.

While contemplating the possibilities, or more accurately, potential power generation options, the question came up, "Just how much power does wind contain and, more importantly, how can that quantity be estimated?"

The solution to that question can be worked out from first principles.  Wind contains kinetic energy - remember that a rock sitting on a mountain side has potential energy and the same rock falling off the mountain side has kinetic energy because it is moving.  Same deal with wind - it is moving so it has kinetic energy in proportion to its mass and and its velocity.

So here goes.....

To Begin

Power, which what we actually want to estimate, is kinetic energy divided by time as defined by Eq. 1a:

Now we must think about a 'slice' of wind.  We need to get expressions for mass and velocity so we need a three dimensional control volume so that our slice has mass (two dimensional things have no mass).  Also, because our control volume will be moving, it will also have a time (t) component, which we suspect we will need in order to get an expression for velocity.  What should the dimensions of our control volume be?  How about the same diameter (d) as a turbine we might use to capture some power from the wind and some height (h) - yes, a cylinder with a horizontal axis still has height, which is the length of the horizontal axis.

The velocity term in Eq. 1 is easy - it is the wind speed.  Our control volume will move with the same velocity as the wind, right?

Let's derive an expression for the mass term in Eq. 1.  We want mass and, let us see..., our control volume has volume and mass density is defined as mass divided by volume.  Can we calculate, or lookup, the density of air?  Of course we can!  Recall the Ideal Gas Law, PV=nRT?  To simplify things a bit, we can look up a dry air density given temperature and pressure, here for example.  A cubic metre of dry air at a temperature of 20^oC and 101.325 kPa has mass of 1.2204 kg.

Mass density is defined by Eq. 2:

The volume, V, we are concerned with is that of a cylinder so we can modify Eq.2:

We will develop two expressions for mass, m.  One for constant density, i.e., constant temperature and pressure, and one for variable temperature and pressure.  First, mass at constant temperature and pressure from Eq. 2a:

Wind Power Density at Constant Temperature and Pressure

So, at constant temperature and pressure... Note that we did not get an expression for time, t, the time it takes our control volume to traverse the turbine - it is trivial because velocity, which would be a known variable, is distance divided by time so that time is distance divided by velocity:

Power is the product of a constant, the dry air density, the square of the turbine diameter and the cube of the wind velocity.  REMEMBER that this equation estimates the total amount of power in wind - not the amount of power the turbine will be able to extract.  If all of the power was extracted from the wind, the wind would not move at the discharge side of the turbine, which would mean no air could traverse the turbine and, obviously, the turbine would sit stationary.  In the real world, the maximum efficiency of a wind turbine is considered to be about 59%.  The actual efficiency of any given turbine depends on the turbine design.

Wind Power Density at Given Temperature and Pressure

Now let us consider an equation to estimate power where air pressure and temperature can vary.  The Ideal Gas Law definition and some algebraic manipulations:

 Eq. 3 can be rearranged, and, with numerical values subsituted for m1, P1 and T1, we get an expression for mass that provides for variable air temperature and/or pressure according to Eq. 3a:

While all of the above ends up looking a bit ugly, it is not as bad as it seems.  Let us put together some equations to estimate the amount of power being carried by wind.  We will consider the case of constant air pressure and temperature first.  Before proceeding, note that the control volume, i.e., 'slice' of air, height will be taken as unity, i.e., 1 m. 

 

Example Calculations

Let us try out our equations for estimating power.  We will do a little checking here by taking the air density as 1.225 kg/m³ in the first equation and P=101.325 kPa and T=288.15 K in the second.  Because these values all correspond to the same set of conditions, we should get the same result in both cases.  The turbine diameter will be taken as unity to simplify the calculations.  Assume the wind velocity is 10 m/s.

Wow!  Not bad, eh?  Notice though that we assumed a one metre diameter turbine which, although not large in terms of a backyard project, neither is it particularly small.  Yet, assuming a turbine efficiency of 50%, only about 240 watts, approximately the power required to operate four 60 watt light bulbs , would be generated.

ABC music notation is a very convenient way to create and store music.  There are several apps freely available but none seemed to do things the way I wanted to so I created my own.  It is written in Pascal using Lazarus.  For the most part, my app simply provides a GUI wrapper around several command line utilities such as ABCM2PS, ABC2MIDI and MIDI2ABC.  It includes:

• a text editor to enter ABC notation,
• a flat file database for music storage (several databases can be used if desired in order, for example, to store music according to genre),
• faculties to create output in postscript (PS), portable document (PDF) and JPEG formats,
• will generate a MIDI file on demand and play it back,
• will output a user specified selection of tunes as a tune book,
• can import to the currently selected database ABC files containing one or more tunes, and,
• can do limited MIDI to ABC conversion.

The app works on Debian Linux and probably other Linux flavours.  It works in a limited way on Windows XP - windows is not nearly as accommodating as Linux when it comes to building applications that incorporate existing applications.

My ABC app is a bit clunky but serves my purposes quite well.  If it might be useful to you, drop me a note using the Contact form. 

I composed a jig for my grand son using my ABC app,  The JPEG output is shown below.