19.2 Solution Dosing, Specific Gravity & Pump Drawdown Math

Liquid Chemical Solutions, Density, and Specific Gravity

While dry chemicals are quantified directly on scales or volumetric hoppers, modern drinking water treatment plants increasingly feed liquid chemicals. Common liquid chemical solutions include liquid aluminum sulfate (alum), ferric chloride ($\text{FeCl}_3$), sodium hypochlorite ($\text{NaOCl}$), sodium hydroxide (caustic soda, $\text{NaOH}$), and hydrofluorosilicic acid ($\text{H}_2\text{SiF}_6$). Liquid chemicals eliminate airborne dust hazards, dissolve immediately upon injection, and are easily metered using automated positive displacement diaphragm or peristaltic pumps.

           +-------------------------------------------------------+
           |                Specific Gravity (SG)                  |
           |             (Density relative to water)               |
           +-------------------------------------------------------+
                                      |
                                      v  × 8.34 lb/gal
           +-------------------------------------------------------+
           |            Solution Density (lb/gal)                  |
           +-------------------------------------------------------+
                                      |
                                      v  × Concentration Decimal (%)
           +-------------------------------------------------------+
           |       Active Chemical per Gallon (lb active/gal)      |
           +-------------------------------------------------------+
                                      |
                                      v  lb/day required ÷ lb active/gal
           +-------------------------------------------------------+
           |            Liquid Feed Rate (gal/day)                 |
           +-------------------------------------------------------+

Unlike water, liquid chemical solutions have densities greater than $8.34\ lb/gal$. Calculating liquid dosage requires determining two physical properties:

1. The density of the bulk solution in pounds per gallon ($lb/gal$).
2. The mass of active chemical constituent contained within each gallon of solution.

Specific Gravity (SG) Defined

Specific Gravity ($SG$) is a dimensionless ratio comparing the mass density of a substance to the mass density of pure water at standard reference temperature ($4^\circ\text{C}$ or $39.2^\circ\text{F}$, where water density is $8.34\ lb/gal$):

$\text{Specific Gravity }(SG) = \frac{\text{Density of Liquid Solution }(lb/gal)}{\text{Density of Water }(8.34\ lb/gal)}$

Multiplying the specific gravity of any chemical by $8.34\ lb/gal$ yields the actual density of that liquid solution:

$\text{Solution Density }(lb/gal) = SG \times 8.34\ lb/gal$

A liquid chemical with an $SG$ of $1.33$ weighs $1.33 \times 8.34 = 11.09\ lb/gal$. A concentrated solution with an $SG$ of $1.53$ ($50%$ caustic soda) weighs $1.53 \times 8.34 = 12.76\ lb/gal$.

Determining Active Chemical Weight per Gallon

Commercial liquid chemicals are manufactured and invoiced as percentage concentrations by weight. A gallon of solution contains both active chemical molecules and water. Multiplying the total weight of one gallon of solution by the decimal concentration yields the pounds of active chemical per gallon:

$\text{Active Chemical Concentration }(lb/gal) = \text{Solution Density }(lb/gal) \times \text{Concentration Decimal}$

$\text{Active Chemical Concentration }(lb/gal) = SG \times 8.34\ lb/gal \times \text{Concentration Decimal}$

Standard Liquid Solution Calculations:

1. Commercial Liquid Alum ($48.5%$ dry alum basis, $SG = 1.33$):
   $\text{Bulk Solution Density} = 1.33 \times 8.34\ lb/gal = 11.092\ lb/gal$ $\text{Active Dry Alum per Gallon} = 11.092\ lb/gal \times 0.485 = \mathbf{5.380\ lb\text{ dry alum/gal}}$ (Operators standardly use $5.38\ lb\text{ dry alum/gal}$ for liquid alum calculations).

2. Commercial Sodium Hypochlorite ($12.5%$ available chlorine, $SG = 1.20$):
   $\text{Bulk Solution Density} = 1.20 \times 8.34\ lb/gal = 10.008\ lb/gal$ $\text{Available Chlorine per Gallon} = 10.008\ lb/gal \times 0.125 = \mathbf{1.251\ lb\ Cl_2/gal}$ (Operators standardly use $1.25\ lb\text{ available }\text{Cl}_2/gal$ for industrial strength bleach).

