CivilSolve

Environmental

Stream Mixing & Dilution

Conservative steady-state mixing of two or more streams: mixed flow Q = ΣQi, flow-weighted concentration C = ΣQiCi/ΣQi, mass loading of each stream in kg/day, and an optional completely-mixed first-order decay C/(1 + kθ). Handles the mg/L ≡ g/m³ identity explicitly. Use for: 'concentration downstream of an outfall', mixing two flows, dilution problems, mass loading in kg/day or lb/day. Not for: plug-flow decay (exponential) or non-conservative reactions during mixing.

Direct solving is free and needs no account. Have a word problem instead? Submit it as text.

Inputs

Load a sample problem:
streamsThe streams being mixed (river + discharge, two pipes, …) — at least two
streams 1
first order decayOptional completely-mixed (CSTR) first-order decay applied AFTER mixing: C = C_mix / (1 + kθ)

Worked examples

These are the solver's own reference problems — the answers come from published sources or independent hand computation, never from the solver itself. The solution below is the live solver output for each.

Example 1

SI river + outfall: 10 m³/s at 5 mg/L + 1 m³/s at 50 mg/L

Given

streams
  1. 1.flowrate 10 m^3/sconcentration 5 mg/L
  2. 2.flowrate 1 m^3/sconcentration 50 mg/L

Assumptions

  • Steady state with complete mixing at the confluence; the constituent is conservative during mixing (no reaction, settling or volatilization).

Solution steps

  1. The mg/L ≡ g/m³ identity

    A liter is a thousandth of a cubic meter and a milligram is a thousandth of a gram, so the factors cancel: concentrations in mg/L ARE g/m³. Multiplying m³/s by mg/L therefore gives g/s directly — no conversion factor needed.

    1 mg/L = 1 g/m³

  2. Flow balance

    Water volume is conserved: flows simply add.

    Qmix = ΣQi

    Qmix = 10 m^3/s + 1 m^3/s = 11 m^3/s

    = 11 m^3/s

  3. Conservative mass balance

    Constituent mass is conserved, so the mixed concentration is the flow-weighted average of the stream concentrations.

    Cmix = Σ(Qi·Ci) / ΣQi

    Cmix = (10 m^3/s × 5 mg/L + 1 m^3/s × 50 mg/L) / 11 m^3/s = 9.091 mg/L

    = 9.091 mg/L

  4. Mass loading of each stream

    Q·C is a mass flux: (m³/s)·(g/m³) = g/s, reported in kg/day — the units regulators and design guides use.

    ṁi = Qi·Ci

    per stream: 4320 kg/day; 4320 kg/day; total = 8640 kg/day

    = 8640

Results

Mixed flow rate Qmix

11m^3/s

Mixed (flow-weighted) concentration Cmix

9.091mg/L

Mass loading of stream 1 (kg/day)

4320

Mass loading of stream 2 (kg/day)

4320

Total mass loading (kg/day)

8640

Where this answer was checked
source
Conservative mass balance (NCEES FE Reference Handbook, mixing/dilution) — Qmix = ΣQ; Cmix = ΣQC/ΣQ; loadings Q·C in kg/day
verified by
hand-recomputed
derivation
Qmix = 11 m³/s. Cmix = (10·5 + 1·50)/11 = 100/11 = 9.0909 mg/L. Loads: 10·5 = 50 g/s = 50·86.4 = 4320 kg/day; 1·50 = 50 g/s = 4320 kg/day; total 8640 kg/day.

Example 2

US wastewater blend: 8 MGD at 2 mg/L + 2 MGD at 25 mg/L

Given

streams
  1. 1.flowrate 8 MGDconcentration 2 mg/L
  2. 2.flowrate 2 MGDconcentration 25 mg/L

Assumptions

  • Steady state with complete mixing at the confluence; the constituent is conservative during mixing (no reaction, settling or volatilization).

Solution steps

  1. The mg/L ≡ g/m³ identity

    A liter is a thousandth of a cubic meter and a milligram is a thousandth of a gram, so the factors cancel: concentrations in mg/L ARE g/m³. Multiplying m³/s by mg/L therefore gives g/s directly — no conversion factor needed.

    1 mg/L = 1 g/m³

  2. Flow balance

    Water volume is conserved: flows simply add.

    Qmix = ΣQi

    Qmix = 8 mgd + 2 mgd = 10 mgd

    = 10 mgd

  3. Conservative mass balance

    Constituent mass is conserved, so the mixed concentration is the flow-weighted average of the stream concentrations.

    Cmix = Σ(Qi·Ci) / ΣQi

    Cmix = (8 mgd × 2 mg/L + 2 mgd × 25 mg/L) / 10 mgd = 6.6 mg/L

    = 6.6 mg/L

  4. Mass loading of each stream

    Q·C is a mass flux: (m³/s)·(g/m³) = g/s, reported in kg/day — the units regulators and design guides use.

