Water tools

RO pressure vessel performance by element integration

Pressure, osmotic pressure and polarisation all change along an RO element, so the honest answer is an integration. This tool walks the feed through each element of a vessel with the solution-diffusion model.

Feed flow to the vesselm³/d

Feed pressurebar

Permeate back-pressurebar

Feed TDS (as NaCl)mg/L

Temperature°C

Elements per vessel

Water permeability kWL/(m²·h·bar)Product specific; brackish elements about 1 to 4.

Salt permeability kSm/hProduct specific; about 0.0003 to 0.001 m/h.

Membrane area per elementm²

Element lengthm

Feed channel heightmmEffective height after the spacer.

Head loss per element at the reference velocitybar

Reference velocity for that head lossm/s

Osmotic coefficient φ

Vessel

Permeate flow

155.3m³/d

Recovery

57.5%

Permeate TDS

76mg/L

Rejection

96.21%

Average flux

28.4L/m²·h

Concentrate

Concentrate flow

114.7m³/d

Concentrate TDS

4,604mg/L

Concentrate pressure

12.50bar

Pressure drop over the vessel

1.70bar

Highest polarisation factor

1.213

Per element

Element 1 flux

33.6L/m²·h

Element 2 flux

31.8L/m²·h

Element 3 flux

30.1L/m²·h

Element 4 flux

28.5L/m²·h

Element 5 flux

26.8L/m²·h

Element 6 flux

25.1L/m²·h

Element 7 flux

23.1L/m²·h

Polarisation factor above 1.2 somewhere in the vessel: raise crossflow or lower flux.

How it works

Each element is cut into ten slices. In each slice the water flux and the salt flux are

J_{W} = k_{W} [(P_{F C} - P_{P}) - (β π_{F C} - π_{P})] J_{S} = k_{S} (β C_{F C} - C_{P})

(1)

with the polarisation factor solved together with the flux:

β = R e j \exp (J_{W} / k_{C P}) + (1 - R e j)

(2)

Mass and flow balances carry the concentrate to the next slice, and the feed channel loses pressure in proportion to velocity squared:

Q_{F C, z + 1} = Q_{F C, z} - J_{W} w d z C_{F C, z + 1} = \frac{Q_{F C, z} C_{F C, z} - J_{S} w d z}{Q_{F C, z + 1}} h_{L} = δ v^{2} L

(3)

where

$J_{W}$: water flux, L/m²·h
$k_{W}$: water permeability, L/m²·h·bar
$P_{F C}, P_{P}$: feed-concentrate channel and permeate pressure, bar
$π_{F C}, π_{P}$: osmotic pressure in the channel and the permeate, bar
$β$: concentration polarisation factor
$J_{S}$: salt flux, mg/m²·h
$k_{S}$: salt permeability, m/h
$C_{F C}, C_{P}$: salt concentration in the channel and the permeate, mg/L
$R e j$: rejection, fraction
$k_{C P}$: boundary layer mass transfer coefficient, m/s
$Q_{F C, z}$: channel flow at position z, m³/s
$w$: effective membrane width, area over length, m
$d z$: slice length, m
$h_{L}$: head loss, bar
$δ$: head loss coefficient, bar·s²/m³
$v$: channel velocity, m/s
$L$: element length, m

Elements in a vessel are chained, the concentrate of one feeding the next; permeate quality is the flow-weighted average. The polarisation mass-transfer coefficient uses a spacer-filled channel correlation on Reynolds and Schmidt numbers. Permeabilities k_W and k_S are product specific and come from the supplier's projection software or pilot data; fouling and ageing are not modelled, and real designs fix recovery first and iterate on pressure and array.

RO pressure vessel performance by element integration

How it works

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