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Allowable RO recovery from the limiting salt

As recovery rises the concentrate, and the boundary layer at the membrane wall, concentrate every ion. The first salt to pass its solubility product limits recovery unless an antiscalant or pretreatment deals with it.

Calciummg/L

Bariummg/L

Strontiummg/L

Sulfatemg/L

Fluoridemg/L

Carbonate (as CO₃)mg/LCarbonate ion only; pH dependent. Use LSI for calcium carbonate at the operating pH.

Orthophosphate (as HPO₄)mg/L

Silica (as SiO₂)mg/L

Concentration polarisation factor βTypical manufacturer limit 1.2; from the polarisation calculator.

Allowable recovery

48%

Limited by barium sulfate.

Recovery allowed by each salt

Barium sulfate

48% recovery

Calcium sulfate (gypsum)

74% recovery

Activities are taken as concentrations and ion pairs are ignored, which understates allowable recovery, often by a wide margin. Treat the result as a screening value and confirm with a speciation model or the membrane supplier's projection.

How it works

At the concentrate end the ion concentration at the membrane wall, for a fully rejected ion, is

C_{M} = \frac{β C_{F}}{1 - r}

(1)

For a salt A_mB_n with solubility product K_sp and N = m + n ions, saturation at the wall is reached when

\frac{β^{N} [A]^{m} [B]^{n}}{(1 - r)^{N}} = K_{s p} r_{a l l o w} = 1 - {(β^{N} [A]^{m} [B]^{n} / K_{s p})}^{1 / N}

(2)

where

$C_{M}$: ion concentration at the membrane wall at the concentrate end, mol/L
$C_{F}$: feed concentration of the ion, mol/L
$β$: concentration polarisation factor
$r$: recovery, fraction
$[A], [B]$: feed molar concentrations of the cation and anion, mol/L
$m, n$: stoichiometric coefficients of the salt A_mB_n
$N$: ions per formula unit, m + n
$K_{s p}$: solubility product, (mol/L)ᴺ
$r_{a l l o w}$: allowable recovery for the salt

The lowest allowable recovery across the salts present is the system limit. Solubility products at 25 °C (pK_sp): calcium carbonate 8.2, calcium fluoride 10.3, calcium orthophosphate 6.6, calcium sulfate 4.6, strontium sulfate 6.2, barium sulfate 9.7, amorphous silica 2.7.

Carbonate and phosphate depend on pH; full rejection is slightly conservative for RO and wrong for nanofiltration. Activities are taken as concentrations and ion pairs are ignored, which understates the real limit; temperature, antiscalant allowance and ionic strength all raise it.

Allowable RO recovery from the limiting salt

How it works

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