Example data table
Sample scenarios use complete mixing (100%) and no decay. Concentrations are in mg/L; flows are in m³/s.
| Scenario | Qr | Cr | Qd | Cd | Mixed C |
|---|---|---|---|---|---|
| 1 | 40 | 1.2 | 1.5 | 120 | 5.49 |
| 2 | 55 | 2 | 2 | 80 | 4.74 |
| 3 | 30 | 0.5 | 0.8 | 60 | 2.05 |
| 4 | 75 | 1 | 3 | 50 | 2.88 |
| 5 | 20 | 3 | 0.5 | 200 | 7.8 |
Formula used
This tool uses a steady-state mass balance for complete mixing at a control point:
- Effective discharge mixing: Qd,eff = Qd·η, where η is mixing efficiency (0–1).
- Dilution factor: DF = (Qr + Qd,eff) / Qd,eff.
- River:discharge ratio: R = Qr / Qd,eff.
If decay is enabled, the calculator applies first-order decay only to the incremental concentration above background:
This keeps the background concentration stable while the added load attenuates with travel time.
How to use this calculator
- Enter upstream river flow and concentration, then choose units.
- Enter discharge flow and concentration from the outfall or tributary.
- Set mixing efficiency to reflect your mixing assumption.
- Optionally enable decay and enter k and travel time to your compliance point.
- Add a target limit to get a PASS/FAIL indicator and allowable thresholds.
- Press Calculate to see results above the form, then export CSV or PDF.
Notes and good practice
- Use consistent sampling conditions: the same flow regime and comparable time windows.
- When mixing is uncertain, run multiple efficiencies (for example 60%, 80%, 100%).
- Decay is parameter-sensitive; cite sources or site measurements for k and travel time.
- This calculator supports quick screening and documentation, not site design sign-off.
Mass balance at the mixing control point
At the mixing location, the calculator applies conservation of mass using flows in m³/s and concentrations in mg/L. The total flow Qt equals Qr plus the effective discharge Qd,eff. For example, Qr=50 and Qd=2 gives Qt=52. If Cr=2 and Cd=80, the mixed value becomes (50×2+2×80)/52=5.0 mg/L, showing how a small discharge can still raise levels. A 1 m³/s increase in Qr raises Qt by 1 and lowers Cmix, which is most visible during low‑flow seasons in practice.
Interpreting dilution factor and ratios
Dilution factor (DF) is Qt divided by Qd,eff, so it increases as the river gets larger or mixing gets weaker. With Qr=50 and Qd,eff=2, DF=26, meaning one part discharge is blended into about twenty‑six parts total flow. The river:discharge ratio (Qr/Qd,eff) is 25 in the same case. These indicators help compare sites when flows change seasonally.
Mixing efficiency and near-field uncertainty
Real rivers rarely mix instantly across the full channel. The tool uses mixing efficiency η to scale the discharge that actually blends at the control point. If η=60%, a 2 m³/s discharge acts like 1.2 m³/s for mixing, reducing Qt and raising Cmix. This is a conservative screen for near‑field conditions, where jets, stratification, and channel geometry control dilution.
First-order decay for reactive pollutants
Some constituents decay, volatilize, or transform as water travels downstream. The calculator applies first‑order decay only to the increment above background: Cdown = Cr + (Cmix−Cr)e^(−kt). If k=0.2 1/day and travel time is 12 hours, kt=0.1 and e^(−kt)=0.905. A 3.0 mg/L increment at mixing becomes 2.72 mg/L at the compliance point.
Using results for compliance and planning
Compare Cdown to your target limit to get PASS or FAIL and to estimate allowable discharge limits. The tool also reports Cd,max and Qd,max that meet the target under current assumptions. Use these outputs to document screening decisions, test scenarios (higher flows, lower Cd, different η), and prioritize field data collection before detailed modeling or permit submittals.
FAQs
What does mixing efficiency change in the calculation?
Mixing efficiency scales the discharge flow that actually blends at the control point. A value of 70% means only 0.70·Qd is used in the mass balance, producing higher mixed concentrations than full mixing.
Why is decay applied only to the increment above background?
Upstream background often represents stable ambient conditions. Applying decay to only (Cmix−Cr) models attenuation of the added load while keeping the river’s baseline concentration unchanged in the downstream estimate.
Can I mix units like m³/day and µg/L?
Yes. The tool converts common flow and concentration units to a consistent basis before computing results, so field measurements can be entered without manual conversion.
What should I do if the result fails the target limit?
Try realistic scenario tests: higher river flow, lower discharge concentration, improved treatment, or a lower mixing efficiency for conservatism. Use the reported Cd,max and Qd,max as screening thresholds, then confirm with site data.
How are Cd,max and Qd,max determined?
Cd,max is rearranged from the downstream limit equation using your current flows, mixing efficiency, and decay factor. Qd,max solves the same relationship for discharge flow at the current Cd; infeasible cases return 0 or N/A.
Is this calculator suitable for permit-grade design?
It is best for screening, documentation, and quick sensitivity checks. Permitting and design typically require site-specific hydrodynamics, geometry, and validated parameters, which may need specialized models and regulatory guidance.