Salt Flux through the Membrane

Environmental Engineering · FE Reference Handbook section

Environmental Engineering

9 formulas

10 exam-style examples

~60 min

All Environmental Engineering lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Salt Flux through the Membrane within Environmental Engineering. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what salt flux through the membrane describes physically and when it applies.
State every one of the 9 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: mg/L × MGD × 8.34 = lb/day is the single most used conversion.

Lecture

Why this section exists. Salt Flux through the Membrane is the part of Environmental Engineering that lets you connect a treatment unit or receiving water body to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a mass balance across one reactor or one unit process. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. mg/L × MGD × 8.34 = lb/day is the single most used conversion. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Aeration basin at a wastewater treatment plant with churning aerated water and walkways. — Photo 1. Where this shows up in practice: salt flux through the membrane.
Capstone Studio instructional photograph

Environmental Engineering — Salt Flux through the Membrane: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a treatment unit or receiving water body. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 9 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Js	Quantity produced by "Js = (Ds Ks / ∆Z)(Cin – Cout)" — read its definition and unit from the handbook line directly above the equation.
Ds	Quantity produced by "Ds = diffusivity of the solute in the membrane (m2/s)" — read its definition and unit from the handbook line directly above the equation.
Ks	Quantity produced by "Ks = solute distribution coefficient (dimensionless)" — read its definition and unit from the handbook line directly above the equation.
C	Quantity produced by "C = concentration (kmol/m3)" — read its definition and unit from the handbook line directly above the equation.
∆Z	Quantity produced by "∆Z = membrane thickness (m)" — read its definition and unit from the handbook line directly above the equation.
Kp	Quantity produced by "Kp = membrane solute mass-transfer coefficient" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

where

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Completely mixed stream blending — Salt Flux through the Membrane

A stream flowing 8.0 cfs at 14.0 mg/L receives a discharge of 11.0 cfs at 85 mg/L. Find the fully mixed concentration.

Given

Q₁ = 8.0 cfs, C₁ = 14.0 mg/L
Q₂ = 11.0 cfs, C₂ = 85 mg/L

Find

Mixed concentration C

Start with the thinking

Mass in equals mass out at steady state.
Weight by flow, never a simple average.

Step-by-step solution

Mass balance
$Q_{1} C_{1} + Q_{2} C_{2} = (Q_{1} + Q_{2}) C$
Loads
$8.0 (14.0) = 112.0; 11.0 (85) = 935.0$
Total flow
$8.0 + 11.0 = 19.0 c f s$
Solve
$C = (112.0 + 935.0) /19.0 = 55.11 m g / L$

Answer: C ≈ 55.1 mg/L

Why the other options are there

49.5 mg/L (unweighted average)
99.0 mg/L (concentrations added)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 2

Landfill volume required for a community's solid waste — Salt Flux through the Membrane

A community of 224,129 people generates 2.4 kg/person/day of MSW and diverts 25% through recycling. Compacted in place at 886 kg/m³ with 10% additional daily cover volume, find the airspace needed for 28 years.

Given

Population = 224,129
Generation = 2.4 kg/cap/d
Diversion = 0.25
Compacted density = 886 kg/m³
Cover = 10%, design life 28 yr

Find

Annual and total landfill airspace

Start with the thinking

Only the landfilled fraction consumes airspace — diverted material is subtracted first.
Daily cover soil is real volume and must be added to the waste volume.

Step-by-step solution

Formula
$l an df i l l e d ma ss = p o p u l a t i o n \times r a t e \times (1 - d i v er s i o n)$
Substituting
$224129 (2.4) (1 - 0.25) = 403, 432 k g / d a y$
Formula
$V = ma ss / d e n s i t y$
Substituting
$V = 403, 432/886 = 455.3 m^{3} / d a y$
Annual with cover
$455.3 \times 365 \times 1.10 = 182, 819 m^{3} / y r$
Design life
$182, 819 \times 28 = 5, 118, 944 m^{3}$

Answer: ≈ 182,819 m³/yr, or 5,118,944 m³ over 28 years

Why the other options are there

6,204,781 m³ (diversion and cover ignored)
4,123,077,084 m³ (mass reported as volume)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 3

Chlorine dose, CT value and daily chemical demand — Salt Flux through the Membrane

A water plant treating 29,401 m³/d applies a chlorine dose of 3.0 mg/L against a demand of 3.8 mg/L, with 22 minutes of contact time. Compute the free residual, the CT value, the daily chlorine mass required, and the concentration remaining after a 2.0-log pathogen reduction of an initial 10⁶ organisms/100 mL.

