Introduction & Context

The concentration ratio (CR) quantifies how much a solute is enriched in the concentrate (reject) stream of a reverse-osmosis (RO) stage when the membrane is assumed to exhibit perfect rejection. In process engineering, this idealised metric is used to:

Size downstream ion-exchange or thermal brine concentrators.
Estimate osmotic pressure rise and scaling potential.
Set recovery limits before CaSO₄, BaSO₄, or silica precipitation occurs.

Because no real membrane rejects 100% of every species, the value obtained here is an upper-bound used for conservative design. The calculation is routinely embedded in HAZOP studies, process simulators, and membrane manufacturers’ projection software.

Methodology & Formulas

Overall mass balance on water
\[ Q_{\text{feed}} = Q_{\text{permeate}} + Q_{\text{concentrate}} \]
Recovery fraction
\[ R = \frac{Q_{\text{permeate}}}{Q_{\text{feed}}} \quad \text{(dimensionless)} \]
Solute mass balance (ideal total rejection)
Because the membrane passes no solute, every kilogram of salt entering in the feed exits in the concentrate: \[ Q_{\text{feed}}\,C_{\text{feed}} = Q_{\text{concentrate}}\,C_{\text{concentrate}} \] Re-arranging gives the concentrate concentration: \[ C_{\text{concentrate}} = C_{\text{feed}}\,\frac{Q_{\text{feed}}}{Q_{\text{concentrate}}} = C_{\text{feed}}\,\frac{1}{1 - R} \]
Concentration ratio
\[ \text{CR} = \frac{C_{\text{concentrate}}}{C_{\text{feed}}} = \frac{1}{1 - R} \]

Empirical operating limits for spiral-wound brackish-water elements
Parameter	Threshold	Consequence if exceeded
Recovery, \( R \)	≤ 0.45	High axial salinity gradient, increased scaling risk
Concentration factor, \( \text{CF} = 1/(1-R) \)	≤ 1.8	Excessive osmotic back-pressure, rapid fouling

The above limits are independent of feed salinity and are used as first-pass design rules before detailed geochemical modelling.

The concentration ratio (CR) quantifies how much salts or contaminants are concentrated in the concentrate stream relative to the feed. It is calculated as:

CR = C_concentrate ÷ C_feed

where C is any conservative parameter such as conductivity, chloride, or silica. In practice, CR is also estimated from flow rates:

CR ≈ Q_feed ÷ Q_concentrate = 1 ÷ (1 – Y)

with Y = system recovery = Q_permeate ÷ Q_feed.

Use the parameter that is:

Conservative (non-reactive, non-precipitating)
Stable across pretreatment chemicals
Measurable on both sides of the membrane with existing probes

Conductivity is the default; chloride is preferred when sulfate-reducing biocides or antiscalants skew conductivity. Silica works if antiscalant dose is constant and known.

The maximum recovery is set by the solubility limits of the least soluble salt. Convert the saturation index (SI) at feed conditions to the SI at the concentrate using:

SI_concentrate ≈ SI_feed × CRⁿ

where n is the stoichiometric coefficient (2 for BaSO₄, 3 for CaF₂, etc.). Keep SI_concentrate below 1 with antiscalant or limit CR by reducing recovery.

Common causes:

Membrane degradation—salt passage increases, lowering apparent CR
Flow meter drift—calibrate mag-meters or coriolis devices quarterly
Temperature changes—viscosity alters concentrate flow more than permeate
Manual concentrate valve movement—lock valve position or use flow-controlled concentrate

Trend CR over weekly averages and cross-check with grab-sample lab analyses to isolate the root cause.

Worked Example: Concentration Ratio for a Brackish-Water RO Skid

A small municipal RO plant is being commissioned to treat 100 m³ h⁻¹ of brackish well water. The design calls for 30% recovery and a maximum concentrate TDS of 5.0 wt%. We need to verify that the expected concentration ratio (and the derived concentration factor) is within membrane-vendor limits.

Knowns

Feed flow rate, Q_feed = 100.0 m³ h⁻¹
Recovery, R = 30% (0.30)
Feed TDS, C_feed = 3.5 wt%
Target concentrate TDS, C_ret = 5.0 wt%

Step-by-Step Calculation

Calculate permeate flow rate:
Q_w = R · Q_feed = 0.30 · 100.0 = 30.0 m³ h⁻¹
Calculate retentate (concentrate) flow rate:
Q_ret = Q_feed − Q_w = 100.0 − 30.0 = 70.0 m³ h⁻¹
Compute concentration ratio (CR) from flow terms:
CR = Q_feed / Q_ret = 100.0 / 70.0 = 1.429
Verify against salt balance (concentration factor, CF):
CF = C_ret / C_feed = 5.0 / 3.5 = 1.429
Check against system limit:
Maximum allowed CF = 1.8 (given). 1.429 ≤ 1.8 → design acceptable.

Final Answer

Concentration ratio (CR) = 1.429
Concentration factor (CF) = 1.429
Both values are below the skid limit of 1.8, so the 30% recovery target is acceptable for this membrane array.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle