23.1 Chemical Feed Calculations & Dosage Problem Solving

23.1 Chemical Feed Calculations & Dosage Problem Solving

[!NOTE] Core Competency Foundation: Mathematical proficiency is among the most heavily weighted domains on Arizona Department of Environmental Quality (ADEQ) operator certification examinations for Water Treatment, Water Distribution, Wastewater Treatment, and Wastewater Collection. Every chemical addition process—coagulation, pH adjustment, fluoridation, chlorination, and dechlorination—depends on an operator's ability to accurately calculate dosages and feeder settings.

In water and wastewater treatment facilities, adding too little chemical leads to regulatory violations, pathogen breakthrough, or permit exceedances. Adding too much chemical wastes utility revenue, risks hazardous disinfection byproduct (DBP) formation, or causes severe chemical burns and toxic water quality conditions. Operators must execute multi-step dosage calculations with zero error under demanding field conditions.

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The Universal Pounds Formula

The most important mathematical relationship in water and wastewater engineering is the Pounds Formula (also known as the mass-loading equation). It links volumetric flow rate, chemical concentration (dosage), and mass:

$\text{Pounds per Day (lbs/day)} = \text{Flow Rate (MGD)} \times \text{Concentration (mg/L)} \times 8.34 \text{ lbs/gal}$

Dimensional Derivation of 8.34

To understand why this formula works, consider the underlying dimensional units:

1. One liter of pure water has a mass of exactly $1,000\text{ grams} = 1,000,000\text{ milligrams}$. Therefore, a concentration of $1\text{ mg/L}$ equals one milligram of chemical in one million milligrams of water: one part per million (1 ppm).
2. One gallon of water weighs 8.34 pounds at standard temperature ($20^\circ\text{C}$ / $68^\circ\text{F}$).
3. One million gallons (1 MG) of water weighs: $1,000,000 \text{ gal} \times 8.34 \text{ lbs/gal} = 8,340,000 \text{ lbs} = 8.34 \text{ million pounds}$
4. If water contains $1\text{ mg/L}$ of a substance, each million pounds of water contains $1\text{ pound}$ of that substance. Consequently, a flow of $1.0\text{ MGD}$ carrying $1.0\text{ mg/L}$ of chemical delivers exactly $8.34\text{ pounds}$ of pure chemical per day.

The Davidson Pie Chart

Operators frequently visualize this formula using the "Davidson Pie" or formula wheel:

          +-------------------------------+
          |          POUNDS/DAY           |
          +---------------+---------------+
          |   Flow (MGD)  | Dosage (mg/L) |
          |               |  * 8.34       |
          +---------------+---------------+

• To solve for Pounds/Day: Multiply $\text{Flow (MGD)} \times \text{Dosage (mg/L)} \times 8.34$.
• To solve for Dosage (mg/L): Divide $\text{Pounds/Day} / [\text{Flow (MGD)} \times 8.34]$.
• To solve for Flow (MGD): Divide $\text{Pounds/Day} / [\text{Dosage (mg/L)} \times 8.34]$.

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Essential Flow & Hydraulic Conversions

Exam questions rarely present flow directly in Million Gallons per Day (MGD). Operators must convert seamlessly between flow units before applying the pounds formula.

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Dry Chemical Feeder Calculations & Purity Adjustments

Dry chemicals—such as hydrated lime ($\text{Ca(OH)}_2$), calcium hypochlorite ($65%\text{ HTH}$), soda ash ($\text{Na}_2\text{CO}_3$), and dry polymer—are rarely $100%$ pure active chemical. When calculating feed rates for dry chemicals, the pounds of pure active chemical required must be adjusted upward to account for chemical purity.

$\text{Dry Feeder Setting (lbs/day)} = \frac{\text{Pure Chemical Required (lbs/day)}}{\text{Chemical Purity (as a decimal)}}$

Step-by-Step Worked Example: Dry Chemical Feed

Problem: An Arizona surface water treatment plant treating Central Arizona Project (CAP) water operates at a flow rate of $3.6\text{ MGD}$. Jar testing indicates an optimal hydrated lime dose of $14.0\text{ mg/L}$ for corrosion control and pH adjustment. The utility purchases commercial hydrated lime certified at $92%$ purity. Calculate the required dry chemical feeder setting in pounds per day.

• Step 1: Calculate pure chemical demand per day $\text{Pure Lime (lbs/day)} = 3.6 \text{ MGD} \times 14.0 \text{ mg/L} \times 8.34 \text{ lbs/gal} = 420.336 \text{ lbs/day}$
• Step 2: Adjust for chemical purity $\text{Commercial Lime Feed Rate} = \frac{420.336 \text{ lbs/day}}{0.92} = 456.89 \approx 456.9 \text{ lbs/day}$
• Step 3: Hourly feed rate (for gravimetric or volumetric feeder calibration) $\text{Hourly Feed Rate} = \frac{456.9 \text{ lbs/day}}{24 \text{ hrs/day}} = 19.04 \text{ lbs/hr}$

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Liquid Chemical Feed Calculations: Specific Gravity & Concentration

Liquid chemicals—such as liquid alum ($48.5%$), ferric chloride ($40%$), sodium hypochlorite ($12.5%$ trade bleach), and sodium hydroxide ($50%\text{ NaOH}$)—require a two-step adjustment because they are solutions rather than pure solids:

1. Specific Gravity (SG): The density of the liquid relative to water ($1.00$). The weight of one gallon of the solution equals $8.34 \times \text{SG}$.
2. Concentration / Percent Active Strength: The fraction of the liquid solution that is the actual active chemical.