Given

Q = 29,401 m³/d
Dose = 3.0 mg/L
Demand = 3.8 mg/L
t = 22 min
Target = 2.0 log

Find

Residual, CT, kg/day of chlorine and surviving organisms

Start with the thinking

Residual = dose − demand; the residual, not the dose, drives disinfection credit.
Each log of removal divides the surviving organism count by ten.

Step-by-step solution

Formula
$r es i d u a l = d ose - d e man d$
Substituting
$3.0 - 3.8 = 0.20 m g / L$
Formula
$C T = C \times t$
Substituting — CT = 0.20(22) = 4.4 mg·min/L
Formula
$ma ss = d ose \times Q$
Substituting
$3.0 g / m^{3} \times 29401 m^{3} / d \div 1000 = 88.20 k g / d a y$
Log removal
$N = 1 0^{6} /1 0^{2} .0 = 1.00 e + 4 or g ani s m s /100 m L$

Answer: Residual = 0.20 mg/L, CT = 4 mg·min/L, 88.2 kg Cl₂/day, survivors 1.0e+4/100 mL

Why the other options are there

CT = 66 (dose used instead of residual)
88,203 kg/day (unit conversion missed)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 4

Population projection and design water demand — Salt Flux through the Membrane

A city of 36,849 grows at 3.1% per year. Project the population in 14 years by both geometric and arithmetic growth, then compute the average and peak day water demand at 333 L/capita/day.

Given

P₀ = 36,849
i = 3.1%/yr
n = 14 yr
Per capita use = 333 L/cap/d

Find

Projected population and design flows

Start with the thinking

Geometric growth compounds; arithmetic growth adds a fixed increment and always gives a smaller value over long horizons.
Water systems are sized on peak day (and peak hour) flow, never on the average.

Step-by-step solution

Formula (geometric)
$P = P_{0} (1 + i)^{n}$
Substituting
$P = 36849 (1 + 0.031)^{1} 4 = 56, 500 p eo pl e$
Formula (arithmetic)
$P = P_{0} (1 + in)$
Substituting
$P = 36849 (1 + 0.031 \times 14) = 52, 841 p eo pl e$
Formula
$Q_{a} v g = P \times p er c a p i t a u se$
Substituting
$Q_{a} v g = 56, 500 (333) /1000 = 18, 814 m^{3} / d$
Peak day
$Q_{p} e ak = 2.6 \times 18, 814 = 48, 918 m^{3} / d$

Answer: P = 56,500 (geometric) vs 52,841 (arithmetic); Q_avg = 18,814 m³/d, Q_peak = 48,918 m³/d

Why the other options are there

15,992 (growth increment reported as population)
Q sized on the average day only

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 5

Chronic daily intake, cancer risk and radioactive decay — Salt Flux through the Membrane

Drinking water contains 0.0620 mg/L of a carcinogen. An adult of 80 kg drinks 1.6 L/day. With a cancer slope factor of 0.26 (mg/kg·d)⁻¹, compute the chronic daily intake and the incremental lifetime cancer risk. Also determine the fraction of a radionuclide with a 21-year half-life remaining after 81 years.

Given

C = 0.0620 mg/L
IR = 1.6 L/d
BW = 80 kg
CSF = 0.26 (mg/kg·d)⁻¹
t½ = 21 yr, t = 81 yr

Find

CDI, lifetime risk and the remaining activity fraction

Start with the thinking

CDI normalises exposure to body weight so a dose-response slope can be applied.
A risk above 10⁻⁶ to 10⁻⁴ is typically the regulatory action range.

Step-by-step solution

Formula
$C D I = (C \times I R) / B W$
Substituting — CDI = (0.0620 × 1.6)/80 = 1.240e-3 mg/kg·d
Formula
$R i s k = C D I \times C S F$
Substituting
$R i s k = 1.240 e - 3 (0.26) = 3.22 e - 4$
Interpretation — above the 10⁻⁶–10⁻⁴ risk range
Formula
$N / N_{0} = (½)^{(} t / t ½)$
Substituting
$(½)^{(} 81/21) = 6.901$

Answer: CDI = 1.24e-3 mg/kg·d, risk = 3.22e-4, 6.90% of the radionuclide remains

Why the other options are there

Risk = 0.01612 (body weight and intake ignored)
-92.9% remaining (linear decay)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 6

BOD exerted after t days — Salt Flux through the Membrane

A wastewater has ultimate BOD L₀ = 256 mg/L and k = 0.25 /day (base e). How much BOD is exerted in 8 days?

Given

L₀ = 256 mg/L
k = 0.25 /day
t = 8 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.25 (8) = 2.00$
Exponential
$e^{(} - 2.00) = 0.1353$
Substituting
$B O D_{t} = 256 (1 - 0.1353) = 221.4 m g / L$
Remaining
$L_{0} - B O D_{t} = 34.6 m g / L$

Answer: BOD_8 ≈ 221.4 mg/L

Why the other options are there

35 mg/L (remaining reported as exerted)
512.0 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 7

Hydraulic detention time in a tank — Salt Flux through the Membrane

A treatment tank holds 120,429 gal and treats 3.7 MGD. Find the hydraulic detention time.

