Logical Math #4: Getting the Witchcraft out of the So-Called “Pounds Formula”
by Donald Proctor
When trying to calculate the amount of a constituent (for example, 140 mg/L of suspended solids) that would be contained in a quantity of wastewater (for example, 6 million gallons), you may have encountered instructions that tell you to multiply the concentration of that constituent, expressed in milligrams per liter, by the quantity of wastewater, expressed in million gallons, and then to multiply the result by 8.34 to get the desired result expressed in pounds.
Amount = 140 mg/L × 6 million gal × 8.34 = 7005.6 lb
Is the answer correct? Yes. Why? Because I said so, that’s why. Now quit asking stupid questions and just do what I tell you. Witchcraft and voodoo dolls are not to be questioned.
Does that somehow not sit well in your mind? It shouldn’t, so let’s explore the process just a bit. First, it is entirely logical that both the volume of wastewater and the concentration of the constituent (mass or weight per unit volume) should be used as multiplying factors. That is, you would expect a larger amount of suspended solids if either the solids concentration or the volume of wastewater were to increase. What is not at all logical, though, is that the weight of only 1 gallon of water should be a factor in what might be contained in 1 million gallons of wastewater or that somehow the answer ends up being in pounds when the concentration was expressed in milligrams per liter.
To start our investigation into why this voodoo doll (often called the “pounds formula”) works, lets start with a simple equation, which is nothing more than a statement that two expressions are equal to each other:
1.0 mg/L = 1.0 mg/L
Now, let’s multiply the right side of this equation by several conversion factors, which is the same as multiplying by 1.0 several times:
1.0 mg/L = (1.0 mg/L) × (1.0 g/1000 mg) × (3.785 L/gal) × (2.204 lb/1000 g) = 8.34 lb/million gal
This indicates that both 1.0 mg/L and 8.34 lb/million gal are, in fact, exactly equal concentrations. Now let’s divide the two “equal things” by 1.0 mg/L:
(1.0 mg/L) ÷ (1.0 mg/L) = (8.34 lb/million gal) ÷ (1.0 mg/L)
Since the left side equals 1.0, it follows that the right side must also equal 1.0, which is to say that (8.34 lb/million gal ÷ 1.0 mg/L) is also a conversion factor.
Now let’s revisit the problem of calculating the amount of suspended solids in the 6 million gallons of wastewater:
Amount = (amount ÷ volume) × volume × conversion factor
Amount = (140 mg/L) × 6.0 million gal × (8.34 lb/million gal ÷ 1.0 mg/L) = 7005.6 lb
Keep in mind that a proper conversion factor does not change the absolute value of any quantity — it only changes the way that absolute quantity is expressed. You could have used this new conversion factor to express the concentration differently before starting the solution and then multiplied by the volume.
140 mg/L × (8.34 lb/million gal ÷ 1.0 mg/L) = 1167.6 lb/million gal
Amount = 1167.6 lb/million gal × 6 million gal = 7005.6 lb
Now does this mean that you should never use the “pounds formula” again? Of course not, but when you use it, remember what kind of units of measure are needed to make it work and what units will apply to the answer. The formula isn’t going to tell you if you do it wrong. At least now, you understand how the witchcraft works.
You will often encounter the statement that 1.0 part per million (ppm) is the same as 1.0 mg/L. This is not exactly correct (but close enough for government work) because 1.0 ppm implies a weight-to-weight ratio, while 1.0 mg/L indicates a weight-to-volume (mass-to-volume, actually) ratio. Generally speaking, we can get by very nicely without such nit-picking unless we are dealing with a solution or suspension that is decidedly heavier or lighter than water.
Some equivalencies are provided below from which a multitude of conversion factors can be constructed as may be needed (see box below). Just remember that any fraction that consists of something in the numerator that is exactly equal to whatever is in the denominator can be used as a valid conversion factor. For example, 30.48 cm divided by 12 in. is a valid conversion factor. So too would be 12 in. divided by 30.48 cm because any conversion factor inverted (turned upside down) is another conversion factor.
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
Practice Problems
You should begin to challenge yourself with problems of your own creation. The following are only suggestions. Convince your best buddy to solve the problems, too, and then compare answers. If you don’t agree, compare solution steps until you can both agree on the same answers. And remember, no biting!
Get the necessary measurements of an available dumpster. What would be the dumpster’s volume when full, expressed in cubic feet, gallons, and cubic meters? If the dumpster were filled with sand weighing 85 lb/ft3, what would be the weight of the sand in tons and in kilograms? |
Tabulated below are numerous units of measure for which several alternate equivalent values are listed. Under the “Volume” heading, for example, you will find that 1.0 ac-ft of volume is also equal to 1233 m3, or 325,900 gal.
