12.7 Process Control Mathematics: Loading, Efficiency, Removal & Solids

12.7 Process Control Mathematics: Loading, Efficiency, Removal & Solids

Every North Carolina needs-to-know document lists mathematics as a required subject, and the examinations provide a state formula sheet. Knowing which formula applies — and what each variable means — is the actual skill being tested.

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1. Removal efficiency

$\text{Percent Removal} = \frac{\text{Influent} - \text{Effluent}}{\text{Influent}} \times 100$

Example. Influent BOD 240 mg/L, effluent BOD 12 mg/L:

(240 − 12) ÷ 240 × 100 = 95.0 percent removal

The same relationship works on mass (pounds per day), which is what you use when flows differ between the two sampling points. North Carolina secondary treatment permits typically require a minimum 85 percent removal of BOD and TSS in addition to the concentration limits.

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2. Loading rates

Worked example — solids loading rate on a secondary clarifier. A 65-foot diameter clarifier receives 2.4 MGD of plant flow plus 1.2 MGD of RAS, with MLSS at 3,100 mg/L.

1. Area = 0.785 × 65² = 0.785 × 4,225 = 3,316.6 ft²
2. Solids applied = (2.4 + 1.2) MGD × 3,100 mg/L × 8.34 = 3.6 × 3,100 × 8.34 = 93,074 lb/day
3. SLR = 93,074 ÷ 3,316.6 = 28.1 lb/day/ft²

That sits at the upper end of the usual 20 to 30 lb/day/ft² design range — a number worth watching if the sludge blanket is rising.

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3. Sludge mass and volume

$\text{lbs of dry solids} = \text{gallons} \times 8.34 \times \frac{\%\text{ solids}}{100} \times \text{specific gravity}$

Example. 12,000 gallons of primary sludge at 4.5 percent solids (specific gravity taken as 1.02):

12,000 × 8.34 × 0.045 × 1.02 = 4,593 lb of dry solids

Volume change with thickening uses the inverse relationship between concentration and volume:

$V_2 = V_1 \times \frac{P_1}{P_2}$

Thickening 30,000 gallons of 1.0 percent WAS to 5.0 percent gives 30,000 × (1.0 ÷ 5.0) = 6,000 gallons — an 80 percent volume reduction.

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4. Volatile solids reduction (the Van Kleeck formula)

Because digestion destroys volatile matter but leaves fixed solids behind, a simple before-and-after comparison of volatile percentages understates the destruction. The standard correction is:

$\%\text{VSR} = \frac{(V_{in} - V_{out})}{V_{in} - (V_{in} \times V_{out})} \times 100$

where V is the volatile fraction expressed as a decimal.

Example. Feed sludge is 72 percent volatile (0.72); digested sludge is 52 percent volatile (0.52):

• Numerator: 0.72 − 0.52 = 0.20
• Denominator: 0.72 − (0.72 × 0.52) = 0.72 − 0.3744 = 0.3456
• %VSR = 0.20 ÷ 0.3456 × 100 = 57.9 percent

That comfortably exceeds the 38 percent minimum used for the volatile solids reduction option under the federal vector attraction reduction requirements.

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5. Blending, dilution, and chemical strength

The blending (mixture) equation:

$(V_1 \times C_1) + (V_2 \times C_2) = (V_1 + V_2) \times C_{mix}$

Example. 4,000 gallons of 12.5 percent hypochlorite is blended with 1,000 gallons of water:

(4,000 × 12.5) + (1,000 × 0) = 5,000 × C → 50,000 = 5,000C → C = 10.0 percent

Chemical usage tracking. Pounds of chemical used per day divided by pounds of a target parameter removed per day gives a chemical efficiency figure that is far more useful for budgeting and troubleshooting than raw consumption. A rising alum-per-pound-of-turbidity figure means the water changed or the feed system is losing accuracy.

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6. Detention time and hydraulic checks

$\text{Detention time (hours)} = \frac{\text{Volume (gallons)} \times 24}{\text{Flow (gpd)}}$

Use it to sanity-check process problems: if a clarifier's detention time has fallen from 2.4 hours to 1.1 hours because the plant is passing wet weather flow, the solids carryover is hydraulic, not biological, and no amount of wasting adjustment will fix it.

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7. Working an exam math problem

1. Write down what is asked, with units.
2. List the given values, converting to the units the formula expects — MGD, mg/L, feet, and gallons are the usual offenders.
3. Select the formula from the state formula sheet, and confirm the variables mean what you think.
4. Solve stepwise, carrying units through the arithmetic.
5. Sanity-check the answer: is an F/M of 12 plausible? Is a detention time of 0.02 hours plausible? A wrong unit conversion almost always produces an answer that is off by a factor of 10, 8.34, 1,440, or 7.48 — learn to recognize those fingerprints.

[!NOTE] Common conversions worth memorizing: 8.34 lb/gal; 7.48 gal/ft³; 62.4 lb/ft³; 1,440 min/day; 694.4 gpm per MGD; 1.547 cfs per MGD; 43,560 ft² per acre; 325,851 gallons per acre-foot; 2.31 ft of head per psi; 0.433 psi per foot of head.