8.1 Mass Balance and Hydraulic Loading
Key Takeaways
- Mass balance questions start with a control volume: identify every flow, concentration, storage change, removal term, and recycle stream crossing the boundary.
- For conservative mixing, the mixed concentration is a flow-weighted average, Cmix = sum(QC) / sum(Q), not an arithmetic average of concentrations.
- The PE WRE loading shortcut lb/day = MGD x mg/L x 8.34 applies when water density is appropriate and the flow is in million gallons per day.
- Hydraulic loading rate is Q / A, while detention time is V / Q; using area where volume is required is a common exam trap.
- A reasonableness check should confirm that blended concentrations, loading rates, and detention times match the physical process described.
Mass Balance and Hydraulic Loading
The April 2024 PE Civil Water Resources and Environmental specification lists mass balance and hydraulic loading under Analysis and Design, and those topics also appear inside treatment, distribution, collection, and stormwater questions. Treat them as design bookkeeping. Before choosing an equation, decide whether the question is asking for a concentration, mass per time, hydraulic rate, detention time, or required area.
Control-Volume Setup
A control volume is the boundary around the process you are analyzing. It may enclose a tank, clarifier, reservoir, basin, junction, pipe reach, or whole treatment train. Label all streams that cross the boundary and ignore internal recirculation unless it crosses the boundary you selected. On the PE exam, a quick sketch often prevents double counting a return stream or losing a small but concentrated sidestream.
| Quantity | Common PE WRE form | Meaning |
|---|---|---|
| Flow | Q, MGD, cfs, gpm | Water or wastewater volume per time |
| Concentration | C, mg/L | Constituent mass per volume |
| Load | Q x C x conversion | Mass per time, often lb/day or kg/day |
| Hydraulic loading | Q / A | Flow applied per plan or media area |
| Detention time | V / Q | Average time fluid remains in a volume |
Calculation Workflow
- Draw the control volume and mark each inflow, outflow, bypass, recycle, and loss.
- Convert units to one basis before combining terms.
- Write the balance in words: accumulation = inputs - outputs + generation - removal.
- Simplify based on the problem statement. For steady conservative mixing, inputs equal outputs.
- Solve for the requested variable and check whether the magnitude is physically plausible.
Typical Exam Calculations
For a conservative blend with no storage change, reaction, or removal, use a flow-weighted concentration:
Cmix = sum(Qi Ci) / sum(Qi)
If the question asks for mass load in U.S. Customary units, use:
Load, lb/day = Q, MGD x C, mg/L x 8.34
For hydraulic design checks, use the denominator named by the criterion. A clarifier overflow rate uses surface area. A filter hydraulic loading rate uses filter area. A detention basin residence time uses the active volume. If a basin is 100 ft by 40 ft by 8 ft deep, its plan area is 4,000 ft^2 and its volume is 32,000 ft^3; the right value depends on the requested check.
Reasonableness Checks
- A blended concentration should normally fall between the input concentrations unless a reaction, chemical addition, settling, or removal process is stated.
- A small high-strength recycle can dominate a mass balance even when it barely changes total flow.
- A hydraulic loading rate increases with flow and decreases with area.
- Detention time decreases when peak flow is used instead of average flow.
- A design answer should be stated in the same type of unit as the criterion: gpd/ft^2, lb/day, hours, or mg/L.
PE WRE Trap Pattern
Many wrong answers come from averaging concentrations, forgetting the 8.34 conversion, mixing cfs with gallons, or using average daily flow when the prompt asks for peak hydraulic loading. Read the final sentence first, then build the calculation around the requested design check.
A treatment basin receives 2.0 MGD at 18 mg/L ammonia and a sidestream of 0.25 MGD at 210 mg/L ammonia. Assuming steady conservative mixing with no removal, what is the mixed ammonia concentration?
A rectangular settling basin is 90 ft long, 30 ft wide, and 10 ft deep. If the peak flow is 1.6 MGD, which hydraulic loading rate should be used for surface overflow checking?