Question 1 of 7: Mass and Energy Balance, Mixture Properties and Ecological Engineering

Question 1: Mass and Energy Balance, Mixture Properties and Ecological Engineering (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 Outlet Concentration from a First-Order CSTR

Given. A completely stirred tank reactor (CSTR) operating at steady state, treating a contaminant that decays by first-order kinetics $dC/dt = -kC$ inside the tank.

Find. The steady-state outlet (= in-tank) contaminant concentration $C_{out}$.

Approach. Write an unsteady mass balance on the completely-mixed reactor (accumulation = in − out − decay), apply the steady-state condition, and solve for $C_{out}$.

1. Write the unsteady-state mass balance. For a completely-mixed reactor, accumulation equals mass in minus mass out minus in-tank decay: $$V\frac{dC}{dt} = QC_{in} - QC_{out} - kVC_{out}.$$
2. Apply the steady-state condition. At steady state $dC/dt = 0$, so: $$0 = QC_{in} - QC_{out} - kVC_{out} \;\;\Rightarrow\;\; C_{out}(Q+kV) = QC_{in} \;\;\Rightarrow\;\; C_{out} = \dfrac{QC_{in}}{Q+kV}.$$
3. Substitute the given values and evaluate. First the decay term $kV$, then the full expression: $$kV = (0.5)(2000) = 1000\ \text{m}^3/\text{d}, \qquad Q+kV = 200+1000 = 1200\ \text{m}^3/\text{d}.$$ $$C_{out} = \dfrac{(200)(500)}{1200} = \boxed{83.3\ \text{mg/L}}.$$
4. Cross-check via hydraulic retention time. $\text{HRT} = V/Q = 2000/200 = 10$ days, so the same balance can be written $C_{out} = C_{in}/(1+k\cdot\text{HRT}) = 500/(1+0.5\times10) = 500/6 = 83.3\ \text{mg/L}$ — matching Step 3 and confirming the arithmetic.

(ii) Homogeneous versus Heterogeneous Mixtures

Two key differences between the physical properties of homogeneous and heterogeneous mixtures, both important to how an environmental engineer characterizes and treats a waste stream:

1. Number of phases and uniformity of composition. A homogeneous mixture is a single phase with uniform composition and properties (density, concentration) at every point, down to the molecular scale — there is no visible interface between its components. A heterogeneous mixture contains two or more distinguishable phases with a visible boundary between them, and its local composition varies from point to point (e.g., more solids near the bottom of a settling tank than near the surface).
2. Separability by simple physical means. Because a heterogeneous mixture has a real phase boundary, its components can usually be separated by straightforward mechanical/physical unit operations that exploit that boundary directly — settling, filtration, screening, centrifugation, cyclone separation. A homogeneous mixture has no such boundary to exploit; separating its components requires processes that act on molecular-scale differences in physical or chemical properties — boiling point (distillation), molecular size (membrane filtration/reverse osmosis), or ionic charge (ion exchange) — rather than a simple mechanical barrier.

Examples found in the environment. A homogeneous mixture: atmospheric air itself, or an industrial flue-gas stream after particulate removal — N₂, O₂, CO₂, SO₂ and water vapour are fully and uniformly intermixed as gases, with no visible phase boundary between them, whether the air is of natural origin or has industrial combustion gases added to it. A heterogeneous mixture: raw municipal wastewater, which contains a liquid (water) phase together with distinct visible solid particulates (grit, fecal solids, floatable grease and fats) and, often, a separate oil/grease film — a mixture generated by everyday domestic and industrial activity that is heterogeneous by its very nature and is treated precisely by exploiting that phase separation (screening, grit removal, sedimentation).

(iii) Ecological Engineering Examples under the Mitsch & Jorgensen Definition

Mitsch and Jorgensen’s definition requires a design that is (a) a societal service, (b) beneficial to both society and nature, (c) systems-based, (d) sustainable, and (e) integrative of society with its natural environment. Two environmental engineering applications that satisfy all five criteria:

1. Constructed (treatment) wetlands for wastewater or stormwater polishing. A constructed wetland engineered ahead of, or in place of, a conventional mechanical treatment step uses natural wetland processes — plant nutrient uptake, microbial degradation in the root zone, and physical settling — to reduce BOD, nutrients and suspended solids at low energy and operating cost (the societal service). At the same time it creates genuine wildlife habitat, supports biodiversity, provides flood-flow attenuation and sequesters carbon in wetland soils (the benefit to nature), it is inherently systems-based (hydrology, vegetation, microbiology and hydraulics designed together as one functioning system) and sustainable (self-maintaining once established, unlike a purely mechanical plant), and it physically integrates the community’s infrastructure with a functioning natural ecosystem rather than isolating the two.
2. Bioengineered (vegetated/living) shoreline and streambank stabilization. Rather than a hard engineered structure (rip-rap or a concrete seawall), a bioengineered shoreline uses native vegetation, root-reinforced soil lifts and, where needed, biodegradable or minimal hard structure to stabilize a bank against erosion — protecting adjacent infrastructure and property (the societal service) while restoring riparian/littoral habitat, improving water quality through vegetative filtering, and maintaining the natural sediment-transport and hydrologic connectivity that a hard structure would block (the benefit to nature). It is systems-based (root strength, hydraulics and ecology co-designed), self-sustaining once vegetation is established, and directly integrates a piece of built infrastructure with the natural shoreline system it replaces.