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Most plumbing engineers are familiar with the storm water flow calculation:
Q = ciA
Q is flow in CFS
c is the coefficient of imperviousness
i is the rainfall intensity in inches/hour
A is the area in acres
The units of this equation have always eluded me, and I have often wondered if this equation was actually the definition of an acre — the amount of land that will produce 1 CFS per inch of rainfall. But, a little research has proven that not to be the case.
Since c is generally taken as 1 for rooftops and since there are 43,560 square feet per acre (a nice round number), this equation is typically rewritten:
Q = i x SF / 43,560
Simplifying this even further, since there are 448 gpm per CFS the equation can be written:
q (gpm) = i x SF / 97.23
That’s a pretty simple equation. As an example, if you have a roof that is 10,000 square feet in area, and a local rainfall rate of 4 inches per hour, the storm flow will be:
q = 4 x 10,000 / 97.23
q = 411 gpm
Getting back to the subject of an acre, I still do not know why the units of this equation work. How does CFS equate to inches per hour times acres? Is it pure chance? Ironically this equation is called the rational equation, but there doesn’t seem to be anything rational about it.
An acre was defined in the Middle Ages as the size of a field that could be ploughed in one day by a yoke of oxen. It was measured as one chain wide (22 yards) and one furlong (10 chains or 220 yards) in length. There is an international acre and a U.S. Survey acre. But, they differ in size by only 25 square inches, or about one quarter of a sheet of paper. So, the difference between the two is insignificant.
Other quirky facts about the acre are:
• It’s about 40 percent of a hectare
• It’s about 90percent of a football field
• It’s 1/640 of a square mile
• It’s exactly 4046.8564224 m2
• It’s 160 perches
• It’s 160 square rods
• It’s 4 roods
• It’s 44,000 square feet less 1 percent
If you quarter a square mile and then quarter each quarter, you end up with a parcel that is ¼ mile on each side and 1/16 of a square mile in area, or 40 acres. In the U.S., farmland was often divided this way, and “the back 40” referred to the 40 acre parcel at the back of the farm.
More on acres. There are other types of acres that include:
The customary acre, which is similar to an international acre but varied in size by location, similar to the variations found in curacates (the amount of land tillable by a team of eight oxen in a ploughing season), virgates (the amount of land tillable by a team of two oxen in a ploughing season), bovates or oxgangs (the amount of land tillable by one ox in a ploughing season), nooks (one quarter of a virgate), and farundels (one quarter of a nook).
A builder’s acre is sometimes used in construction and real estate. It measures 40,000 square feet and is intended to simplify math for marketing. It is approximately 10 percent smaller than a survey acre.
A Scottish acre is one of many obsolete Scottish units of measure, whereas an Irish acre is 7,840 square yards – or nearly two international acres. Meanwhile, a Cheshire acre is 10,240 square yards (more than two acres), and a Roman acre was 1,260 square meters (about 1/3 of an international acre).
Prior to the development of the acre of measure, the Byzantines defined an area of farmland as a zeugarion, which could be ploughed by a pair (zeugos) of oxen. The Ottomans later retained this measure after winning the Byzantine-Ottoman wars.
So, you now know way more than you ever wanted to about the acre, assuming you are still awake. However, the units of the rational equation still elude me. If anyone can enlighten me, please do. Meanwhile, I have an urge to buy an acre of land to see if I can plough it in a day with a yoke of oxen.
Timothy Allinson is a senior professional engineer with Murray Co., Mechanical Contractors, in Long Beach, Calif. He holds a bsme from Tufts University and an mba from New York University. He is a professional engineer licensed in both mechanical and fire protection engineering in various states, and is a leed accredited professional. Allinson is a past-president of aspe, both the New York and Orange County Chapters. He can be reached at email@example.com.