3. Commercial Ferric Chloride ($40.0%,\text{FeCl}_3$, $SG = 1.42$):
   $\text{Bulk Solution Density} = 1.42 \times 8.34\ lb/gal = 11.843\ lb/gal$ $\text{Active }\text{FeCl}_3\text{ per Gallon} = 11.843\ lb/gal \times 0.400 = \mathbf{4.737\ lb\,\text{FeCl}_3/gal}$

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Liquid Chemical Feed Rate and Unit Conversion Mathematics

To operate a liquid chemical metering pump, the daily mass requirement ($lb/day$) calculated from the Pounds Formula must be converted into a volumetric pumping rate: Gallons per Day ($gpd$), Gallons per Hour ($gph$), or Milliliters per Minute ($mL/min$).

1. Determining Liquid Feed Rate in Gallons per Day ($gpd$)

Dividing daily pure chemical mass demand by active concentration per gallon yields volumetric feed rate in gallons per day:

$\text{Liquid Feed Rate }(gpd) = \frac{\text{Pure Chemical Required }(lb/day)}{\text{Active Chemical Concentration }(lb/gal)}$

$\text{Liquid Feed Rate }(gpd) = \frac{\text{Flow }(MGD) \times \text{Dose }(mg/L) \times 8.34\ lb/gal}{SG \times 8.34\ lb/gal \times \text{Concentration Decimal}}$

Notice that the water density constant $8.34$ cancels in the numerator and denominator, simplifying the direct relationship to:

$\text{Liquid Feed Rate }(gpd) = \frac{\text{Flow }(MGD) \times \text{Dose }(mg/L)}{SG \times \text{Concentration Decimal}}$

Exam Tip: While the $8.34$ constant algebraically cancels when solving directly for $gpd$, working through the intermediate steps—first calculating $lb/day$ and then dividing by $lb/gal$—is strongly recommended on certification exams to prevent conceptual confusion and ensure correct units.

2. Converting Gallons per Day ($gpd$) to Gallons per Hour ($gph$)

Chemical metering pump capacities and stroke dials are typically calibrated in gallons per hour ($gph$). Dividing daily gallons by 24 hours yields hourly pump requirements:

$\text{Feed Rate }(gph) = \frac{\text{Feed Rate }(gpd)}{24\ hr/day}$

3. Converting Gallons per Day ($gpd$) to Milliliters per Minute ($mL/min$)

When calibrating small chemical pumps or verifying delivery using a graduated cylinder, operators measure delivery in milliliters per minute ($mL/min$). Establishing the mathematical conversion requires two volume-time equivalents:

• $1.0\text{ gallon} = 3,785.41\text{ milliliters } (mL)$ (standardized on exams as $3,785\ mL/gal$).
• $1.0\text{ day} = 24\text{ hr} \times 60\text{ min/hr} = 1,440\text{ minutes}$.

Synthesizing these factors yields the universal conversion factor:

$\text{Feed Rate }(mL/min) = \frac{\text{Feed Rate }(gpd) \times 3,785\ mL/gal}{1,440\ min/day} = \text{Feed Rate }(gpd) \times 2.62847$

$\mathbf{\text{Feed Rate }(mL/min) \approx \text{Feed Rate }(gpd) \times 2.628}$

To convert directly from gallons per hour ($gph$) to milliliters per minute ($mL/min$):

$\text{Feed Rate }(mL/min) = \frac{\text{Feed Rate }(gph) \times 3,785\ mL/gal}{60\ min/hr} = \text{Feed Rate }(gph) \times 63.083$

$\mathbf{\text{Feed Rate }(mL/min) \approx \text{Feed Rate }(gph) \times 63.09}$

Conversely, to convert a measured $mL/min$ rate back to gallons per day ($gpd$):

$\text{Feed Rate }(gpd) = \frac{\text{Feed Rate }(mL/min)}{2.628} = \text{Feed Rate }(mL/min) \times \frac{1,440}{3,785} \approx \mathbf{\text{Feed Rate }(mL/min) \times 0.380}$

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Metering Pump Drawdown Calibration Cylinder Verification

Chemical metering pumps (mechanical diaphragm, hydraulic diaphragm, peristaltic, or progressive cavity) are subject to mechanical wear, check valve fouling, suction head variation, discharge line backpressure, and chemical off-gassing (such as sodium hypochlorite vapor-locking). Operators must verify pump delivery using a drawdown calibration cylinder.

