Scientific & Mathematical Fundamentals
1. Engineering Principles & Definitions
In municipal potable drinking water plants, industrial effluent wastewater remediation, and reverse osmosis (RO) desalination facilities, chemical reagents (such as liquid alum, polyaluminum chloride, sodium hypochlorite, and caustic soda) are delivered via positive displacement metering diaphragm pumps. Sizing the dosing pump throughput requires converting target concentration (parts per million or mg/L) into volumetric delivery factoring in commercial solution purity and specific gravity.
2. Mathematical Formulas & Governing Equations
The chemical feed rate and positive displacement pump flow rate are governed by mass conservation:
\[ \dot{m}_{pure} = Q_{\text{water}} \times C_{\text{ppm}} \times 10^{-3} \text{ (kg/hr)} \]\[ Q_{\text{pump}} (\text{L/hr}) = \frac{\dot{m}_{pure}}{\left(\frac{\text{Purity}\%}{100} \right) \times \rho_{\text{solution}}} \]3. Reference Standards & Data Table
| Chemical Name | Standard Form | Typical Commercial Purity | Specific Gravity (g/cm³) | Target Function |
|---|---|---|---|---|
| Alum (Aluminum Sulfate) | Liquid Solution | 48%: 50% | 1.32: 1.34 | Primary coagulant for turbidity removal |
| Sodium Hypochlorite (Bleach) | Liquid Solution | 10%: 15% | 1.18: 1.22 | Disinfection & biological oxidation |
| Sodium Hydroxide (Caustic Soda) | Liquid Solution | 50% | 1.52 | pH elevation & alkalinity adjustment |
| Sulfuric Acid | Concentrated Liquid | 93%: 98% | 1.83 | pH reduction & RO acid dosing |
| Polyaluminum Chloride (PAC) | Liquid Solution | 30%: 32% | 1.20: 1.25 | High-efficiency flocculant |
📐 Field Case Study: Industrial Wastewater Sodium Hypochlorite Dosing
Real-World Chemical Engineering ProblemAn industrial wastewater facility discharges 1,500 m³/day requiring a continuous active chlorine disinfectant dose of 4.0 mg/L (ppm).
The stream requires 6.00 kg of pure active chlorine element daily.
The site uses bulk commercial liquid sodium hypochlorite supplied at 12.5% active chlorine by mass.
At solution specific gravity of 1.21 kg/L, calibrate the positive-displacement metering pump to 27.55 mL/min (1.65 L/hr). Size delivery piping with our Pipe Sizing Calculator and explore our Chemical Engineering Hub.
Principles of Stoichiometric Chemical Feed & Dosing Engineering
In municipal water treatment, industrial effluent neutralization, and cooling tower biocidal maintenance, chemical metering pumps inject coagulants, oxidants, and pH buffers at precise volumetric delivery rates. Accurate metering prevents either treatment failure (turbidity breakthrough, pathogen survival) or hazardous over-dosing (trihalomethane disinfection byproduct formation, chemical wasting).
Continuous Active Chemical Feed Rate Formulation
The universal mass-balance equation linking target chemical dosage ($D$ in $\text{mg/L}$ or $\text{ppm}$) to continuous volumetric plant process flow ($Q$) is derived from mass conservation:
Where:
- $Q$ = Volumetric raw fluid process flow rate ($\text{m}^3/\text{hr}$ or $\text{L/min}$).
- $D$ = Target analytical dosage in parts per million ($\text{mg/L}$).
- $C_{\%}$ = Active chemical purity percentage by weight ($\%\text{ w/w}$, e.g. $12.5\%$ for commercial sodium hypochlorite bleach, $40\%$ for ferric chloride).
- $ ho_{\text{chem}}$ = Specific gravity or liquid density of the chemical solution ($\text{kg/L}$ or $\text{g/mL}$).
Coagulation Chemistry: Alum vs Ferric Metal Salts
Primary chemical coagulation neutralizes negative zeta potentials on colloidal clay, silt, and humic acid particles. When aluminum sulfate ($\text{Al}_2(\text{SO}_4)_3 \cdot 14\text{H}_2\text{O}$) dissolves in water, it consumes natural bicarbonate alkalinity according to the stoichiometric reaction:
Every $1.0\text{ mg/L}$ of commercial alum added to water consumes approximately $0.50\text{ mg/L}$ of natural alkalinity expressed as $\text{CaCO}_3$. If baseline water alkalinity drops below $30\text{ mg/L}$, supplemental hydrated lime ($\text{Ca}(\text{OH})_2$) or soda ash ($\text{Na}_2\text{CO}_3$) dosing must be metered concurrently to maintain optimum coagulation pH ($6.2 - 6.8$).
| Chemical Reagent | Primary Water Treatment Purpose | Standard Solution Purity | Solution Density (kg/L) | Stoichiometric Alkalinity Impact |
|---|---|---|---|---|
| Sodium Hypochlorite ($\text{NaOCl}$) | Primary disinfection, bio-oxidation | $12.5\%\text{ w/w}$ | 1.20 kg/L | Slightly basic; increases alkalinity |
| Aluminum Sulfate (Alum) | Colloidal destabilization, coagulation | $48.5\%\text{ liquid}$ | 1.32 kg/L | Consumes $0.50\text{ mg/L CaCO}_3$ per mg Alum |
| Ferric Chloride ($\text{FeCl}_3$) | Coagulation, phosphorus precipitation | $40.0\%\text{ liquid}$ | 1.43 kg/L | Consumes $0.92\text{ mg/L CaCO}_3$ per mg $\text{FeCl}_3$ |
| Sodium Hydroxide ($\text{NaOH}$) | pH elevation, corrosion control | $50.0\%\text{ caustic}$ | 1.52 kg/L | Adds $1.25\text{ mg/L CaCO}_3$ per mg NaOH |
| Sulfuric Acid ($\text{H}_2\text{SO}_4$) | pH depression, scaling prevention | $93.0\%\text{ commercial}$ | 1.83 kg/L | Consumes $1.02\text{ mg/L CaCO}_3$ per mg $\text{H}_2\text{SO}_4$ |