Product production at steady state, single substrate limiting

Example 1

First-order BOD decay — solve for remaining BOD — Product production at steady state, single substrate limiting

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 380.0 mg/L; rate constant (k) = 0.1800 1/day; time (t) = 6.5000 day, determine the remaining BOD (Lt) in mg/L.

Given

$ultimate BOD (L_{0}) = 380.0 mg/L$
$rate constant (k) = 0.1800 1/day$
$time (t) = 6.5000 day$

Find

remaining BOD (Lt), in mg/L

Start with the thinking

The governing relation printed in this handbook section is First-order BOD decay.
Everything except Lt is given, so isolate Lt symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$L_t = L_0 e^{-k t}$
Step 2 — Rearrange the relation so that Lt stands alone on the left-hand side.
Step 3
$List the givens: ultimate BOD (L_{0}) = 380.0 mg/L, rate constant (k) = 0.1800 1/day, time (t) = 6.5000 day$
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$Lt = 117.9\ \text{mg/L}$
Step 6 — Check: returning Lt = 117.9 mg/L to
$L_t = L_0 e^{-k t}$
reproduces the given quantities, and both sides carry the same units.

Answer:

Lt = 117.9\ \text{mg/L}

Why the other options are there

235.9 — kept a factor of two that cancels in the correct rearrangement.
58.9697 — dropped that same factor in the other direction.
129.7 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 2

Steady-state mass balance — solve for blended concentration — Product production at steady state, single substrate limiting (2)

A environmental engineering problem uses Steady-state mass balance. Given flow 1 (Q1) = 10.0000 MGD; concentration 1 (C1) = 22.5000 mg/L; flow 2 (Q2) = 15.5000 MGD; concentration 2 (C2) = 37.5000 mg/L, determine the blended concentration (C) in mg/L.

Given

$flow 1 (Q_{1}) = 10.0000 MGD$
$concentration 1 (C_{1}) = 22.5000 mg/L$
$flow 2 (Q_{2}) = 15.5000 MGD$
$concentration 2 (C_{2}) = 37.5000 mg/L$

Find

blended concentration (C), in mg/L

Start with the thinking

The governing relation printed in this handbook section is Steady-state mass balance.
Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
Step 2 — Rearrange the relation so that C stands alone on the left-hand side.
Step 3 — List the givens: flow 1 (Q1) = 10.0000 MGD, concentration 1 (C1) = 22.5000 mg/L, flow 2 (Q2) = 15.5000 MGD, concentration 2 (C2) = 37.5000 mg/L.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$C = 31.6176\ \text{mg/L}$
Step 6 — Check: returning C = 31.6176 mg/L to
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
reproduces the given quantities, and both sides carry the same units.

Answer:

C = 31.6176\ \text{mg/L}

Why the other options are there

63.2353 — kept a factor of two that cancels in the correct rearrangement.
15.8088 — dropped that same factor in the other direction.
34.7794 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 3

First-order BOD decay — solve for ultimate BOD — Product production at steady state, single substrate limiting (3)

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2500 1/day; time (t) = 9.0000 day; remaining BOD (Lt) = 344.0 mg/L, determine the ultimate BOD (L0) in mg/L.

Given

$rate constant (k) = 0.2500 1/day$
$time (t) = 9.0000 day$
$remaining BOD (Lt) = 344.0 mg/L$

Find

ultimate BOD (L0), in mg/L

Start with the thinking

The governing relation printed in this handbook section is First-order BOD decay.
Everything except L0 is given, so isolate L0 symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$L_t = L_0 e^{-k t}$
Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.
Step 3
$List the givens: rate constant (k) = 0.2500 1/day, time (t) = 9.0000 day, remaining BOD (Lt) = 344.0 mg/L$
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$L_{0} = 3264\ \text{mg/L}$
Step 6 — Check: returning L0 = 3,264 mg/L to
$L_t = L_0 e^{-k t}$
reproduces the given quantities, and both sides carry the same units.

Answer:

L_{0} = 3264\ \text{mg/L}

Why the other options are there

6,528 — kept a factor of two that cancels in the correct rearrangement.
1,632 — dropped that same factor in the other direction.
3,590 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 4

Steady-state mass balance — solve for concentration 1 — Product production at steady state, single substrate limiting (4)

A environmental engineering problem uses Steady-state mass balance. Given flow 1 (Q1) = 14.0000 MGD; flow 2 (Q2) = 16.5000 MGD; concentration 2 (C2) = 29.0000 mg/L; blended concentration (C) = 8.7300 mg/L, determine the concentration 1 (C1) in mg/L.

