How the Mass Balance Works
An RO system operating under steady-state conditions must satisfy two independent conservation laws simultaneously:
- Volumetric (Water) Mass Balance: the total volume of water entering equals the total leaving. Qf = Qp + Qc.
- Solute (TDS) Mass Balance: the total dissolved-solids load entering equals the total leaving. Qf x Cf = Qp x Cp + Qc x Cc.
Simplified TDS-based dissolved-solids balance. This calculator applies both balances sequentially, starting from the four user inputs (feed flow, feed TDS, recovery, and rejection), and reports both a volumetric balance check and a solute mass balance check so you can verify the calculation is internally consistent.
This is a simplified calculation intended for preliminary engineering checks and system-level balancing. It is not detailed membrane-element projection or simulation software and should not be used as a substitute for element-level performance modeling, fouling analysis, or project-specific process simulation.
The dissolved-solids load (kg/day) is derived from: Flow (m3/day) x Concentration (mg/L) / 1000. This is the standard unit conversion used in process engineering design.
Governing Equations
Qf * Cf = Qp * Cp + Qc * Cc Where:
-
Qf= Feed water flow rate entering the RO system [m3/day (or selected unit)] -
Qp= Permeate (product water) flow rate [m3/day] -
Qc= Concentrate (brine / reject) flow rate [m3/day] -
Cf= Feed water total dissolved solids (TDS) concentration [mg/L] -
Cp= Permeate TDS concentration (product water quality) [mg/L] -
Cc= Concentrate TDS concentration (brine salinity) [mg/L] -
Y= System recovery percentage [%] -
R= Salt rejection percentage [%] -
SP= Salt passage percentage (100 - R) [%]
Derived Equations:
Qp = Qf x (Y / 100) Qc = Qf - Qp Cp = Cf x (1 - R / 100) Load_f = (Qf x Cf) / 1000 Load_p = (Qp x Cp) / 1000 Load_c = Load_f - Load_p Cc = (Load_c x 1000) / Qc CF_water = Qf / Qc CF_tds = Cc / Cf Worked Engineering Example
Design Scenario: Seawater RO (SWRO) Train
A seawater desalination train processes a feed flow of 100 m3/day with a TDS of 35,000 mg/L, operating at 40% recovery and 99% salt rejection.
- Permeate Flow (Qp):
Qp = 100 x (40/100) = 40 m3/day - Concentrate Flow (Qc):
Qc = 100 - 40 = 60 m3/day - Salt Passage (SP):
SP = 100 - 99 = 1% - Permeate TDS (Cp):
Cp = 35,000 x (1 - 99/100) = 350 mg/L - Feed Dissolved-Solids Load:
Load_f = (100 x 35,000) / 1000 = 3,500 kg/day - Permeate Dissolved-Solids Load:
Load_p = (40 x 350) / 1000 = 14 kg/day - Concentrate Dissolved-Solids Load:
Load_c = 3,500 - 14 = 3,486 kg/day - Concentrate TDS (Cc):
Cc = (3,486 x 1000) / 60 = 58,100 mg/L - Water Concentration Factor:
CF_water = 100 / 60 = 1.6667x - TDS Concentration Factor:
CF_tds = 58,100 / 35,000 = 1.66x
The two concentration factors are similar but not equal because the permeate carries a small dissolved-solids load (14 kg/day). If salt rejection were 100%, CF_water and CF_tds would be identical.
Engineering Notes
Water Concentration Factor vs. TDS Concentration Factor
These two factors are distinct and should not be confused:
- Water Concentration Factor (CF_water = Qf / Qc) describes the volumetric ratio between the feed and the concentrate stream. It is determined purely by recovery. At Y = 40%, CF_water = 1.667x.
- TDS Concentration Factor (CF_tds = Cc / Cf) describes how much the actual concentrate TDS exceeds the feed TDS. It is determined by the interplay between recovery and salt rejection. At perfect rejection (R = 100%), CF_tds equals CF_water. In practice, some salt passes through (SP = 1-3%), so CF_tds is slightly lower than CF_water.