A calibration cylinder is a clear, vertically mounted cylinder graduated in milliliters ($mL$), installed on the suction manifold of the metering pump between the chemical storage tank and the pump inlet.

Standard Calibration Procedure

1. Open the isolation valve to fill the calibration cylinder with chemical solution from the bulk storage tank.
2. Isolate the bulk storage tank by closing the tank suction valve, forcing the chemical feed pump to draw liquid exclusively from the graduated cylinder.
3. With the pump operating at its designated stroke and speed settings, start a stopwatch as the liquid meniscus passes an initial reference mark ($0\ mL$ or an even increment).
4. Allow the pump to draw down the chemical for exactly 60 seconds (1.0 minute) or a known time duration.
5. Record the total volume displaced in milliliters ($mL$).
6. Re-open the storage tank isolation valve and close the calibration cylinder valve.

Calculating Actual Delivery and Percent Error

The measured drawdown volume in $1.0\text{ minute}$ represents the pump's actual delivery in $mL/min$. Converting to daily delivery:

$\text{Actual Feed Rate }(gpd) = \frac{\text{Drawdown Volume }(mL) \times 1,440\ min/day}{\text{Test Duration }(min) \times 3,785\ mL/gal}$

$\text{Actual Feed Rate }(gpd) = \text{Drawdown }(mL/min) \times 0.3804$

To evaluate pump accuracy, calculate the Percent Error between actual delivery and expected setpoint:

$\%\text{ Error} = \left( \frac{\text{Actual Feed Rate} - \text{Expected Feed Rate}}{\text{Expected Feed Rate}} \right) \times 100$

• Acceptable Operating Limit: Most regulatory agencies and utility standard operating procedures require chemical feed pumps to deliver within $\pm 5%\text{ to }\pm 10%$ of target setpoints.
• Negative Error (Underfeeding): If actual delivery is significantly below expected, inspect for clogged suction strainers, worn pump diaphragm or tubing, debris lodged in suction/discharge check valve balls, vapor lock from chemical off-gassing, or excessive suction lift.
• Positive Error (Overfeeding): If actual delivery significantly exceeds expected, inspect for siphoning past check valves caused by low discharge line pressure, a torn anti-siphon valve diaphragm, or improperly adjusted backpressure valves.

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Step-by-Step Worked Practice Calculations

Example 1: Liquid Alum Feed Rate Determination

Problem Statement: A conventional surface water plant treats a steady flow of $6.0\text{ MGD}$. The operator sets the coagulant dose to $24.0\text{ mg/L}$ based on streaming current and jar test data. The utility receives commercial liquid alum with a specific gravity of $1.34$ and an active concentration of $49.0%$ dry aluminum sulfate by weight. Calculate:

1. The dry alum mass requirement in pounds per day ($lb/day$).
2. The active chemical content per gallon of liquid alum ($lb/gal$).
3. The liquid alum feed rate in gallons per day ($gpd$) and gallons per hour ($gph$).
4. The expected pump calibration delivery in milliliters per minute ($mL/min$).