$\text{Pounds of Solution per Gallon} = 8.34 \text{ lbs/gal} \times \text{Specific Gravity (SG)}$ $\text{Pounds of Pure Active Chemical per Gallon} = 8.34 \text{ lbs/gal} \times \text{SG} \times \text{Concentration (decimal)}$

Once the pounds of active chemical per gallon are determined, the liquid volumetric feed rate is calculated:

$\text{Liquid Feed Rate (Gallons per Day, GPD)} = \frac{\text{Pounds Pure Chemical Required/day}}{\text{Pounds Pure Chemical per Gallon}}$

Converting GPD to Milliliters per Minute (mL/min)

Chemical metering pumps (peristaltic or diaphragm) are calibrated in milliliters per minute (mL/min). The mathematical conversion between GPD and mL/min is:

$\text{Feed Rate (mL/min)} = \frac{\text{Feed Rate (GPD)} \times 3,785 \text{ mL/gal}}{1,440 \text{ min/day}} = \text{Feed Rate (GPD)} \times 2.6285$

[!TIP] The 2.6285 Constant: Memorize the conversion factor $2.6285$. Multiplying GPD by $2.6285$ yields mL/min directly. Dividing mL/min by $2.6285$ converts back to GPD immediately, saving valuable calculation time during examination testing.

Step-by-Step Worked Example: Liquid Sodium Hypochlorite Feed

Problem: A groundwater well in Pinal County pumps at $850\text{ GPM}$ directly into the distribution system. The ADEQ-approved disinfection protocol requires a free chlorine dosage of $1.8\text{ mg/L}$. The utility utilizes $12.5%$ commercial sodium hypochlorite (NaOCl) solution with a specific gravity of $1.20$. Determine:

1. The daily pure chlorine demand in lbs/day.
2. The available pounds of chlorine per gallon of bleach.
3. The liquid feed rate in Gallons per Day (GPD).
4. The chemical metering pump calibration rate in mL/min.

• Step 1: Convert flow from GPM to MGD $\text{Flow (MGD)} = \frac{850 \text{ GPM} \times 1,440 \text{ min/day}}{1,000,000} = 1.224 \text{ MGD}$
• Step 2: Calculate daily pure chlorine requirement $\text{Pounds Pure } \text{Cl}_2\text{/day} = 1.224 \text{ MGD} \times 1.8 \text{ mg/L} \times 8.34 \text{ lbs/gal} = 18.375 \text{ lbs/day}$
• Step 3: Calculate pounds of active chlorine per gallon of liquid bleach $\text{Pounds Active } \text{Cl}_2\text{/gal} = 8.34 \text{ lbs/gal} \times 1.20 \text{ (SG)} \times 0.125 \text{ (strength)} = 1.251 \text{ lbs/gal}$
• Step 4: Calculate liquid feed rate in GPD $\text{Liquid Feed Rate} = \frac{18.375 \text{ lbs/day}}{1.251 \text{ lbs/gal}} = 14.688 \approx 14.69 \text{ GPD}$
• Step 5: Convert GPD to mL/min $\text{Pump Delivery Rate} = 14.688 \text{ GPD} \times 2.6285 = 38.61 \approx 38.6 \text{ mL/min}$

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Metering Pump Calibration: The Drawdown Cylinder Test

Regardless of manufacturer pump curves or digital stroke displays, chemical metering pumps must be physically calibrated under actual system head conditions using a graduated cylinder drawdown test.

   +------------------+          +-----------------------------------+
   | Chemical Day Tank|          |        Drawdown Cylinder          |
   +--------+---------+          +-----------------+-----------------+
            |                                      |                  
       [Valve A]                              [Valve B]               
            |                                      |                  
            +------------------+-------------------+                  
                               |                                      
                               v                                      
                    [Chemical Metering Pump]                          
                               |                                      
                               v To Injection Quill                   

Drawdown Calibration Protocol

1. Fill a graduated calibration column connected directly to the pump suction piping.
2. Close Valve A (isolating the main bulk or day tank) and open Valve B (feeding strictly from the drawdown cylinder).
3. Operate the metering pump at the desired stroke and speed setting.
4. Using a precision stopwatch, measure the exact volume of chemical drawn down (in milliliters) over a measured period (typically 60 seconds).
5. Calculate actual flow output:

$\text{Actual Feed Rate (mL/min)} = \frac{\text{Volume Drawn Down (mL)}}{\text{Time Elapsed (seconds)}} \times 60 \text{ sec/min}$ $\text{Actual Feed Rate (GPD)} = \frac{\text{Actual Feed Rate (mL/min)}}{2.6285} = \frac{\text{mL}}{\text{sec}} \times 22.82$

If the measured delivery differs from the calculated dosage requirement, adjust the pump stroke length or speed setting proportionally:

$\text{New Setting (\%)} = \frac{\text{Desired Feed Rate}}{\text{Actual Measured Feed Rate}} \times \text{Current Setting (\%)}$

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Polymer Solution Preparation & Dilution Kinetics ($C_1V_1 = C_2V_2$)

Flocculant and coagulant aid polymers are typically supplied as concentrated, viscous liquid emulsions ($30%\text{ to }50%$ active polymer) or dry granular powders. They must be prepared into dilute working solutions ($0.10%\text{ to }0.50%$) before injection to prevent uncoiled polymer chains from agglomerating into useless "fish eyes."

Calculations for batching and diluting follow the universal mass-balance dilution formula:

$C_1 \times V_1 = C_2 \times V_2$

Where:

• $C_1$ = Initial concentration of neat chemical (stock solution)
• $V_1$ = Volume of neat chemical required
• $C_2$ = Final concentration of dilute working solution
• $V_2$ = Final total volume of dilute working solution

Step-by-Step Worked Example: Polymer Dilution

Problem: An operator needs to prepare $400\text{ gallons}$ of a $0.25%$ active polymer solution in a mix tank. The neat liquid polymer has a concentration of $40%$ active ingredient and a specific gravity of $1.15$. Assuming water weighs $8.34\text{ lbs/gal}$, how many gallons of neat polymer are needed?

• Step 1: Apply dilution balance equation $C_1 \times V_1 = C_2 \times V_2$ $40\% \times V_1 = 0.25\% \times 400 \text{ gallons}$
• Step 2: Solve for neat volume $V_1$ $V_1 = \frac{0.25 \times 400}{40} = \frac{100}{40} = 2.50 \text{ gallons of neat polymer}$
• Step 3: Verification of water addition $\text{Water Volume Required} = 400 \text{ gal} - 2.5 \text{ gal} = 397.5 \text{ gallons of potable water}$

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Chlorine Chemistry, Dosage, Demand, and Residual

Chlorine is the primary disinfectant used in Arizona drinking water systems and wastewater reclamation facilities. The fundamental relationship governing chlorination is:

$\text{Chlorine Dosage (mg/L)} = \text{Chlorine Demand (mg/L)} + \text{Chlorine Residual (mg/L)}$ $\text{Chlorine Demand (mg/L)} = \text{Chlorine Dosage (mg/L)} - \text{Chlorine Residual (mg/L)}$ $\text{Chlorine Residual (mg/L)} = \text{Chlorine Dosage (mg/L)} - \text{Chlorine Demand (mg/L)}$

Components of Chlorine Chemistry

• Chlorine Dosage: The total concentration of chlorine added to the water.
• Chlorine Demand: The concentration of chlorine consumed by inorganic reducing agents (iron, manganese, hydrogen sulfide, nitrite) and organic compounds (natural organic matter, bacteria, algae) during a specified contact time.
• Chlorine Residual: The concentration of chlorine remaining in the water after the demand has been satisfied.
• Total Chlorine Residual: Composed of two distinct chemical forms: $\text{Total Chlorine} = \text{Free Available Chlorine} + \text{Combined Available Chlorine}$
  • Free Available Chlorine: Hypochlorous acid ($\text{HOCl}$) and hypochlorite ion ($\text{OCl}^-$). $\text{HOCl}$ is $80\text{ to }100$ times more powerful as a disinfectant than $\text{OCl}^-$. At higher pH values ($>7.5$), the equilibrium shifts toward the weaker $\text{OCl}^-$ ion.
  • Combined Available Chlorine: Chloramines (monochloramine, dichloramine, trichloramine) formed when chlorine reacts with ammonia nitrogen.

Arizona Environmental Challenges

In Arizona, high ambient summer temperatures exceeding $40^\circ\text{C}$ ($104^\circ\text{F}$) heat raw surface water and shallow distribution pipes to temperatures over $30^\circ\text{C}$ ($86^\circ\text{F}$). Elevated water temperatures accelerate chemical reaction kinetics, drastically increasing chlorine demand and speeding the decay of free chlorine residuals in storage reservoirs. Operators must balance higher dosages to maintain the ADEQ-mandated minimum residual ($0.2\text{ mg/L}$ free chlorine or $0.5\text{ mg/L}$ total chloramine throughout the entire distribution system) while staying below the Maximum Residual Disinfectant Level (MRDL) of $4.0\text{ mg/L}$ to minimize trihalomethane (THM) and haloacetic acid (HAA5) disinfection byproducts.