Practical Concentration Limits
| RO Application | Typical Recovery | CF_water | Practical Limit Driver |
|---|---|---|---|
| Seawater (SWRO) | 35% - 50% | 1.54x - 2.0x | High osmotic pressure; pump pressure limits |
| Brackish (BWRO) | 65% - 85% | 2.86x - 6.67x | Mineral scaling (CaSO4, BaSO4, silica) |
| Wastewater reuse | 70% - 80% | 3.33x - 5.0x | Organic fouling; silica; chloramine |
| High recovery / ZLD | 85% - 95% | 6.67x - 20x | Requires softening, antiscalants, interstage pumps |
Assumptions and Limitations
Engineering Assumptions:
- Steady-state operation: Feed flow, TDS, recovery, and rejection are assumed constant over the calculation period.
- Single-stage system: This calculator models a single RO train (single pass). Multi-stage or multi-pass configurations require separate stage-by-stage analysis.
- Simplified TDS model: TDS is treated as a single composite parameter. Ion-specific chemistry, osmotic pressure, activity coefficients, and scaling potential are not individually modeled.
- Negligible density differences: Feed, permeate, and concentrate stream densities are assumed equal, making volumetric and mass flow balances equivalent. This is valid for brackish and most seawater applications in preliminary design.
- System-level rejection: The salt rejection value is applied uniformly across all dissolved solids. Real systems have ion-specific rejection profiles; monovalent ions typically pass at higher rates than divalent ions.
Limitations:
- Does not calculate operating pressure, net driving pressure, or transmembrane pressure.
- Does not account for concentration polarization modulus.
- Does not model temperature effects on membrane permeability or osmotic pressure.
- Scaling saturation indices (LSI, Stiff-Davis, CCPP) must be calculated separately using ion-specific water chemistry analysis.
- Not a substitute for manufacturer membrane projection software (e.g., DuPont WAVE, Hydranautics IMSDesign, Toray DS2).
Continue in the RO Engineering Workbench
Carry feed flow, feed TDS, recovery, and rejection into the RO Engineering Workbench to extend this mass balance with temperature-driven osmotic pressure, permeate-quality checks, and scaling-risk screening — collated into a saved project with auto-generated engineering reports.
Frequently Asked Questions
Why do the water and TDS concentration factors differ?
CF_water (Qf / Qc) is set purely by recovery, while CF_tds (Cc / Cf) also depends on salt rejection. Because a small dissolved-solids load always escapes with the permeate, CF_tds is slightly lower than CF_water unless rejection is 100%.
How should concentrate TDS be verified for design?
Treat the computed concentrate TDS as a screening value. Confirm with membrane projection software (DuPont WAVE, Hydranautics IMSDesign, Toray DS2), laboratory analysis of the actual feed, and ion-specific water chemistry because real systems reject monovalent and divalent ions at different rates.
Can this balance be applied to each pressure vessel?
The calculator models a single-train, single-pass balance. Multi-stage or multi-pass configurations must be analyzed stage by stage, because interstage feed concentration changes recovery and rejection at each stage.
Engineering Disclaimer
Preliminary Engineering Estimate Only. This calculator uses simplified system-level flow and TDS relationships. Actual RO performance depends on feed-water chemistry, temperature, pressure, membrane characteristics, recovery, concentration polarization, pretreatment, and operating conditions. Verify results against project-specific data, manufacturer projections, applicable standards, and qualified engineering review before design or operation.
Technical References
- DuPont FilmTec™ Reverse Osmosis and Nanofiltration Technical Manual (Form No. 45-D01504-en).
- AWWA Manual M46: Reverse Osmosis and Nanofiltration, American Water Works Association.
- Crittenden, J.C. et al., MWH's Water Treatment: Principles and Design, 3rd ed., Wiley, 2012.
- Metcalf & Eddy / AECOM, Water Reuse: Issues, Technologies, and Applications, McGraw-Hill, 2007.