Fundamental Principles

A. Population projections and demand calculations

Population Projections and Demand Calculations

Project a service population by arithmetic, geometric, and declining-growth models, then convert it to design-year water, wastewater, and solid-waste flows with peaking factors.

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B. Reactors

Ideal Reactors: Batch, CSTR, and Plug Flow

The three ideal reactors, the general mass balance, hydraulic residence time, and why a PFR always beats a CSTR for first-order conversion.

Nearly every unit process an environmental engineer touches — a chlorine contact tank, an activated-sludge basin, a lagoon, a GAC column, an anaerobic digester — is some approximation of one of three idealized reactors. Master those three idealizations and a single bookkeeping statement (mass in = mass out plus what reaction destroys or creates), and you can size or rate almost any process the FE throws at you. The reactor relations below follow the FE Reference Handbook — Chemical Engineering (Reactor Design / Flow Reactors), with residence-time forms echoed throughout the Environmental Engineering chapter. This concept is the ideal-reactor taxonomy — the three reactor types and how each behaves; the companion 'Mass Balances and Reactor Sizing' concept applies the same balance to blending and to reactor networks (series and parallel). The trap that costs the most points is treating a completely-mixed tank like a tube: for the same volume and reaction, the two give very different conversions, and knowing which is which is the whole game.

The general mass balance

Everything starts from a control volume drawn around the reactor and a balance on the species of interest. Rate of accumulation equals what flows in, minus what flows out, plus what the reaction generates (negative for a reactant being consumed). For a constant-density liquid this is the master equation; every reactor model is just a special case obtained by deciding which terms vanish. At steady state the accumulation term is zero — the workhorse simplification for continuous-flow processes.

\underbrace{V\dfrac{dC}{dt}}_{\text{accumulation}} = \underbrace{Q\,C_0}_{\text{in}} - \underbrace{Q\,C}_{\text{out}} + \underbrace{r_A V}_{\text{generation}}

Batch reactor

A batch reactor is filled, sealed, reacted, and emptied — no flow in or out during the reaction, so the in and out terms drop and the balance reduces to

dC/dt = r_A

. Concentration changes with time, not position. For a first-order decay

r_A = -kC

, integration gives the familiar exponential decline, and the reaction 'time' is simply the clock. Batch is the model for jar tests, BOD bottles, and any fill-and-draw operation.

\dfrac{dC_A}{dt} = r_A \;\Rightarrow\; C_A = C_{A0}\,e^{-kt}\ \ (\text{1st order})

Continuous-stirred tank reactor (CSTR)

A CSTR (also called a completely-mixed flow reactor, CMFR) is so well mixed that the contents are uniform and the effluent leaves at the exact tank concentration — the reaction therefore proceeds at the low exit concentration everywhere. At steady state the balance rearranges to give the design relation directly. Crucially,

-r_A

is evaluated at the EXIT condition, which is why a single CSTR is the least efficient ideal reactor for a normal (positive-order) reaction: all of its volume works at the worst-case low driving concentration.

V = \dfrac{Q\,(C_0 - C)}{-r_A}\Bigg|_{\text{exit}}, \qquad \dfrac{C}{C_0} = \dfrac{1}{1 + k\tau}\ \ (\text{1st order})

Plug-flow reactor (PFR)

In a plug-flow reactor every fluid element moves through like a piston with no mixing in the flow direction, so concentration falls steadily along the length. Each slice behaves like a tiny batch reactor whose elapsed time is its travel time — which is why, for the same residence time and a first-order reaction, the PFR achieves identical conversion to a batch reactor and strictly more than a CSTR. Long, narrow channels (chlorine contact tanks, sand filters) are designed to approach plug flow precisely to exploit this advantage.

\tau = -\!\int_{C_0}^{C}\dfrac{dC}{-r_A}, \qquad \dfrac{C}{C_0} = e^{-k\tau}\ \ (\text{1st order})

Hydraulic residence time and space-time

The hydraulic residence time (HRT, or detention time)

\tau = V/Q

is the average time a fluid parcel spends in the reactor — the single most important sizing parameter. In chemical-engineering language this is the space-time, and its reciprocal is the space-velocity. For continuous reactors

\tau

replaces the clock

t

of the batch reactor in every first-order expression. Keep it distinct from solids retention time (sludge age,

\theta_c

), which can be far longer than the HRT in any process that recycles or retains biomass.

\tau = \dfrac{V}{Q}

CSTR vs. PFR for the same job

Set the two continuous reactors side by side at equal

\tau

and first-order

k

: the CSTR leaves

C/C_0 = 1/(1+k\tau)

while the PFR leaves

C/C_0 = e^{-k\tau}

. Because

e^{-k\tau} < 1/(1+k\tau)

for all

k\tau>0

, the PFR always wins — and the gap widens as you demand higher conversion. Equivalently, to reach a target conversion a CSTR needs a larger volume than a PFR. The exam loves this comparison: a CSTR sized for

90\%

removal of a first-order pollutant needs several times the volume a PFR would.

