Question 1 of 7: Mass Balance, Partitioning and Disinfection

Question 1: Mass Balance, Partitioning and Disinfection (20 marks)

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.

(i) Steady-State Outflow PO₄ Concentration from the Lake

Given. A well-mixed lake at steady state, fed by an upstream river and losing water to both evaporation and an outflow stream, with the phosphate ($PO_4$) it carries subject to first-order decay:

Find. The steady-state $PO_4$ concentration in the outflow stream, $C_o$.

Approach. Write an unsteady mass balance on $PO_4$ for the completely-mixed lake, note that evaporation removes only water (zero $PO_4$ mass flux), apply the steady-state condition, and solve for $C_o$.

1. Write the unsteady-state $PO_4$ mass balance. Evaporation is a water flux, not a $PO_4$ flux, so it does not appear as a removal term for $PO_4$ mass; only the outflow stream and in-lake decay remove $PO_4$: $$V\frac{dC}{dt} = Q_iC_i - Q_oC_o - KVC_o.$$
2. Apply the steady-state condition. At steady state $dC/dt=0$, so: $$0 = Q_iC_i - Q_oC_o - KVC_o \;\;\Rightarrow\;\; C_o(Q_o+KV) = Q_iC_i \;\;\Rightarrow\;\; C_o = \dfrac{Q_iC_i}{Q_o+KV}.$$
3. Substitute the given values and evaluate. Convert $C_i$ to mg/m$^3$ ($1$ mg/L $=1000$ mg/m$^3$) so all quantities share consistent units, then compute the decay term and the denominator: $$C_i = 15\ \text{mg/L} = 1.5\times10^4\ \text{mg/m}^3, \qquad KV = (0.07)(10^6) = 7\times10^4\ \text{m}^3/\text{yr}.$$ $$Q_o+KV = 5\times10^4+7\times10^4 = 1.2\times10^5\ \text{m}^3/\text{yr}.$$
4. Solve for the outflow concentration. $$C_o = \dfrac{(10^5)(1.5\times10^4)}{1.2\times10^5} = \dfrac{1.5\times10^9}{1.2\times10^5} = 1.25\times10^4\ \text{mg/m}^3 = \boxed{12.5\ \text{mg/L}}.$$

(ii) The Organic Carbon Adsorption Coefficient $K_{OC}$ and Its Governing Factors

The organic carbon adsorption (partition) coefficient, $K_{OC}$, is the equilibrium ratio of an organic chemical’s concentration sorbed onto the organic-carbon fraction of a soil or sediment to its concentration remaining dissolved in the surrounding water, $K_{OC} = C_{\text{sorbed, per g organic carbon}}/C_{\text{water}}$. Because it is normalized to organic-carbon content rather than to the whole soil mass, $K_{OC}$ is treated as an intrinsic, largely soil-independent property of the chemical itself — a high $K_{OC}$ chemical (e.g., many chlorinated pesticides, PAHs) partitions strongly onto organic matter and is relatively immobile in groundwater, while a low $K_{OC}$ chemical (e.g., many small polar solvents) stays largely dissolved and migrates readily with the water.

Two environmental factors that affect how strongly an organic chemical actually partitions to the soil in the field (as distinct from the chemical-specific $K_{OC}$ value itself):

1. Soil organic-carbon content, $f_{OC}$. The whole-soil, whole-sediment partition coefficient is $K_d = K_{OC}\times f_{OC}$. A soil with a high fraction of organic carbon (e.g., a peaty or organic-rich topsoil) sorbs far more of the chemical for the same $K_{OC}$ than a mineral soil with little organic matter — so even though $K_{OC}$ is defined to be a property of the chemical, the amount of the chemical that actually leaves the water phase in a given soil depends directly on how much organic carbon that soil contains.
2. pH (for ionizable organic chemicals). Many organic contaminants (weak acids or bases — e.g., phenols, chlorinated anilines) exist partly in a neutral, hydrophobic form and partly in a charged, hydrophilic form, and the ratio between the two is set by the soil-water pH relative to the chemical’s $pK_a$. The neutral form partitions strongly to organic carbon (behaving as its $K_{OC}$ predicts), while the ionized form is far more water-soluble and partitions poorly; shifting the pH shifts the neutral/ionized fraction and therefore shifts the effective partitioning even though the chemical’s intrinsic $K_{OC}$ has not changed.

(iii) Designing an Efficient Chlorine Disinfection System

An efficient chlorine disinfection system is designed around four linked decisions — the chemical form of chlorine, the dose, the contact time, and how success is verified — all governed by Chick–Watson disinfection kinetics.

1. Form of chlorine. Free chlorine (as $HOCl$/$OCl^-$ from gas $Cl_2$ or hypochlorite) is the strongest and fastest-acting disinfectant form and is preferred for primary disinfection; $HOCl$ dominates and is most effective at pH below about 7.5, since it is a far stronger oxidant/disinfectant than the dissociated $OCl^-$ ion. Where a persistent distribution-system residual is also needed, chloramines (formed by adding ammonia after free-chlorine contact) are used for secondary disinfection because they decay much more slowly, even though they disinfect far more slowly per unit dose than free chlorine.
2. Dose and demand. The applied dose must first satisfy the chlorine demand — chlorine consumed reacting with ammonia, organic matter and reduced inorganics in the water (the classic breakpoint-chlorination curve) — before any disinfecting residual concentration $C$ remains. The dose is therefore sized as demand plus the target residual $C$ required to achieve disinfection over the design contact time.
3. Contact time and the $Ct$ concept. Disinfection follows Chick–Watson kinetics, $\ln(N/N_0) = -k\,C^n\,t$ (with $n\approx1$ for chlorine), so inactivation is governed by the product of residual concentration and contact time, $Ct$. Regulators specify a minimum $Ct$ to achieve a target log-inactivation for a given pathogen at a given temperature and pH; the system is designed (contact-basin volume, baffling to avoid short-circuiting so the effective $t_{10}$ approaches the theoretical detention time) to guarantee that $Ct$ is met even under peak-flow conditions.
4. Pathogen indicators and target organisms. Because pathogens themselves are not routinely measured, Escherichia coli and total/fecal coliforms serve as indicator organisms for recent fecal contamination, while heterotrophic plate counts assess general water quality. The disinfection design must be sized for the most chlorine-resistant organism realistically present, not just bacteria: viruses (e.g., enteric viruses) are more resistant than vegetative bacteria, and protozoan cysts/oocysts (Giardia, and especially Cryptosporidium, which is highly chlorine-resistant) are the most resistant of all — systems with a credible Cryptosporidium risk therefore add a physical/UV barrier rather than relying on chlorine $Ct$ alone.