Operator math fundamentals

The spreadsheet is gone on exam day

Most plants do the daily math for you. A spreadsheet figures the dosage, SCADA shows the detention time, the chart on the wall has the loading rate. You keep the plant compliant for years without working the arithmetic by hand. Then you sit down for a closed book exam and the spreadsheet is gone.

The exam hands you a formula and conversion sheet, so you are not memorizing factors. What it tests is whether you can pick the right formula, put your numbers in the right places, handle the units, and solve. That is a skill, and it comes back fast with reps. Two things trip people more than the math itself: choosing the formula, and converting units before you plug in.

Units are where most points leak

These conversions are on the formula sheet, but you move faster when you know them cold:

• gpm to MGD: multiply by 1440, then divide by 1,000,000
• MGD to cfs: multiply by 1.547
• cubic feet to gallons: multiply by 7.48
• psi to feet of head: multiply by 2.31

The single most common mistake on the whole exam is mixing MGD with gpm, or square feet with cubic feet. Write the units next to every number and cancel them as you go. If the units do not come out right, the number is wrong.

Area and volume

Rectangle area is length times width. Circle area is 0.785 times diameter times diameter, where 0.785 is pi divided by 4 so you can work straight from the diameter. Box volume is length times width times depth. Cylinder volume is 0.785 times diameter times diameter times depth. To turn cubic feet into gallons, multiply by 7.48.

For a pipe, the length is the depth. Convert the diameter from inches to feet first, by dividing by 12, or the answer will be off by a wide margin.

Flow, velocity, and detention time

Flow equals velocity times area, so velocity is flow divided by area and area is flow divided by velocity. Know all three. Detention time is volume divided by flow, with the units kept consistent: if the volume is in gallons, put the flow in gallons per day, get the answer in days, then multiply by 24 for hours. Detention time is one of the most asked calculations on both exams, so drill it until the unit handling is automatic.

The 8.34 relationship, the one to learn

This is the heart of operator math. Pounds per day equals flow in MGD times concentration in mg/L times 8.34, because 8.34 is the weight of one gallon of water in pounds. Dosing, loading, and feed rate all run through that single line. Rearranged, mg/L is pounds per day divided by MGD times 8.34, and MGD is pounds per day divided by mg/L times 8.34.

Keep dose, demand, and residual straight, because the exam mixes them on purpose. The dose is what you add, the residual is what is left, and the demand is the difference. And when a chemical is only partly available, say calcium hypochlorite at 65 percent available chlorine, divide the pounds of chlorine you need by 0.65 to get the pounds of product to feed.

Loading and percent removal

Loading spreads flow or mass over an area or a length. Surface overflow rate is flow in gpd divided by surface area in square feet. Weir overflow rate is flow in gpd divided by weir length in feet. Filtration rate is flow in gpm divided by filter area in square feet, so convert MGD to gpm first. Organic loading is the 8.34 line again, in pounds per day.

Percent removal is influent minus effluent, divided by influent, times 100. It shows up for BOD and TSS on the wastewater exam and ties straight to the 85 percent secondary treatment floor. Watch the trap where the question gives you what is left and asks for what was removed.

Pumping horsepower

Water horsepower is gpm times head in feet, divided by 3960. Brake horsepower is water horsepower divided by the pump efficiency. Motor input horsepower is brake horsepower divided by the motor efficiency. The number gets bigger as you move from water to brake to motor, because more input covers the losses. If your number got smaller, you divided the wrong way.

Two worked the exam's way

A plant treats 1.5 MGD and needs a 2.5 mg/L chlorine dose. Pounds per day is 1.5 times 2.5 times 8.34, which is 31.3 pounds per day.

A 100,000 gallon tank takes 0.5 MGD. That flow is 500,000 gallons per day, so detention time is 100,000 divided by 500,000, which is 0.2 day, times 24 is 4.8 hours.

Where operators lose points

• mixing MGD and gpm: decide which one the formula wants and convert first
• mixing square feet and cubic feet: area is two dimensions, volume is three
• confusing dose, demand, and residual
• dropping the 8.34: if a dosing answer is off by about a factor of eight, that is why
• multiplying by the available fraction instead of dividing, for product
• dividing the wrong way on efficiency, which should raise the number
• leaving a pipe diameter in inches before using 0.785

Practice until it is automatic

The fastest way to lock this in is reps. The free practice tool has 120 of these problems worked out, with no login and nothing to sign up for, and the operator calculator back-solves in any direction so you can check your work. If you want the lay of the whole exam, what's on the operator exam maps the nine domains. Fees, application windows, and renewal clocks differ by state and are listed on your state's page.