Solution Procedure:

• Step 1: Calculate Pure Dry Alum Mass Demand
  $\text{Feed Rate }(lb/day) = 6.0\ MGD \times 24.0\ mg/L \times 8.34\ lb/gal = 1,200.96\ lb\text{ dry alum/day}$

• Step 2: Determine Active Alum Content per Gallon
  $\text{Solution Density} = 1.34 \times 8.34\ lb/gal = 11.1756\ lb/gal$ $\text{Active Dry Alum} = 11.1756\ lb/gal \times 0.490 = 5.4760\ lb\text{ dry alum/gal}$

• Step 3: Calculate Volumetric Liquid Feed Rate
  $\text{Liquid Feed Rate }(gpd) = \frac{1,200.96\ lb/day}{5.4760\ lb/gal} = 219.313\ gpd \approx 219.3\ gpd$ $\text{Liquid Feed Rate }(gph) = \frac{219.313\ gpd}{24\ hr/day} = 9.138\ gph \approx 9.14\ gph$

• Step 4: Convert to Pump Calibration Rate in $mL/min$
  $\text{Delivery }(mL/min) = 219.313\ gpd \times 2.62847 = 576.45\ mL/min \approx 576.5\ mL/min$ (Verification: $9.138\ gph \times 63.09 = 576.5\ mL/min$).

Example 2: Liquid Ferric Chloride Feed Rate

Problem Statement: A treatment plant dosing ferric chloride ($\text{FeCl}_3$) treats a flow of $3.5\text{ MGD}$ at an applied dosage of $15.0\text{ mg/L}$. Bulk ferric chloride solution is delivered at $40.0%$ concentration by weight with a specific gravity of $1.42$. Calculate the required volumetric liquid feed rate in gallons per day ($gpd$).

Solution Procedure:

• Step 1: Calculate Pure Active $\text{FeCl}_3$ Required ($lb/day$)
  $\text{Feed Rate }(lb/day) = 3.5\ MGD \times 15.0\ mg/L \times 8.34\ lb/gal = 437.85\ lb/day$

• Step 2: Calculate Active $\text{FeCl}_3$ per Gallon
  $\text{Solution Density} = 1.42 \times 8.34\ lb/gal = 11.8428\ lb/gal$ $\text{Active }\text{FeCl}_3\text{ per Gallon} = 11.8428\ lb/gal \times 0.400 = 4.7371\ lb\,\text{FeCl}_3/gal$

• Step 3: Calculate Daily Volumetric Liquid Feed Rate
  $\text{Liquid Feed Rate }(gpd) = \frac{437.85\ lb/day}{4.7371\ lb/gal} = 92.4299\dots\ gpd \approx 92.4\ gpd$

Example 3: Drawdown Calibration Verification and Percent Error

Problem Statement: An operator sets a hypochlorite metering pump to deliver $45.0\text{ gallons per day}$ of $12.5%$ sodium hypochlorite into a finished water transmission main. During a scheduled pump calibration drawdown test, the liquid level in the suction cylinder drops exactly $130\text{ mL}$ in $60\text{ seconds}$. Calculate:

1. The expected drawdown delivery rate in $mL/min$.
2. The actual delivered feed rate in gallons per day ($gpd$).
3. The percent error of the pump setpoint.

Solution Procedure:

• Step 1: Calculate Expected Delivery in $mL/min$
  $\text{Expected Rate }(mL/min) = 45.0\ gpd \times 2.62847 = 118.28\ mL/min \approx 118.3\ mL/min$

• Step 2: Calculate Actual Pumping Rate in $gpd$ from Measured Drawdown
  $\text{Actual Rate }(gpd) = \frac{130\ mL/min \times 1,440\ min/day}{3,785\ mL/gal} = \frac{187,200}{3,785} = 49.458\ gpd \approx 49.5\ gpd$

• Step 3: Calculate Percent Error
  $\%\text{ Error} = \left( \frac{\text{Actual} - \text{Expected}}{\text{Expected}} \right) \times 100 = \left( \frac{49.46 - 45.00}{45.00} \right) \times 100$ $\%\text{ Error} = \left( \frac{+4.46}{45.00} \right) \times 100 = +9.91\% \approx +9.9\%$
  (The pump is overfeeding by $9.9%$. The operator should inspect the anti-siphon valve for proper seating and adjust the pump stroke/speed).

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Reference Tables: Solution Densities, Multipliers, and Calibration Troubleshooting

Table 19.2.1: Liquid Chemical Densities and Active Concentrations

Table 19.2.2: Liquid Chemical Feed Conversion Multipliers

Table 19.2.3: Chemical Metering Pump Calibration Troubleshooting Matrix