\dfrac{C}{C_0}\bigg|_{\text{CSTR}} = \dfrac{1}{1+k\tau} \;>\; e^{-k\tau} = \dfrac{C}{C_0}\bigg|_{\text{PFR}}

Exam strategy

Identify the reactor first: 'completely mixed' or 'effluent equals tank concentration' means CSTR (

1/(1+k\tau)

); 'long contact tank', 'tubular', or 'no axial mixing' means PFR (

e^{-k\tau}

); 'fill and draw' or a closed bottle means batch (

e^{-kt}

). Then nail the order of reaction — these clean forms are FIRST-order only; a zero-order reaction proceeds at constant rate regardless of concentration. Compute

\tau = V/Q

in consistent units and confirm

k

carries the matching inverse-time units so

k\tau

is dimensionless. When a problem asks 'which reactor is smaller' or 'why use a baffled tank', the answer is almost always the PFR/plug-flow advantage. Quick check: for the same

\tau

, CSTR conversion should be lower than PFR — if your numbers say otherwise, you have swapped the formulas.

Key equations

General mass balance (constant V)

V\dfrac{dC}{dt} = Q\,C_0 - Q\,C + r_A V

Accumulation = in − out + generation. $r_A<0$ for a consumed reactant; set the left side to zero for steady state.

Batch reactor, first order

C_A = C_{A0}\,e^{-kt}

No flow; concentration decays with reaction time $t$ . $k$ in $\text{time}^{-1}$ .

CSTR design equation

V = \dfrac{Q\,(C_0 - C)}{-r_A}

$-r_A$ evaluated at EXIT (tank) conditions. $V$ = volume, $Q$ = volumetric flow.

CSTR, first order

\dfrac{C}{C_0} = \dfrac{1}{1 + k\tau}

Fraction remaining for a single completely-mixed tank; $\tau=V/Q$ .

PFR design integral

\tau = -\int_{C_0}^{C}\dfrac{dC}{-r_A}

Each slice acts as a batch element; reduces to $e^{-k\tau}$ for first order.

PFR, first order

\dfrac{C}{C_0} = e^{-k\tau}

Fraction remaining for plug flow; identical to a batch reactor with $t=\tau$ .

Hydraulic residence time

\tau = \dfrac{V}{Q}

Mean detention time (space-time). Reciprocal is space-velocity $SV=1/\tau$ .

Zero-order conversion

C = C_0 - k\tau \quad (\text{CSTR or PFR})

For a zero-order reaction the rate is constant, so CSTR and PFR give the SAME effluent; $k$ in mass/(vol·time).

Damköhler number

Da = k\tau

Dimensionless ratio of reaction rate to throughput; governs conversion in both continuous reactors.

Worked examples

Effluent from a completely-mixed tank

Problem. A $500\ \text{m}^3$ completely-mixed tank treats $100\ \text{m}^3/\text{h}$ of water carrying a pollutant that decays first-order at $k = 0.30\ \text{h}^{-1}$ . Find the residence time and the fraction of pollutant remaining. Compare with a PFR of the same size.

Solution. Residence time:

\tau = V/Q = 500/100 = 5.0\ \text{h}

, so

k\tau = 0.30(5.0) = 1.5

. CSTR:

C/C_0 = 1/(1+k\tau) = 1/(1+1.5) = 1/2.5 = 0.400

— i.e.

60.0\%

removed. PFR (same

\tau

):

C/C_0 = e^{-k\tau} = e^{-1.5} = 0.223

— i.e.

77.7\%

removed. CSTR leaves

\mathbf{40.0\%}

(

60.0\%

removal); the identical-size PFR leaves

\mathbf{22.3\%}

(

77.7\%

removal). Sanity check:

k\tau

is dimensionless (

\text{h}^{-1}\cdot\text{h}

), and the PFR outperforms the CSTR at equal volume, exactly as theory requires.

\dfrac{C}{C_0}\Big|_{\text{CSTR}} = \dfrac{1}{1+1.5}=0.400 \;>\; e^{-1.5}=0.223 = \dfrac{C}{C_0}\Big|_{\text{PFR}}

Tank volume for 90% conversion — CSTR vs. PFR

Problem. A first-order reaction with $k = 0.50\ \text{h}^{-1}$ must achieve $90\%$ conversion ( $X=0.90$ ). Find the required residence time for a CSTR and for a PFR, and the ratio of their volumes.

