Logical Math #5: Areas and Volumes of Basins, Pipelines, Sludge Application Sites
by Donald Proctor
Many calculations that are necessary in our vocation involve either areas or volumes (or both) of treatment basins, pipelines, sludge drying beds, land-application sites, buildings, or other facilities. For example, calculating detention time in a chlorine contact basin requires, first, knowledge about the volume of liquid contained in the basin when in operation. This may require that we obtain all of the appropriate dimensions and calculate the volume. Even if the basin volume is listed in the operation and maintenance manual, it would be nice to get those dimensions and calculate that volume yourself. Mistakes have sometimes been made and recorded, and it is a chance to practice your math skills.
Mutual Perpendicularity
I don’t think the term mutual perpendicularity actually exists except in Proctor’s mind — but it ought to! In every calculation of an area, you will always have to multiply two linear measurements, and those two measurements must always be mutually perpendicular (that is, at right angles to each other, forming a 90° angle). In every calculation of a volume, you will always need to multiply three linear measurements, and each measurement must be mutually perpendicular to each of the other two measurements.
Even in cases of obtaining the area of a circle, where you end up multiplying radius × radius or radius squared, for any selected radius there is another radius of exactly the same length that is perpendicular to the first one. And when you multiply any two mutually perpendicular dimensions (for area calculations), remember the good habit of using both the numerical value and the unit of measure in all calculations — as in 2.5 ft × 2.2 ft = 5.5 ft2.
To start, let’s consider the formulas for areas of common shapes. The most common shape for which you may need to calculate area is the rectangle, for which
Area = length × width.
Recall that this formula is only a very logical expression of common sense, as was shown for the picnic table problem in installment #1.
| Also, note that a square is only a special case of a rectangle in which both length and width are equal. In both cases, the length and width must be mutually perpendicular. If the two ends and two bases are parallel but not mutually perpendicular, the figure or shape is called a parallelogram. To determine the area of a parallelogram, you must somehow determine the perpendicular distance that separates two parallel sides (shown as "H" in Figure 1). Fortunately, we don’t encounter the parallelogram figure very often in our vocation. |
Figure 1.
 |
If two sides of a four-sided figure are parallel but not equal in length, and the other two sides are not parallel, the figure is called a trapezoid. Use the two parallel side dimensions as "bases" in the formula:
Area = 1/2 (basea + baseb) × height.
Note that the value [1/2 (basea + baseb)] is the same as the average base length, as if we were determining the area of a rectangle.
We encounter trapezoids as the sloping walls of basins, as cross sections of such basins, as pieces of chemical hopper walls, and perhaps as land areas where sludge might be applied or a grass plot might need to be mowed.
Next, let’s consider a triangle. The formula for the area of a triangle is
Area = 1/2 base × height.
Two triangles are shown in Figure 2 just to indicate that a trapezoid is, in reality, simply two triangles.
Therefore, it is logical to expect that the trapezoid area formula amounts to the sum of two triangular area formulas. That might indicate that using logic can reduce the memorizing burden.
The next geometric formula for which we might need to calculate area is the circle. Either of two formulas can be used:
Area = πr2 or Area = π/4D2
Don’t let the Greek letter π (pi) disturb you. It is nothing more or less than a symbol to represent a dimensionless ratio that is numerically equal to 3.14159265 (however, 3.14 is good enough for our purposes). The value of π/4 is 0.7854, or just a smidge more (well 3.5% more) than the fraction 3/4. |
Figure 2.
 |
Several relationships fit that value of π.
First, the circumference of a circle is π times the length of the circle’s diameter. If it helps, remember that you have to walk 3.14 times as far (π times as far) to walk completely around a circular clarifier as you would to walk straight across the clarifier bridge. (Don’t try that shortcut if the bridge only goes halfway.)
|
Figure 3.
 |
Two squares are superimposed upon a drawn circle in Figure 2 to help illustrate two other relationships involving π. Notice that the area of the small square is equal to the radius squared, r2, so we see that the circular area (πr2) is π times as big as the square that would just contain one quadrant, or quarter, of the circle. Also notice that the area of the large square is D2, so the area of the circle is π/4 times as big as the square that would just contain it.
Volumes of Common Shapes
Recall the expression mutual perpendicularity. In almost all commonly encountered figures, calculating volume will involve finding a cross-sectional area and multiplying by a length dimension that is perpendicular to that cross section.
|
Figure 4.
 |
For the rectangular solid shown in Figure 4, for example, any two dimensions multiplied together would be a cross-sectional area, and the third dimension would be a perpendicular dimension. Now consider the skewed cylindrical figure in Figure 4. If you can get the horizontal base area (elliptical) and multiply by the vertical height, it would yield the volume. Alternatively, if |
you can get the circular cross-sectional area (π/4D2) and multiply by the perpendicular length dimension, it will also yield the volume.
When it comes to cones and pyramids, the mutual perpendicularity rule still applies, but the volume is only one-third of the product of the area of the base times the perpendicular height. It might help to remember this by thinking of a cone or pyramid being carved from a solid prism or cylinder of soap, starting from a flat base. When the cone or pyramid is finally carved, two-thirds of the soap will be flakes on the floor, and only one-third will be left in the figure.
Another geometric shape for which you may need to calculate the volume is a sphere. High-pressure-gas storage vessels are frequently spheres. (So are rising bubbles, but we seldom need to know their volume.) More often, you may encounter half-spheres as the ends of pressure vessels that are otherwise cylindrical. The volume of a sphere is given by the formula
V = 4/3πr3
where r is the radius of the sphere.
Quite frequently, basins in treatment plants or pump stations are a composite of more than one geometric shape. Circular basins and anaerobic digesters often have conical bottom sections. Sludge hoppers are frequently truncated pyramids (that is, pyramids with the pointed ends cut off). You may have to figure two volumes and either add them together or subtract one from the other to get the appropriate basin volume.
Consider how seldom you are going to need to calculate areas or volumes of treatment vessels, pipelines, sludge piles, shipping crates, or whatever. There is very little reason to perform such calculations in a hurry. Take your time, plan your solution, look up formulas if you must — but don’t panic. It often helps to draw a sketch, add in the dimensions, and pretend that you’re building Rome — and remember, Rome wasn’t built in a day.
Donald Proctor, Ph.D., was director of the California Water Quality Control Institute (San Marcos) and held a Grade V wastewater treatment plant operator certificate until his retirement in 1994. He is a registered engineer in Washington state and serves as an ad hoc member of the advisory committee for wastewater treatment plant operator certification in Washington. The author would like to thank the Yakima, Wash., section of the Pacific Northwest Clean Water Association (Caldwell, Idaho) for sharing this information.
©2006 Water Environment Federation. All rights reserved.