18.2 Chemical Feed, Dosage Calculations & the Pounds Formula

9.2 Chemical Feed, Dosage Calculations & the Pounds Formula

Water and wastewater treatment processes rely on precise chemical addition to facilitate coagulation, flocculation, pathogen disinfection, pH stabilization, and fluoridation. Over-dosing chemicals squanders public utility funds, creates hazardous chemical residuals (such as disinfection byproducts), and risks permit violations. Under-dosing fails to achieve pathogen inactivation, destabilize colloidal turbidity, or precipitate phosphorus.

To control these processes accurately, operators rely on the Pounds Formula—the most frequently utilized calculation in environmental utility operations.

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The Foundational Pounds Formula

The pounds formula calculates the daily mass of chemical required to achieve a target concentration in a given flow stream, or conversely, calculates the mass of pollutants (such as BOD5 or TSS) entering a treatment facility:

$\text{Chemical Feed (lbs/day)} = \text{Flow (MGD)} \times \text{Dosage (mg/L)} \times 8.34\text{ lbs/gal}$

Scientific Derivation & Unit Cancellation

To understand why this formula works, examine the definition of concentration. One milligram per liter ($1\text{ mg/L}$) expresses a mass-to-volume ratio in the metric system. Because $1\text{ liter of water}$ has a mass of $1,000\text{ grams} = 1,000,000\text{ milligrams}$, a concentration of $1\text{ mg/L}$ represents one part per million ($1\text{ ppm}$):

$1\text{ mg/L} = 1\text{ ppm} = \frac{1\text{ lb chemical}}{1,000,000\text{ lbs of water}}$

Now consider $1.0\text{ Million Gallons (MG)}$ of water. Since one gallon of water weighs $8.34\text{ lbs}$, one million gallons of water weighs:

$1,000,000\text{ gal} \times 8.34\text{ lbs/gal} = 8,340,000\text{ lbs}$

If we apply a chemical dosage of $1.0\text{ ppm}$ ($1\text{ lb of chemical per million pounds of water}$) to one million gallons of water:

$\text{Mass} = 8,340,000\text{ lbs of water} \times \left( \frac{1\text{ lb chemical}}{1,000,000\text{ lbs water}} \right) = 8.34\text{ lbs chemical}$

Therefore, $1\text{ mg/L}$ applied to $1\text{ MGD}$ always equals exactly $8.34\text{ lbs/day}$. Setting up the unit cancellation grid confirms this:

$\left( \frac{\text{Million Gallons}}{\text{day}} \right) \times \left( \frac{\text{lbs Chemical}}{\text{Million lbs Water}} \right) \times \left( \frac{8.34\text{ lbs Water}}{\text{Gallon Water}} \right) \times \left( \frac{1,000,000\text{ Gallons}}{\text{Million Gallons}} \right) = \frac{\text{lbs Chemical}}{\text{day}}$

Formula Rearrangements

Depending on the operational data available, the pounds formula can be rearranged algebraically to solve for dosage or flow:

• Solving for Dosage (mg/L): $\text{Dosage (mg/L)} = \frac{\text{Chemical Feed (lbs/day)}}{\text{Flow (MGD)} \times 8.34\text{ lbs/gal}}$
• Solving for Flow (MGD): $\text{Flow (MGD)} = \frac{\text{Chemical Feed (lbs/day)}}{\text{Dosage (mg/L)} \times 8.34\text{ lbs/gal}}$

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Adjusting for Chemical Purity and Active Ingredients

The basic pounds formula assumes that the chemical being fed is $100\%$ pure active ingredient (such as pure gaseous chlorine, $\text{Cl}_2$). However, most water and wastewater treatment chemicals are supplied as dry mixtures, hydrated crystals, or commercial aqueous solutions containing inert binders, carrier water, or stabilizers.

Because commercial products are less than $100\%$ pure, more commercial product must be fed to supply the required mass of pure chemical:

$\text{Commercial Chemical Feed (lbs/day)} = \frac{\text{Pure Chemical Required (lbs/day)}}{\text{Active Ingredient Fraction (decimal)}}$

$\text{Commercial Chemical Feed (lbs/day)} = \frac{\text{Flow (MGD)} \times \text{Dosage (mg/L)} \times 8.34\text{ lbs/gal}}{\text{Purity (as a decimal)}}$

Common Chemical Strengths & Active Fractions

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Liquid Chemical Feed Calculations & Specific Gravity

Liquid chemicals—such as sodium hypochlorite, liquid alum, ferric chloride, caustic soda ($\text{NaOH}$), and aqueous ammonia—are metered volumetrically (gallons per day or milliliters per minute). Calculating liquid chemical dosages requires accounting for Specific Gravity (SG).

