16-Civ-B5 Water Supply and Wastewater Treatment · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 98-Civ-B5 Water Supply and Wastewater Engineering — National Examination, December 2015. Three hours; closed book, one aid sheet written on both sides; an approved calculator is permitted. Question 1 is compulsory and the candidate attempts any five of the remaining six, so 100 marks are written out of the 115 printed (Q1 = 25 marks, Q2 to Q7 = 15 marks each; Q2 splits 12 + 3). Every one of the seven questions is solved below, because this set is a study resource rather than an exam script.
Reference texts.
Check: representative design data. Questions 1, 3, 4 and 7 are discussion questions and print no numbers. Where a number appears in those answers it is a representative Canadian municipal value chosen by the solver to make the argument concrete; it is labelled as such at the point of use, and every one of them. The graded content of those questions is the reasoning, not the arithmetic.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given. Three of the four classical destabilisation mechanisms of coagulation–flocculation theory, to be described. The illustrative ionic strengths, valences and mixing intensities used below are representative water-treatment values chosen by the solver.
Find. A physical description of each mechanism, the conditions under which it dominates, and how the three differ in practice.
Approach. Clay and organic colloids in a natural water carry a net negative surface charge and are held apart by electrostatic repulsion; each mechanism is a different way of defeating that repulsion or of by-passing it altogether, and the three are best understood by asking, for each, what physically holds the aggregate together.
A colloid in water is surrounded by an electrical double layer: a tightly held Stern layer of counter-ions immediately at the surface, and beyond it a diffuse layer in which the excess counter-ion concentration decays exponentially into the bulk solution. The characteristic decay distance is the Debye length $\kappa^{-1}$, and it depends only on the ionic strength of the solution, not on the particle. Two particles repel each other whenever their diffuse layers overlap, so the range of the repulsion is set by $\kappa^{-1}$.
Adding an indifferent electrolyte — one whose ions do not adsorb specifically, such as sodium chloride — raises the ionic strength and compresses the diffuse layer. In a symmetrical electrolyte at 25 °C the Debye length is approximately
$$\kappa^{-1} \approx \frac{0.304}{\sqrt{I}}\ \mathrm{nm}, \qquad I \text{ in mol/L}$$so raising the ionic strength from $10^{-3}$ to $10^{-1}$ M shrinks the double layer from 9.6 nm to 0.96 nm, a tenfold reduction. Once the repulsive barrier is thinner than the range of the ever-present van der Waals attraction, particles that collide by Brownian motion stick, and the suspension coagulates. Note carefully what has not happened: the surface charge itself is unchanged. Compression shortens the reach of the repulsion; it does not neutralise it.
The dependence on counter-ion valence is dramatic and is the content of the Schulze–Hardy rule: the critical coagulation concentration scales approximately as the inverse sixth power of the counter-ion charge.
Given. A negatively charged colloid whose critical coagulation concentration with Na+ is 100 mmol/L.
Find. The predicted CCC for Ca2+ and Al3+.
$$\mathrm{CCC} \propto z^{-6} \;\Rightarrow\; \mathrm{CCC_{Ca^{2+}}} = \frac{100}{2^6} = 1.56\ \mathrm{mmol/L}, \qquad \boxed{\mathrm{CCC_{Al^{3+}}} = \frac{100}{3^6} = 0.137\ \mathrm{mmol/L}}$$A trivalent ion is therefore some 730 times more efficient than a monovalent one, which is the theoretical reason why trivalent aluminium and iron salts became the universal water-treatment coagulants. In practice, however, pure double-layer compression is rarely the operative mechanism in a treatment plant: the ionic strengths needed are far higher than a potable water can tolerate, and it is chiefly of interest in estuaries, where river-borne clay flocculates on meeting seawater and builds the delta.
When a metal salt is dosed well beyond its solubility limit — typically alum at 20 to 60 mg/L held between pH 6.5 and 7.5, or ferric salts between pH 5.5 and 8.5 — the dominant product is not a soluble hydrolysis species but a bulk amorphous precipitate of aluminium or ferric hydroxide. That precipitate nucleates throughout the rapid-mix volume and then grows and settles, and as it does so it physically entraps, adsorbs and drags down the colloids in its path. The colloids are removed as passengers of a precipitate they did not cause.
The name is exactly descriptive: the floc sweeps the water clean. Because removal is by enmeshment rather than by charge neutralisation, the mechanism is remarkably forgiving — it works on very dilute suspensions where collisions between colloids alone would be far too rare, it is insensitive to modest dose errors, and it does not exhibit restabilisation at over-dose. It is the workhorse mechanism of the great majority of conventional water treatment plants, including the one proposed in Question 1. The costs are a substantially larger chemical dose than charge neutralisation requires and a correspondingly larger volume of hydroxide sludge for the residuals train.
The controlling variable is pH, not dose, because pH determines whether the metal exists as soluble cationic hydrolysis products, as the insoluble hydroxide, or as the soluble aluminate anion. Outside the sweep window the precipitate simply does not form, and the dose is wasted.
Bridging is a mechanical mechanism rather than an electrostatic one. A long-chain polymer — a synthetic polyacrylamide of molecular weight in the millions, or a natural polymer such as starch or chitosan — adsorbs onto a particle surface at a number of discrete segments, through hydrogen bonding, charge interaction or specific chemical affinity. The remaining loops and tails extend well beyond the double layer into the solution, and when a second particle drifts within reach, a free segment adsorbs onto it too. The two particles are then physically tied together by the polymer chain.
Three features follow directly and distinguish bridging from the other two mechanisms. First, the floc is much stronger and shear-resistant, because the bond is a covalent-scale physical tether rather than a balance of weak forces — which is why polymers are used as flocculant aids ahead of high-rate clarifiers and as conditioners ahead of centrifuges and belt presses. Second, the polymer need not be oppositely charged to the particles: anionic polymers bridge negative clays perfectly well, because adsorption occurs at specific sites rather than by generalised attraction. Third, and critically for operation, there is an optimum dose. Below it, too few bridges form. Above it, the excess polymer saturates every available surface site, so an approaching particle finds no vacant site to attach to and the suspension is restabilised — sterically this time, not electrostatically. Excessive mixing is equally destructive, because the extended loops and tails fold back onto their own particle and the bridging capacity is lost irreversibly.
All three mechanisms only destabilise; they do not by themselves transport particles into contact. That transport is the flocculation step, and it is quantified by the Camp–Stein velocity gradient.
Given. A rapid-mix tank of 5 m³ drawing 1.5 kW and a flocculation basin of 600 m³ drawing 300 W, both at 15 °C where the dynamic viscosity is $1.139\times10^{-3}$ Pa s, with detention times of 30 s and 25 min respectively.
Find. The velocity gradient and the dimensionless $Gt$ product for each.
$$G = \sqrt{\frac{P}{\mu V}} \;\Rightarrow\; G_{\mathrm{rapid}} = \sqrt{\frac{1500}{1.139\times10^{-3}\times 5}} = 513\ \mathrm{s^{-1}}, \qquad G_{\mathrm{floc}} = \sqrt{\frac{300}{1.139\times10^{-3}\times 600}} = 21.0\ \mathrm{s^{-1}}$$giving $Gt = 1.5\times10^{4}$ for the rapid mix and $Gt = 3.1\times10^{4}$ for the flocculator. The contrast is the point: destabilisation needs a violent, brief mix so that the coagulant is dispersed before it has finished hydrolysing, whereas aggregation needs a gentle, long one so that the floc grows without being sheared apart. Tapering $G$ down through successive flocculation compartments respects the fact that the floc becomes more fragile as it grows.
| Feature | Ionic layer compression | Sweep coagulation | Inter-particle bridging |
|---|---|---|---|
| What holds the aggregate together | Van der Waals attraction, once repulsion is shortened in range | Physical enmeshment in a settling precipitate | A polymer chain adsorbed on two particles |
| Chemical added | Indifferent electrolyte (high ionic strength) | Alum or ferric salt, above the solubility limit | High-molecular-weight polymer, 0.1 to 1 mg/L |
| Governing variable | Ionic strength and counter-ion valence | pH, then dose | Dose and mixing intensity |
| Key relation | $\kappa^{-1}\approx 0.304/\sqrt{I}$ nm; CCC $\propto z^{-6}$ | Metal-hydroxide solubility window | Optimum surface coverage near one half |
| Restabilisation on over-dose | No | No | Yes — steric restabilisation |
| Floc strength | Weak | Moderate, but bulky and dense enough to settle | Strong and shear-resistant |
| Where it dominates | Estuaries; rarely in treatment plants | Conventional water treatment; most plants | Flocculant aid; sludge conditioning before dewatering |
| CCC ratio Na+ : Ca2+ : Al3+ | 1 : 1/64 : 1/729, i.e. 100 : 1.56 : 0.137 mmol/L on the illustrative colloid | ||
| Velocity gradients | Rapid mix $G = 513\ \mathrm{s^{-1}}$, $Gt = 1.5\times10^4$; flocculation $G = 21.0\ \mathrm{s^{-1}}$, $Gt = 3.1\times10^4$ | ||