Length
1.0 mm = 0.1 cm = 0.03937 in. = 0.001 m
1.0 cm = 0.3937 in. = 0.03281 ft = 0.01 m
1.0 in. = 1000 milli-inch = 25.4 mm = 2.54 cm = 0.0833 ft = 0.0254 m
1.0 ft = 30.48 cm = 12 in. = 0.3333 yd = 0.3048 m
1.0 yd = 91.44 cm = 36 in. = 3.0 ft = 0.9144 m
1.0 m = 1000 mm = 100 cm = 3.28 ft = 1.094 yd
1.0 fathom = 72 in. = 6 ft = 1.83 m
1.0 km = 106 mm = 3280 ft = 1094 yd = 1000 m = 0.621 mi
1.0 mi = 5280 ft = 1760 yd = 1609 m = 1.609 km
Area
1.0 cm2 = 100 mm2 = 0.155 in.2 = 10-4 m2
1.0 in.2 = 6.45 cm2 = 0.006944 ft2
1.0 ft2 = 929 cm2 = 144 in.2 = 0.1111 yd2 = 0.0929 m2
1.0 yd2 = 8361 cm2 = 1296 in.2 = 9.0 ft2 = 0.836 m2
1.0 m2 = 104 cm2 = 1550 in.2 = 10.76 ft2 = 1.20 yd2 = 0.01 ac
1.0 ac = 43,560 ft2 = 4840 yd2 = 4074 m2 = 0.0015625 mi2
1.0 ha = 107,600 ft2 = 2.471 ac = 0.01 km2 = 0.00386 mi2
1.0 km2 = 106 m2 = 247 ac = 0.3861 mi2
1.0 mi2 = 27.9 × 106 ft2 = 2.6 × 106 m2 = 640 ac
Volume
1.0 cm3 = 1.0 mL = 0.06102 in.3 = 0.001 L
1.0 in.3 = 16.39 cm3 = 0.01639 L = 0.00433 gal
1.0 qt = 946.4 mL = 57.75 in.3 = 0.9464 L = 0.25 gal
1.0 ft3 = 28320 cm3 = 1728 in.3 = 28.32 L = 7.481 gal
1.0 m3 = 106 mL = 1000 L = 264.2 gal = 35.31 ft3 = 1.0 stere
1.0 ac-ft = 1,233,619 L = 325,900 gal = 43,560 ft3 = 1233 m3
1.0 mg = 3.78 × 106 L = 106 gal = 133,672 ft3 = 3785 m3 = 3.068 ac-ft
Force
1.0 lb = 7000 grains = 16 oz = 4.4482 Newton = 444,823 dynes
1.0 Newton (N) = 100,000 dynes = 0.2248 lb
1.0 ton = 2000 lb
Mass and weight-mass equivalents (in Earth gravity)
1.0 g = 1000 mg = 0.001 kg
1.0 kg = 1000 g = 0.0685 slug
1.0 slug = 14.594 kg
1.0 lb = 453.6 g = 0.4356 kg
1.0 kg = 2.204 lb
1.0 slug of mass = 14.59 kg = 32.174 lb
Time
1.0 d = 24 h = 1440 min = 86,400 s
Flow rates
1.0 gal/min = 63.08 mL/s = 1440 gal/d
1.0 L/s = 22,827 gal/d = 86.4 m3/d = 15.85 gal/min
1.0 ac-ft/d = 325,872 gal/d = 226 gal/min = 14.28 L/s = 0.325 mgd
1.0 ft3/s = 448.86 gal/min = 1.98 ac-ft/d = 0.646 mgd
1.0 mgd = 694 gal/min = 43.8 L/s = 3.07 ac-ft/d = 1.55 ft3/s
1.0 m3/s = 15,852 gal/min = 1000 L/s = 70 ac-ft/d = 22.8 mgd
Pressure–head relationships
1.0 ft of water = 62.4 lb/ft2 = 22.4 mm of mercury = 0.4335 lb/in.2 = 0.0295 atmosphere (atm)
1.0 in. of mercury = 70.5 lb/ft2 = 3.38 kP = 1.133 ft of water = 0.490 lb/in.2
1.0 lb/in.2 = 6894 N/m2 = 144 lb/ft2 = 2.307 ft of water = 0.0680 atm = 6894 P = 6.89 kP
1.0 atm = 760 mm of mercury = 33.8995 ft of water = 29.92 in. of mercury = 14.696 lb/in.2 = 101.3 kP
Energy and/or work
1.0 ft-lb = 13.6 × 106 dyne-cm = 1.356 J = 1.356 N-m = 0.324 cal = 0.00128 Btu
1.0 Btu = 252 cal = 1056 J = 0.293 W-h
1.0 hp-h = 1.98 × 106 ft-lb = 2546 Btu = 0.746 Kw-h = 2.68 X 106 J
1.0 kW-h = 3.6 × 106 J = 2.65 × 106 ft-lb = 1000 W-h = 1.341 hp-h
Power
1.0 W = 1.0 N-m = 1.0 J/s = 0.735 ft-lb/s = 0.057 Btu/min
1.0 hp = 33,000 ft-lb/min = 746 W = 550 ft-lb/s = 0.746 kW
1.0 kW = 1000 W = 1000 J/s =44,250 ft-lb/min = 56.8 Btu/min = 0.95 Btu/s
Concentration or dosage (assuming solution density = 1.0 g/cm3)
1.0 ppm = 1.0 mg/L = 8.34 lb/million gal = 0.058 grain/gal
1.0 grain/gal = 17.1 mg/L
Specific weight and density of water
1.0 gal = 8.34 lb
1.0 ft3 = 62.4 lb
1.0 L = 1.0 kg (mass) = 2.204 lb (weight)
1.0 mL = 1.0 g = 1000 mg
Mercury (Hg) is 13.58 times as dense or heavy as water.
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