    ṁi = Qi·Ci

    per stream: 60.57 kg/day; 189.3 kg/day; total = 249.8 kg/day

    = 249.8

  5. Handbook form check

    The familiar US form lb/day = 8.34 · Q(MGD) · C(mg/L) hides the conversion 1 MGD · 1 mg/L = 8.345 lb/day; the small difference from the exact value is that rounded 8.34.

    ṁ(lb/day) ≈ 8.34 · Q(MGD) · C(mg/L)

    shortcut: 550.4 lb/day vs exact 550.8 lb/day

Results

Mixed flow rate Qmix

10mgd

Mixed (flow-weighted) concentration Cmix

6.6mg/L

Mass loading of stream 1 (kg/day)

60.57

Mass loading of stream 2 (kg/day)

189.3

Total mass loading (kg/day)

249.8

Where this answer was checked
source
Conservative mass balance with US flows (NCEES FE Reference Handbook) — Cmix in MGD·mg/L units; loadings via 1 MGD = 3785.411784 m³/day
verified by
hand-recomputed
derivation
Qmix = 10 MGD. Cmix = (8·2 + 2·25)/10 = 66/10 = 6.6 mg/L. Load 1 = 8·3785.411784 m³/day · 2 g/m³ = 60566.6 g/day = 60.567 kg/day; load 2 = 2·3785.411784·25 = 189270.6 g/day = 189.27 kg/day; total = 249.84 kg/day (folklore check: 8.34·10·6.6 = 550.4 lb/day ≈ 249.8 kg/day ✓).

Example 3

mixing then CSTR decay: Cmix = 40 mg/L, k = 0.5/day, θ = 2 day → 20 mg/L

Given

streams
  1. 1.flowrate 0.3 m^3/sconcentration 20 mg/L
  2. 2.flowrate 0.1 m^3/sconcentration 100 mg/L
first order decay
rate 0.5 1/daydetention time 2 day

Assumptions

  • Steady state with complete mixing at the confluence; the constituent is conservative during mixing (no reaction, settling or volatilization).
  • Decay is applied as a steady-state completely-mixed (CSTR) reactor after mixing; a plug-flow reach would decay further, to C·e^(−kθ).

Solution steps

  1. The mg/L ≡ g/m³ identity

    A liter is a thousandth of a cubic meter and a milligram is a thousandth of a gram, so the factors cancel: concentrations in mg/L ARE g/m³. Multiplying m³/s by mg/L therefore gives g/s directly — no conversion factor needed.

    1 mg/L = 1 g/m³

  2. Flow balance

    Water volume is conserved: flows simply add.

    Qmix = ΣQi

    Qmix = 0.3 m^3/s + 0.1 m^3/s = 0.4 m^3/s

    = 0.4 m^3/s

  3. Conservative mass balance

    Constituent mass is conserved, so the mixed concentration is the flow-weighted average of the stream concentrations.

    Cmix = Σ(Qi·Ci) / ΣQi

    Cmix = (0.3 m^3/s × 20 mg/L + 0.1 m^3/s × 100 mg/L) / 0.4 m^3/s = 40 mg/L

    = 40 mg/L

  4. Mass loading of each stream

    Q·C is a mass flux: (m³/s)·(g/m³) = g/s, reported in kg/day — the units regulators and design guides use.

    ṁi = Qi·Ci

    per stream: 518.4 kg/day; 864 kg/day; total = 1382 kg/day

    = 1382

  5. First-order decay in the mixed volume (CSTR)

    A completely mixed volume at steady state with first-order decay dilutes the mixed concentration by 1/(1 + kθ).

    Cout = Cmix / (1 + k·θ)

    k·θ = 0.5 per day × 2 day = 1; Cout = 40 mg/L / 2 = 20 mg/L

    = 20 mg/L

Results

Mixed flow rate Qmix

0.4m^3/s

Mixed (flow-weighted) concentration Cmix

40mg/L

Mass loading of stream 1 (kg/day)

518.4

Mass loading of stream 2 (kg/day)

864

Total mass loading (kg/day)

1382

Concentration after first-order decay, C/(1 + kθ)

20mg/L

Where this answer was checked
source
Conservative mass balance + steady-state CSTR first-order decay — C = Cmix/(1 + kθ)
verified by
hand-recomputed
derivation
Qmix = 0.3 + 0.1 = 0.4 m³/s. Cmix = (0.3·20 + 0.1·100)/0.4 = (6 + 10)/0.4 = 40 mg/L. kθ = 0.5·2 = 1.0. C = 40/(1 + 1) = 20 mg/L. Loads: 6 g/s = 518.4 kg/day; 10 g/s = 864 kg/day.