Specific Gravity & Solution Weight

Specific gravity is the ratio of the density of a substance to the density of pure water ($8.34\text{ lbs/gal}$ at $4^\circ\text{C}$):

$\text{Weight of Liquid Chemical (lbs/gal)} = \text{Specific Gravity} \times 8.34\text{ lbs/gal}$

For example, commercial $12.5\%$ sodium hypochlorite typically has a specific gravity of $1.20$: $\text{Weight of } \text{NaOCl Solution} = 1.20 \times 8.34\text{ lbs/gal} = 10.008\text{ lbs/gal}$

Active Ingredient Weight per Gallon

Once the total weight per gallon is known, multiply by the percent concentration by weight to determine the active chemical content per gallon:

$\text{Active Chemical (lbs/gal)} = \text{Solution Weight (lbs/gal)} \times \left( \frac{\% \text{ Concentration}}{100} \right)$ $\text{Active } \text{Cl}_2 \text{ in } 12.5\% \text{ NaOCl} = 10.008\text{ lbs/gal} \times 0.125 = 1.251\text{ lbs active } \text{Cl}_2\text{/gal}$

Liquid Feed Rate in Gallons per Day (gpd)

$\text{Liquid Feed (gpd)} = \frac{\text{Pure Chemical Required (lbs/day)}}{\text{Active Chemical per Gallon (lbs/gal)}}$ $\text{Liquid Feed (gpd)} = \frac{\text{Flow (MGD)} \times \text{Dosage (mg/L)} \times 8.34\text{ lbs/gal}}{\text{Specific Gravity} \times 8.34\text{ lbs/gal} \times (\% / 100)}$

Notice that the $8.34$ constant in the numerator and denominator cancel out: $\text{Liquid Feed (gpd)} = \frac{\text{Flow (MGD)} \times \text{Dosage (mg/L)}}{\text{Specific Gravity} \times (\% / 100)}$

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Chemical Pump Calibration: Drawdown Cylinder Testing

Diaphragm and peristaltic chemical metering pumps deliver small flow rates that cannot be verified accurately with plant inline magnetic meters. Operators calibrate metering pumps using a graduated glass or PVC drawdown cylinder mounted on the chemical suction piping.

   ┌──────────────┐
   │ Chemical Day │
   │     Tank     │
   └──────┬───────┘
          │
       [Valve A] ── Normal feed from tank
          │
          ├─── [Valve B] ──► ┌───────────────┐
          │                  │ Drawdown Tube │ (Graduated in mL)
          ▼                  └───────────────┘
     [Metering Pump]
          │
          ▼
     To Injection Point

Drawdown Testing Procedure

1. Fill the drawdown tube with chemical from the storage tank.
2. Close the supply valve from the main storage tank (Valve A) while opening the calibration cylinder valve (Valve B).
3. Start a stopwatch and measure the milliliters ($\text{mL}$) drawn down over a known time (typically $1.0\text{ or } 2.0\text{ minutes}$).
4. Calculate the pump pumping rate in $\text{mL/min}$: $\text{Pump Rate (mL/min)} = \frac{\text{Milliliters Consumed (mL)}}{\text{Time Elapsed (minutes)}}$
5. Convert $\text{mL/min}$ to Gallons per Day (gpd): $\text{gpd} = \frac{\text{mL/min} \times 1,440\text{ min/day}}{3,785.4\text{ mL/gal}} = \frac{\text{mL/min}}{2.6288}$ $\text{mL/min} = \text{gpd} \times 2.6288$

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Solution Dilution and Batching ($C_1 V_1 = C_2 V_2$)

When preparing chemical solutions in a day tank (e.g., preparing a working polymer solution or diluting concentrated caustic soda), the mass of active chemical before dilution equals the mass of active chemical after dilution:

$C_1 \times V_1 = C_2 \times V_2$

Where:

• $C_1 = \text{Concentration of stock / source chemical}$ ($\%,\text{ mg/L},\text{ or lbs/gal}$)
• $V_1 = \text{Volume of stock / source chemical needed}$
• $C_2 = \text{Target concentration of final prepared solution}$
• $V_2 = \text{Total volume of final prepared solution}$

Rearranging to solve for stock chemical volume:

$V_1 = \frac{C_2 \times V_2}{C_1}$ $\text{Dilution Water Volume} = V_2 - V_1$

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Dry Chemical Feeder Calibration: The Catch-and-Weigh Method

Volumetric and gravimetric dry chemical feeders (screw feeders, belt feeders, and oscillating hoppers) deliver dry alum, hydrated lime, or soda ash. Calibration requires physically collecting the discharged chemical over a timed interval:

$\text{Feed Rate (lbs/min)} = \frac{\text{Net Weight Collected (lbs)}}{\text{Collection Time (minutes)}}$ $\text{Feed Rate (lbs/hr)} = \text{Feed Rate (lbs/min)} \times 60\text{ min/hr}$ $\text{Feed Rate (lbs/day)} = \text{Feed Rate (lbs/hr)} \times 24\text{ hr/day}$

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Step-by-Step Dosing Calculations

Worked Example 1: Disinfection with Calcium Hypochlorite

A remote community water system in eastern Oregon treats an average daily well flow of $350,000\text{ gpd}$ ($0.35\text{ MGD}$). The target free chlorine residual after a 30-minute contact period is $1.2\text{ mg/L}$, and water testing indicates a chlorine demand of $0.8\text{ mg/L}$. The utility uses granular calcium hypochlorite containing $65\%$ available chlorine. Calculate the daily chemical feed rate in pounds per day.