Solution. PFR:

\tau_{\text{PFR}} = \dfrac{1}{k}\ln\!\dfrac{1}{1-X} = \dfrac{1}{0.50}\ln(10) = 2.0(2.303) = 4.61\ \text{h}

. CSTR:

\tau_{\text{CSTR}} = \dfrac{X}{k(1-X)} = \dfrac{0.90}{0.50(0.10)} = \dfrac{0.90}{0.050} = 18.0\ \text{h}

. Since

V = Q\tau

at the same

Q

, the volume ratio equals the

\tau

ratio:

18.0/4.61 = 3.9

. The CSTR needs

\tau = \mathbf{18.0\ \text{h}}

versus the PFR's

\mathbf{4.61\ \text{h}}

— about

\mathbf{3.9\times}

the volume. Sanity check: the penalty for back-mixing balloons at high conversion, the hallmark CSTR-vs-PFR result.

\dfrac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \dfrac{X/[k(1-X)]}{(1/k)\ln[1/(1-X)]} = \dfrac{18.0}{4.61} \approx 3.9

Batch decay and a zero-order check

Problem. (a) A sealed BOD-style batch test holds a pollutant that decays first-order at $k = 0.25\ \text{h}^{-1}$ ; find the fraction remaining after $8\ \text{h}$ . (b) A different pollutant degrades at a constant zero-order rate $k_0 = 2.0\ \text{mg/(L·h)}$ from $C_0 = 50\ \text{mg/L}$ ; find the effluent of a CSTR and of a PFR, each with $\tau = 10\ \text{h}$ .

Solution. (a) Batch, first order: the clock

t

plays the role of

\tau

, so

C/C_0 = e^{-kt} = e^{-0.25(8)} = e^{-2.0} = 0.135

— i.e.

86.5\%

removed. (b) Zero order: the rate does not depend on concentration, so both ideal continuous reactors reduce to

C = C_0 - k_0\tau = 50 - 2.0(10) = 30\ \text{mg/L}

. The CSTR and PFR give the IDENTICAL effluent — there is no plug-flow advantage when the order is zero. Batch fraction remaining

\mathbf{0.135}

(

86.5\%

removed); zero-order effluent

\mathbf{30.0\ \text{mg/L}}

for both reactors. Sanity check:

kt

and

k_0\tau

are each dimensionless-consistent (

\text{h}^{-1}\cdot\text{h}

and

\text{mg/(L·h)}\cdot\text{h}=\text{mg/L}

), and the zero-order equality is exactly why the CSTR-vs-PFR contrast vanishes off first-order kinetics.

C_{\text{batch}} = C_0 e^{-kt} = C_0 e^{-2.0}; \qquad C_{\text{CSTR}} = C_{\text{PFR}} = C_0 - k_0\tau = 30.0\ \text{mg/L}

Common pitfalls

•Using the CSTR formula for a plug-flow tank (or vice versa). Completely-mixed $\to 1/(1+k\tau)$ ; tubular/long contact $\to e^{-k\tau}$ . The two give very different volumes for the same job.
•Evaluating the CSTR rate at the inlet concentration. In a CSTR the contents — and therefore the reaction rate — are at the EXIT concentration; that is exactly why it is the least efficient ideal reactor.
•Applying first-order forms to a zero-order reaction. A zero-order rate is constant, so $C = C_0 - k\tau$ and a CSTR and PFR give the SAME effluent — no plug-flow advantage.
•Confusing hydraulic residence time $\tau=V/Q$ with solids retention time (sludge age $\theta_c$ ). In activated sludge $\theta_c$ can be 5–15 days while $\tau$ is only hours.
•Inconsistent units in $k\tau$ : if $k$ is in $\text{day}^{-1}$ but $\tau$ in hours, $k\tau$ is meaningless. Convert so the product is dimensionless before plugging in.
•Reporting 'conversion' when the formula gives 'fraction remaining'. $C/C_0$ is what is LEFT; conversion is $X = 1 - C/C_0$ . Read which the question wants.
•Assuming real basins are ideal. Dead zones and short-circuiting make actual contact time less than $V/Q$ — disinfection credit uses $t_{10}$ , not the theoretical $\tau$ .

References

NCEES FE Reference Handbook — Chemical Engineering (Reactor Design; Flow Reactors, Steady State)
NCEES FE Reference Handbook — Environmental Engineering (hydraulic residence time)
Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery — reactor types and detention time
Davis & Cornwell, Introduction to Environmental Engineering — reactor analysis and CSTR vs PFR

Mass Balances and Reactor Sizing

Draw a control volume, write accumulation = in − out ± reaction, simplify at steady state, and size single, series, and parallel reactors for a target conversion.

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C. Materials science

Materials Science: Properties and Corrosion

Compare metals, polymers, and ceramics, read the stress-strain curve for the key mechanical properties, and use the galvanic series to predict and prevent electrochemical corrosion.

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