Given

V = 120,429 gal
Q = 3.7 MGD

Find

Detention time θ

Start with the thinking

θ = V/Q with consistent volume units.
Express the answer in hours for design comparison.

Step-by-step solution

Detention
$θ = V / Q$
Flow in gal/day
$3.7 \times 1 0^{6} = 3, 700, 000 g a l / d a y$
Substituting
$θ = 120429/3, 700, 000 = 0.0325 d a y$
Convert
$θ = 0.0325 (24) = 0.78 h r$

Answer: θ ≈ 0.78 hr

Why the other options are there

0.033 hr (day/hour conversion missed)
30.72 hr (ratio inverted)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 8

Chemical feed rate from dosage — Salt Flux through the Membrane

A plant treating 2.7 MGD requires a 8.0 mg/L dose. What is the daily chemical demand in pounds?

Given

Q = 2.7 MGD
Dose = 8.0 mg/L
8.34 lb/gal

Find

lb/day of chemical

Start with the thinking

The 8.34 factor converts mg/L·MGD to lb/day.
Adjust for purity if the product is not 100% active.

Step-by-step solution

Feed
$l b / d a y = Q (M G D) \times d ose (m g / L) \times 8.34$
Substituting
$l b / d a y = 2.7 (8.0) (8.34)$
Evaluate — 180.1 lb/day
At 65% available strength — 277.1 lb/day of product

Answer: ≈ 180.1 lb/day

Why the other options are there

21.6 lb/day (8.34 omitted)
2.59 lb/day (factor divided)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 9

Converting ppm to mg/m³ — Salt Flux through the Membrane

A stack gas contains 57.0 ppm of a compound with molecular weight 46 g/mol at 25 °C and 1 atm. Express the concentration in mg/m³.

Given

C = 57.0 ppm
MW = 46 g/mol
Molar volume = 24.45 L/mol

Find

Concentration in mg/m³

Start with the thinking

ppm by volume needs the molar volume to become a mass concentration.
24.45 L/mol applies at 25 °C; use 22.4 at 0 °C.

Step-by-step solution

Conversion
$m g / m^{3} = pp m \times M W /24.45$
Substituting
$m g / m^{3} = 57.0 (46) /24.45$
Evaluate — 107.2 mg/m³
Reverse check
$pp m = 107.2 (24.45) /46 = 57.0 ✓$

Answer: ≈ 107.2 mg/m³

Why the other options are there

30.3 mg/m³ (ratio inverted)
117.1 mg/m³ (0 °C molar volume used)

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Example 10

First-order removal in a CSTR versus a plug-flow reactor — Salt Flux through the Membrane

A reactor of volume 3,115 m³ treats 1.50 m³/s carrying 191 mg/L of a contaminant that decays first-order with k = 0.15 h⁻¹. Compute the hydraulic residence time and the effluent concentration if the tank behaves as a CSTR and as a plug-flow reactor.

Given

V = 3,115 m³
Q = 1.50 m³/s
C₀ = 191 mg/L
k = 0.15 h⁻¹

Find

τ, CSTR effluent and PFR effluent

Start with the thinking

The mass balance for steady state is: in − out − reaction = 0.
For the same volume, plug flow always outperforms a single completely mixed tank for first-order kinetics.

Step-by-step solution

Formula
$τ = V / Q$
Substituting
$τ = 3115/1.50 = 2, 077 s = 0.577 h$
Formula (CSTR)
$C = C_{0} / (1 + k τ)$
Substituting
$C = 191/ (1 + 0.15 \times 0.577) = 175.8 m g / L$
Formula (PFR)
$C = C_{0} e^{(} - k τ)$
Substituting
$C = 191 e^{(} - 0.15 \times 0.577) = 175.2 m g / L$
Comparison — plug flow removes 8.3% versus 8.0% for the CSTR

Answer: τ = 0.58 h; C_CSTR = 175.8 mg/L, C_PFR = 175.2 mg/L

Why the other options are there

174.5 mg/L (linear decay assumed)
175.8 mg/L for both reactors

Reference: FE Reference Handbook — Environmental Engineering → Salt Flux through the Membrane

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a treatment unit or receiving water body, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Salt Flux through the Membrane contains 9 relations; you must be able to find this page in under 15 seconds.
Exam style: a mass balance across one reactor or one unit process.
Unit rule: mg/L × MGD × 8.34 = lb/day is the single most used conversion.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

mg/L × MGD × 8.34 = lb/day is the single most used conversion
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.