1. Calculate total chlorine dosage required (Dosage $=$ Demand $+$ Residual): $\text{Dosage} = 0.8\text{ mg/L} + 1.2\text{ mg/L} = 2.0\text{ mg/L}$
2. Calculate pure chlorine required using the pounds formula: $\text{Pure } \text{Cl}_2 \text{ (lbs/day)} = 0.35\text{ MGD} \times 2.0\text{ mg/L} \times 8.34\text{ lbs/gal} = 5.838\text{ lbs/day}$
3. Adjust for $65\%$ product purity: $\text{Commercial Feed Rate} = \frac{5.838\text{ lbs/day}}{0.65} = 8.98\text{ lbs/day}$

Worked Example 2: Liquid Sodium Hypochlorite Feed & Drawdown Calibration

A municipal wastewater treatment plant discharges $2.8\text{ MGD}$ of treated effluent and must maintain a chlorine dosage of $3.5\text{ mg/L}$. The plant doses $12.5\%$ sodium hypochlorite with a specific gravity of $1.20$. Calculate:

1. The daily feed rate in gallons per day (gpd).
2. The drawdown pump calibration rate in milliliters per minute (mL/min).

Step 1: Calculate pure chlorine required: $\text{Pure } \text{Cl}_2 \text{ (lbs/day)} = 2.8\text{ MGD} \times 3.5\text{ mg/L} \times 8.34\text{ lbs/gal} = 81.732\text{ lbs/day}$

Step 2: Determine active chlorine per gallon of bleach: $\text{Weight per gallon} = 1.20 \times 8.34\text{ lbs/gal} = 10.008\text{ lbs/gal}$ $\text{Active } \text{Cl}_2 \text{ per gallon} = 10.008\text{ lbs/gal} \times 0.125 = 1.251\text{ lbs active } \text{Cl}_2\text{/gal}$

Step 3: Calculate liquid feed rate in gallons per day: $\text{Liquid Feed (gpd)} = \frac{81.732\text{ lbs/day}}{1.251\text{ lbs/gal}} = 65.33\text{ gpd}$

Step 4: Convert to drawdown rate in mL/min: $\text{Feed Rate (mL/min)} = \frac{65.33\text{ gal/day} \times 3,785.4\text{ mL/gal}}{1,440\text{ min/day}} = \frac{247,300.2}{1,440} = 171.74\text{ mL/min}$

Worked Example 3: Fluoridation with Hydrofluorosilicic Acid

A conventional surface water filtration plant treats $4.2\text{ MGD}$. Ambient raw water contains $0.10\text{ mg/L}$ of natural fluoride ion. The plant targets the OHA/CDC recommended optimal fluoride concentration of $0.70\text{ mg/L}$ using commercial hydrofluorosilicic acid ($\text{H}_2\text{SiF}_6$). The commercial acid solution has a specific gravity of $1.23$, an acid concentration of $24\%$ by weight, and contains $79.2\%$ fluoride ion in the pure acid molecule ($19.0\%$ net available fluoride ion by weight). Calculate the required chemical feed rate in gallons per day.

1. Determine required dosage increase: $\text{Dose Increase} = 0.70\text{ mg/L} - 0.10\text{ mg/L} = 0.60\text{ mg/L}$
2. Calculate pure fluoride mass required: $\text{Pure } \text{F}^- \text{ (lbs/day)} = 4.2\text{ MGD} \times 0.60\text{ mg/L} \times 8.34\text{ lbs/gal} = 21.017\text{ lbs/day}$
3. Determine solution weight and net available fluoride per gallon: $\text{Solution Weight} = 1.23 \times 8.34\text{ lbs/gal} = 10.258\text{ lbs/gal}$ $\text{Available } \text{F}^- \text{ per gallon} = 10.258\text{ lbs/gal} \times 0.190 = 1.949\text{ lbs } \text{F}^-\text{/gal}$
4. Calculate liquid acid feed rate: $\text{Liquid Feed (gpd)} = \frac{21.017\text{ lbs/day}}{1.949\text{ lbs/gal}} = 10.78\text{ gpd}$

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Practical Operator Scenarios & Exam Pitfalls

• Multiplying vs. Dividing by Purity: If a problem requires $100\text{ lbs}$ of pure chemical and the product is $65\%$ pure, multiplying $100 \times 0.65 = 65\text{ lbs}$ is an automatic error trap. Feeding $65\text{ lbs}$ provides only $42.25\text{ lbs}$ of active ingredient. Always divide: $100 / 0.65 = 153.8\text{ lbs}$.
• Forgetting Specific Gravity in Liquid Calculations: Water weighs $8.34\text{ lbs/gal}$, but chemical solutions are denser. Failing to multiply by specific gravity underestimates solution weight and causes inaccurate chemical feed settings.
• Drawdown Calibration Time Errors: If a drawdown test is conducted for $2\text{ minutes}$ and yields $300\text{ mL}$, remember to divide by $2$ to obtain $150\text{ mL/min}$ before calculating daily feed rates.