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Illustrated guide · Operating calculations

How are water balance, recovery and concentration factor calculated?

Fix the boundary and time basis, then close water and conservative-solute balances before interpreting RO recovery or concentration.

Direct answer

Direct answer

Draw the control boundary, choose one time basis and list every stream plus storage change. The general water balance is ΣQin−ΣQout=dV/dt. For a steady RO skid without material flush, sample or leak flows, Qf=Qp+Qc and recovery Y=Qp/Qf. For a conservative solute at steady state, QfCf=QpCp+QcCc, hence Cc=(QfCf−QpCp)/Qc. Only when permeate solute is negligible and Qc=Qf(1−Y) does the ideal concentration factor become CF=Cc/Cf≈1/(1−Y): 75% recovery gives about 4×, not 1.75×. Actual species ratios change with salt passage, precipitation, dosing, reaction, leakage and analytical error; conductivity is not an exact mass concentration for every ion. Stage recoveries cannot simply be added. If stage 2 treats only stage-1 concentrate with no recycle, Ytotal=1−(1−Y₁)(1−Y₂). Batch/recirculating systems require initial/final inventory; internal recycle must not be counted as new feed at the overall boundary. Higher recovery cuts concentrate volume but raises terminal concentration, osmotic pressure and scaling risk and lowers concentrate-side flow. Use synchronized totalizers, calibrated meters and tank levels, report closure error and cross-check with a conservative ion or temperature-compensated conductivity trend.

Fix four calculation boundaries

The equations are short; boundary, time and hidden streams cause most errors.

Element, stage, pass, system or plant?

Recycle may cross a local boundary but cancel inside the overall system. Draw and name every crossing.

Snapshot steady state or cumulative period?

Tank changes, startup displacement, flush and intermittent drains require synchronized volumes plus ΔV.

Water concentration or a species?

1/(1−Y) is an ideal volume relationship; each ion needs a Q×C balance including passage and sinks/sources.

Are data on one traceable basis?

Meter range/zero, totalizer reset, level-volume curve, temperature, density and sample time must align.

1

Draw feed, permeate, concentrate and storage boundaries on the real RO plant

The rack, high-pressure pump, online flow instruments and product/concentrate tanks put rate and inventory in one balance.

Draw feed, permeate, concentrate and storage boundaries on the real RO plant：RO feed Qf and high-pressure pump、Permeate Qp and product tank、Concentrate Qc and brine tank、Meters, levels and sample side flows1234

What to identify

  1. 1RO feed Qf and high-pressure pump
  2. 2Permeate Qp and product tank
  3. 3Concentrate Qc and brine tank
  4. 4Meters, levels and sample side flows

What the image proves

Steady RO has one main inlet and two outlets; tanks, flushes, sampling, drains and recycle add ΔV and auxiliary terms.

How to verify it

Number every stream on a P&ID-like sketch and read totalizers plus tank levels at the same start/end times.

2

Bench mass conservation shows that removed water does not remove all salt

A measured feed divides through a small membrane cell into clear permeate and darker concentrate, with cylinders, scales and salt samples.

Bench mass conservation shows that removed water does not remove all salt：Initial Vf and Cf、Membrane control boundary、Permeate Vp and Cp、Remaining Vc and Cc1234

What to identify

  1. 1Initial Vf and Cf
  2. 2Membrane control boundary
  3. 3Permeate Vp and Cp
  4. 4Remaining Vc and Cc

What the image proves

Batch water balance is Vf=Vp+Vc; conservative solute balance is VfCf=VpCp+VcCc.

How to verify it

Measure all three volumes/masses and one species. Missing salt points to precipitation, adsorption, sampling, hold-up or analysis.

3

A transparent continuous pilot synchronizes Qf, Qp, Qc and three quality points

Feed tank and pump supply a clear pressure vessel while separate receivers and sensors identify all streams.

A transparent continuous pilot synchronizes Qf, Qp, Qc and three quality points：Feed tank Qf/Cf、Membrane and operating pressure、Permeate Qp/Cp、Concentrate Qc/Cc1234

What to identify

  1. 1Feed tank Qf/Cf
  2. 2Membrane and operating pressure
  3. 3Permeate Qp/Cp
  4. 4Concentrate Qc/Cc

What the image proves

Recovery Y=Qp/Qf and salt rejection are different metrics; both permeate and concentrate matter.

How to verify it

Wait for stable pressure, flows, conductivities and level slopes; close water and Q×C balances with simultaneous samples.

4

Three rigs show that high recovery, normal recovery and hidden loss can look alike

Independent loop inventories and side hoses differ; abnormal foaming or drainage exposes a missing stream/nonsteady state.

Three rigs show that high recovery, normal recovery and hidden loss can look alike：Lower recovery: larger concentrate、Hidden side flow/leak/nonsteady state、Higher recovery: smaller concentrate、Independent boundaries and storage1234

What to identify

  1. 1Lower recovery: larger concentrate
  2. 2Hidden side flow/leak/nonsteady state
  3. 3Higher recovery: smaller concentrate
  4. 4Independent boundaries and storage

What the image proves

Recovery is comparable only for the same boundary and period. Small visible brine can also be a meter or inventory error.

