4.3: Dimensional Analysis
Often in the Water and Wastewater Operations, we must use the relationship between different quantities to determine the results of various calculations. In other words, we will need to perform calculations while also recognizing the changes in the measurements and dimensions to assure the measurement of the result is reasonable.
Conversions
In the Water Industry, we often convert between units of measurement such as gallons to liters or days to hours. To convert between units of measurement, we first need to know the conversion factors or equivalencies. These equivalencies are usually provided in a chart and over time, you will most likely start to learn the ones that are used often, much like we know there are 24 hours in a day or 7 days in a week or 12 feet in a yard.
When converting between units, the idea is to multiply by a form of 1. We will create a fraction that is equivalent to 1 by doing the following:
- Set up the starting measurement as a fraction.
- The unit we are removing is placed opposite the position of where it is currently
- The “new” unit will be placed in the other position
- Insert the equivalencies
Examples:
- Convert 153 miles per hour to feet per hour. We start by writing the words into a fraction, namely, \(\frac{153\,\, miles}{1\,\, hour}\). Next, since we want to convert miles to feet, we need to eliminate the miles, so we will multiply by a fraction with miles in the denominator (opposite of where the miles are currently) and since we need to replace the miles with feet, we will place the feet in the numerator (to replace the current miles), then insert the equivalencies, 5280 feet = 1 mile, hence we multiply the original measurement by \(\frac{5280\,\, feet}{1\,\, mile}\)\[\frac{153\,\, miles}{1\,\, hour}\cdot \frac{5280\,\, feet}{1\ mile}\nonumber \]Next, we simplify by reducing or canceling the mile unit of measurement and multiply straight across to obtain:\[\frac{153\,\, \cancel{miles}}{1\,\, hour}\cdot \frac{5280\,\, feet}{1\,\, \cancel{mile}}=\frac{153}{1\,\, hour}\cdot \frac{5280\,\, feet}{1}=\frac{807,840\,\, feet}{1\,\, hour} \text{or 807,840 feet per hour}\nonumber \]
- The Plant needs to treat every 60 gallons of water with 7 grams of chlorine. However, the chlorine is packaged in ounces. Convert the required amount from gallons/gram to gallons/ounce. We start by writing the given information as a fraction, namely, \(\frac{60\,\, gallons}{7\,\, grams}\). Next, since we want to convert the grams to ounces and the grams are currently in the denominator, we create a fraction with the grams in the numerator and the ounces in the denominator (so that the grams are opposite each other) and then insert the equivalencies to obtain:\[\begin{aligned} \frac{60\,\, gallons}{7\,\, grams}\cdot \frac{1\,\, gram}{0.03527396195\,\, ounces}&\, \\ \frac{60\,\, gallons}{7\,\, \cancel{grams}}\cdot \frac{1\,\, \cancel{gram}}{0.03527396195\,\, ounces}&=\frac{60\,\, gallons}{0.246917734\,\, ounces} \\ &=\frac{60}{0.246917734}\frac{gal}{oz}=242.9959125 \,\, gal/oz \end{aligned}\]
To convert between metric units, we can use the same process described above or we can use a prefix table and move the decimal point based on the direction and the number of places it takes to move from one metric unit of measure to the others based on the order of the sizes of the prefixes.
Let’s begin by using a mnemonic to learn the basic six prefixes along with the base unit measurement and their order on the ladder.
“King Henry Died By Drinking Chocolate Milk”
This statement can help with remembering the order of six prefixes and the base unit.
For the base, you can have any of the following units of measurement:
- Meters (length)
- Grams (weight)
- Liters (volume)
- Meters\({}^{2}\) (area)
To convert from one prefix to another,
- Determine if we must move left or right
- Determine the number of places to move on the ladder
- Move the decimal place in the number the same number of places in the same direction as we moved on the prefix ladder.
NOTE: Mega is three units to the left of Kilo and Micro is three units to the right of Milli
Examples:
1. Convert 674.325 centimeters to dekameters
Using the ladder, to move from centi to deka, we must move LEFT 3 places, hence, we must move the decimal place 3 places to the left to obtain 0.674325 dekameters.
2. Convert 67 hectoliters to milliliters.
Using the ladder, to move from hector to milli, we must move RIGHT 5 places, hence, we must move the decimal place 5 places to the right, keeping in mind that for a whole number, the decimal is understood to be after the last digit, 67 hectoliters convert to 6,700,000 milliliters.
3. Convert 6.5 kilograms to megagrams.
Since mega is 3 places to the left of kilo, we must move the decimal place 3 places to the left so 6.5 kilograms becomes 0.0065 megagrams.
4. Convert 8765 centimeters to meters.
Remembering that the base represents meters, we need to move from centi to the base (meters) which would be LEFT by 2 places, so move the decimal two places to the left to convert 8765 centimeters to 87.65 meters.