Common Conversion Factors Distance Learning Math Index Websites for Those Who Need Help Homepage Calculations  In order to correcly solve a problem, all units of measure must be converted to the proper units required for the formula.  Examples of some units the operator will be using are grains per gallon, grams, ounces, pounds, cubic feet, square feet, liters, gallons, parts per million and pounds per square inch.  Units of time are important in water calculations.  To convert gallons per minute into gallons per hour, multiply by 60.  For gallons per day when gallons per minute is known, multiply gallons per minute by 1440.  This is the number of minutes in a day (24  x  60).      *Here are some formulas and figures that you will need in water calculations.     Area and Volume of Square or Rectangular Figures:     Example 1)     Area       Example 2)      Area of Triangles:                            The area of a triangle is 1/2 of a square or rectangle.               Hence, the formula for the area is half the base (b) times the altitude or height (a).            A = 1/2b  x  a                           A = 1/2b  x  a            A = 3'  x  6'                              A = 5'  x  6'            A = 18 sq. ft.                            A = 30 sq. ft.          Example 3)     Area and Volume of Circular Figures:   Concerning circles, several terms are used.  The distance across a circle through the center is the diameter (D).  Half the diameter is called the radius (r).  A number pi which is called a constant is 3.1416.  Another term is "squared".  When a number is squared, it is multiplied by itself.  The area of a circle equals the radius (r), which is half the diameter, squared times pi.     Example 4)    Find the area of a circle with a 20 foot diameter:                                  A = pi (r)2                                 A = 3.14  x  102                                 A = 3.14  x  10  x  10                                  A = 314.0  sq. ft.      *The volume of a circular tank is the area of the floor surface multiplied by the depth. This volume equals pi times the radius squared times the depth, or the area of the bottom times the depth.  The formula for the volume is V = pi (r)2  x  depth  (remember pi (r)2 is the area of the circle).       Example 5)     Volume                                 Volume equals length times width times depth.                                            V = L  x  W  x  D                                          V = 10'  x  6'  x  5'                                          V = 300 cubic feet  *Unless great accuracy is desired, the operator should consider 7.5 gallons as being equal to 1 cubic foot.  Actually 1 cubic foot of water is 7.481 gallons.  Thus, the volume of the tank in  Example 5 (6' wide, 10' long, 5' deep) is 2250 gallons. (10'  x  6'  x  5'  x  7.5  =  2250 gallons or 300 cubic feet  x  7.5 gallons  =  2250 gallons).            Example 6) Find the volume of a tank 20 feet in diameter and 20 feet deep.  The area of the tank bottom is 3.14  x  102 or 3.14  x  10  x  10 = 314 square feet.  Next, multiply the area by the depth and obtain the volume:  314 sq. ft.  x  20 ft. = 6280 cu. ft.  To find the number of gallons the tank will hold, multiply the volume in cubic feet by  the number of gallons in one cubic foot:  6280 cu. ft.  x  7.5 gal. = 47,100 gallons.   Circumference of Circles  The distance around a circle is called the circumference.  The circumference, C, may be determined by measuring the diameter and multiplying by pi.      Example 7)   Find the circumference of a circle 20 feet in diameter.                             C = pi(D)                              C = 3.14(20)                              C = 62.8 ft .   Area of Tanks or Cylinders In order to find the area of the walls of a tank or cylinder, the circumference is multiplied by the height.  Such a calculation would be useful in deciding how much paint would be necessary to coat a tank.  The area of the walls of a cylinder is called the "lateral area", A.  If a tank is 20 feet in diameter and 20 feet deep, its circumference is 3.14  x  20 = 62.8 feet.  The area of the wall, the lateral area, is 62.8'  x  20' or 1256 sq. ft.  Thus the formula for lateral area is:                  A = pi(D)(H)                  A = pi  x  20  x  20                  A = 3.14  x  20  x  20                  A = 1256 sq. ft.                                    Lateral Area (A) = pi(D)(H)  The area of the bottom of the cylinder is pi(r)2, or 3.14(10)2 = 314.6 sq. ft.  The area of the top of the cylinder would also be 314.6 sq. ft. so total area of the inside of the cylinder would be 1256 + 314.6 = 1885.2 sq. ft.  If the inside and outside were to be painted, the total area would be 1885.2  x  2 = 3770.4 sq. ft. If one gallon of paint will cover 250 sq. ft., to paint the enclosed cylinder inside and out would require 3770.4/250 = 15.08, or 16 gallons.  Suppose the tank has a conical top which must be painted.  The area of a cone is found by multiplying the circumference of its base by half of the slant height.  The top overhangs the tank  2 feet all around making its diameter 24 feet, rather than 20 feet, as the tank has.  If measured it would be found that the slant height would be about 14.4 feet.  Thus the area is:  A = 2  pi(r)  x  1/2 slant height A = 2  x  3.14  x  12'  x  7.2' = 542 sq. ft. For the inside or outside of the cone.    If the tank has a rounded bottom, its approximate area can be found by using the formula for half a sphere (ball).  Even if the bottom is not quite round, this formula should give a good approximation of the area.  Multiply the radius of the tank by the circumference.  (Remember, circumference = 2pi(r)).                        Area = r  x  2 (pi(r))                      A = 10  x  2  x  3.14  x  10'                      A = 628 sq. ft.  The total area to be painted is the sum of these areas (the walls, top and bottom):  1256 + 542 + 628 = 2,456 sq. ft.  If the tank is to be painted inside and out, the area must be doubled.  The approximate volume of this bottom section, if it is half of a sphere (ball) is 1/2 D3       By adding this figure to the volume of the cylinder, which was figured in Example 6, the total volume of the tank will be 47,000 + 15,000, or approximately 62,000 gallons of water.      Detention Time Detention time, sometimes called the retention period, is the length of time water is retained in a vessel or basin.  It may also be considered the period from the time the water enters a settling basin until it flows out the other end.  To calculate the detention period of a basin, the volume of the basin must be first obtained.  Using a basin 20' wide, 60' long and 10' deep the volume would be:              V = L  x  W  x  D              V = 60  x  20  x  10              V = 12,000 cu. ft.              Gallons = V  x  7.5 gal. per cu. ft.              Gallons = 12,000  x  7.5 = 90,000 gallons    Assume that the plant filters 250 gpm.  Therefore, 90,000/250 = 360 minutes, or 6 hours of detention time.  Stated another way, the detention time is the length of time theoretically required for the coagulated water to flow through the basin.  If chlorine were added to the water as it entered the basin, then the chlorine contact time would be six hours.    Example Problems for Measurement Conversions