Numeric Equations

Numeric equations require the quantities related to have specific units. These equations include an aggregate unit conversion factor whose units usually cancel. To use one of these equations in Calchemy you may be able to just "leave out" the conversion factor and let Calchemy do the conversions for you. However, often the mysterious factor will include an "efficiency" or a geometry like pi, or in some cases even constants of dimension.

As an example, suppose you find the following equation relating the power output of a windmill to its blade radius and the windspeed:

P = 1.19E-3 * R^2 * V^3

in which:
P = Power out in watts
R = Radius of the widmill blade in feet
V = Wind Speed in miles per hour

In this case you can be sure this is not a quantity equation because the units on both sides of the equal sign don't match! This indicates that the mysterious factor actually has real units. Physically the quantity "left out" is the density of air, which you might discover using Calchemy:

1.1e-3 * (4 ft)^2 * (20 mph)^3 ? watt
Mismatch: watt
You can use Calchemy to solve for the mysterious factor, and then use the factor in Calchemy to solve the equation with parameters having different units than the original numeric equation was "designed" for. In general this procedure involves entering a known solution to get the real factor.

In the windmill example we know our missing factor has units of Density, and that a 4 ft radius blade in a 20 mph wind produces 140 watts of power (by plugging just numbers into the original numeric equation), so...

(4 ft)^2 * (20 mph)^3 ; 140 watt ? g/cc
= 0.00013178 g/cc
Now we can use this factor with includes efficiency, pi and who knows what to solve the equation with parameters having any units. First lets just check our original solution...
0.00013178 g/cc ; (4 ft)^2 ; (20 mph)^3 ? watt
= 140 watt
And now for something completely different...
0.00013178 g/cc ; (0.667 fathoms)^2 ; (53760 furlongs/fortnight)^3 ? watt
= 140.14 watt

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Last Updated 6/21/2009
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