How to verify it

Trace valves, hoses, drains, flush, samples and seal leaks; validate online totals against tank-volume change.

5

Field reconciliation checks meters, levels, drains and samples together

Engineers read flow and tank level, capture leakage and use portable analysis on the three streams.

Field reconciliation checks meters, levels, drains and samples together：Feed/permeate/concentrate meters、Tank level-volume change、Leak, flush and drain loss、Three samples and compensated quality1234

What to identify

  1. 1Feed/permeate/concentrate meters
  2. 2Tank level-volume change
  3. 3Leak, flush and drain loss
  4. 4Three samples and compensated quality

What the image proves

A credible account closes water and a suitable conservative solute and explains the residual.

How to verify it

Calibrate or clamp-check key meters, time a volume test, measure starting/ending levels and collect visible drains; investigate rather than allocate residual away.

Nine steps from boundary to accepted closure

Keep the same control volume, period and units.

  1. 1. Boundary

    element/stage/pass/system/plant

    Classify crossing and internal recycle.

  2. 2. Streams

    feed/product/brine/flush/sample/drain/leak/reuse

    Avoid a three-meter-only account.

  3. 3. Time

    stable snapshot or synchronized Δt

    Do not mix operating states.

  4. 4. Storage

    ΔV=Vend−Vstart

    Handle tanks, batch and transients.

  5. 5. Water closure

    ΣVin−ΣVout−ΔV=residual

    Test boundary and metering.

  6. 6. Recovery

    qualified product/fresh feed

    Name stage, pass, system or plant Y.

  7. 7. Solute balance

    Σ(V×C)in−Σ(V×C)out−ΔM

    Find passage, source or sink.

  8. 8. Concentration

    observed Cc/Cf; ideal 1/(1−Y)

    Interpret departures.

  9. 9. Uncertainty

    accuracy+sampling+temperature+closure+outcome

    Make the result reproducible and safe.

Four commonly confused metrics

Write numerator, denominator and boundary.

Water balance

Definition
ΣQin−ΣQout=dV/dt; cumulative form includes inventory.
Does not prove
Closure alone does not prove quality or each meter's accuracy.
Use
Find missing flow, duplicate recycle, meter bias and transient storage.

Water recovery Y

Definition
Qualified product/fresh feed for the named boundary.
Does not prove
Not salt rejection or automatically plant net reuse.
Use
Quantify product-water fraction with brine and risk.

Salt rejection/passage

Definition
Often R=1−Cp/Cf; state the feed concentration basis.
Does not prove
Cannot replace Y; high rejection says nothing about brine volume.
Use
Assess membrane ion separation after normalization.

Concentration factor

Definition
Observed species Cc/Cf; ideal volume CF≈1/(1−Y).
Does not prove
Not every ion equals conductivity ratio when passage/reaction occurs.
Use
Estimate osmotic/scaling load and verify by solute balance.

For two series stages where stage 2 treats only stage-1 concentrate with no other stream, Ytotal=1−(1−Y₁)(1−Y₂). Two 50% stages give 75%, not 100%. Redraw the overall boundary if interstage recycle, bypass or a second pass changes the topology.

Retain four raw evidence sets

Boundary/time

Numbered flow diagram, valve/recycle state, stability criterion, start/end and flush events.

Water quantity

Instantaneous and totalized flows, calibration/range, tank level-volume curve, leaks and collected drains.

Water quality

Simultaneous samples, compensated conductivity, TDS/target ion, pH, precipitation and method limits.

Operating result

Last-stage flow/ΔP, normalized permeate/salt passage, osmotic/scaling prediction, CIP and compliant yield.

Four closure conflicts

Signal
Qf exceeds Qp+Qc while level rises or flushing occurs
Likely cause
Transient storage or omitted auxiliary flow
First action
Use synchronized totals and add ΔV/flush/sample/leak
Signal
Water closes but solute mass disappears with turbidity/scale
Likely cause
Precipitation, adsorption or analysis bias
First action
Check scale, filtered/dissolved ions, pH/T and deposit
Signal
High Y with low last-stage flow and high brine EC/ΔP
Likely cause
Real concentration near hydraulic/scaling limit
First action
Check design, scaling, minimum concentrate flow and normalized data
Signal
Flow recovery stable but EC ratio differs from 1/(1−Y)
Likely cause
Passage, compensation/composition, dosing or nonconservative EC
First action
Use a specific conservative ion and exact Q×C balance

Four misconceptions

75% recovery means a 1.75 concentration factor

Ideal CF is 1/(1−0.75)=4; recovery is removed-water fraction, CF is inverse remaining fraction.

Stage recoveries add

A downstream stage treats the upstream remainder; multiply remaining fractions or redraw a complex boundary.

Conductivity ratio is exact salt concentration

Temperature, composition, ion activity, passage, precipitation and dosing alter it.

Higher recovery is always better

It raises concentration, osmotic pressure, scaling and terminal low-flow risk; optimize safe compliant lifecycle cost.

Continue learning

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