Given

$flow 1 (Q_{1}) = 14.0000 MGD$
$flow 2 (Q_{2}) = 16.5000 MGD$
$concentration 2 (C_{2}) = 29.0000 mg/L$
$blended concentration (C) = 8.7300 mg/L$

Find

concentration 1 (C1), in mg/L

Start with the thinking

The governing relation printed in this handbook section is Steady-state mass balance.
Everything except C1 is given, so isolate C1 symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
Step 2 — Rearrange the relation so that C1 stands alone on the left-hand side.
Step 3 — List the givens: flow 1 (Q1) = 14.0000 MGD, flow 2 (Q2) = 16.5000 MGD, concentration 2 (C2) = 29.0000 mg/L, blended concentration (C) = 8.7300 mg/L.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$C_{1} = -15.1596\ \text{mg/L}$
Step 6 — Check: returning C1 = -15.1596 mg/L to
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
reproduces the given quantities, and both sides carry the same units.

Answer:

C_{1} = -15.1596\ \text{mg/L}

Why the other options are there

-30.3193 — kept a factor of two that cancels in the correct rearrangement.
-7.5798 — dropped that same factor in the other direction.
-16.6756 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 5

First-order BOD decay — solve for rate constant — Product production at steady state, single substrate limiting (5)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 250.0 mg/L; time (t) = 7.0000 day; remaining BOD (Lt) = 143.7 mg/L, determine the rate constant (k) in 1/day.

Given

$ultimate BOD (L_{0}) = 250.0 mg/L$
$time (t) = 7.0000 day$
$remaining BOD (Lt) = 143.7 mg/L$

Find

rate constant (k), in 1/day

Start with the thinking

The governing relation printed in this handbook section is First-order BOD decay.
Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$L_t = L_0 e^{-k t}$
Step 2 — Rearrange the relation so that k stands alone on the left-hand side.
Step 3
$List the givens: ultimate BOD (L_{0}) = 250.0 mg/L, time (t) = 7.0000 day, remaining BOD (Lt) = 143.7 mg/L$
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$k = 0.0791\ \text{1/day}$
Step 6 — Check: returning k = 0.0791 1/day to
$L_t = L_0 e^{-k t}$
reproduces the given quantities, and both sides carry the same units.

Answer:

k = 0.0791\ \text{1/day}

Why the other options are there

0.1582 — kept a factor of two that cancels in the correct rearrangement.
0.0396 — dropped that same factor in the other direction.
0.0870 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 6

Steady-state mass balance — solve for blended concentration (case 2) — Product production at steady state, single substrate limiting (6)

A environmental engineering problem uses Steady-state mass balance. Given flow 1 (Q1) = 10.0000 MGD; concentration 1 (C1) = 24.5000 mg/L; flow 2 (Q2) = 10.0000 MGD; concentration 2 (C2) = 26.0000 mg/L, determine the blended concentration (C) in mg/L.

Given

$flow 1 (Q_{1}) = 10.0000 MGD$
$concentration 1 (C_{1}) = 24.5000 mg/L$
$flow 2 (Q_{2}) = 10.0000 MGD$
$concentration 2 (C_{2}) = 26.0000 mg/L$

Find

blended concentration (C), in mg/L

Start with the thinking

The governing relation printed in this handbook section is Steady-state mass balance.
Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
Step 2 — Rearrange the relation so that C stands alone on the left-hand side.
Step 3 — List the givens: flow 1 (Q1) = 10.0000 MGD, concentration 1 (C1) = 24.5000 mg/L, flow 2 (Q2) = 10.0000 MGD, concentration 2 (C2) = 26.0000 mg/L.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$C = 25.2500\ \text{mg/L}$
Step 6 — Check: returning C = 25.2500 mg/L to
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
reproduces the given quantities, and both sides carry the same units.

Answer:

C = 25.2500\ \text{mg/L}

Why the other options are there

50.5000 — kept a factor of two that cancels in the correct rearrangement.
12.6250 — dropped that same factor in the other direction.
27.7750 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 7

First-order BOD decay — solve for remaining BOD (case 2) — Product production at steady state, single substrate limiting (7)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 375.0 mg/L; rate constant (k) = 0.1900 1/day; time (t) = 6.5000 day, determine the remaining BOD (Lt) in mg/L.

Given

$ultimate BOD (L_{0}) = 375.0 mg/L$
$rate constant (k) = 0.1900 1/day$
$time (t) = 6.5000 day$

Find

remaining BOD (Lt), in mg/L

Start with the thinking

The governing relation printed in this handbook section is First-order BOD decay.
Everything except Lt is given, so isolate Lt symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$L_t = L_0 e^{-k t}$
Step 2 — Rearrange the relation so that Lt stands alone on the left-hand side.
Step 3
$List the givens: ultimate BOD (L_{0}) = 375.0 mg/L, rate constant (k) = 0.1900 1/day, time (t) = 6.5000 day$
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$Lt = 109.1\ \text{mg/L}$
Step 6 — Check: returning Lt = 109.1 mg/L to
$L_t = L_0 e^{-k t}$
reproduces the given quantities, and both sides carry the same units.

Answer:

Lt = 109.1\ \text{mg/L}

Why the other options are there

218.1 — kept a factor of two that cancels in the correct rearrangement.
54.5315 — dropped that same factor in the other direction.
120.0 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 8

Steady-state mass balance — solve for concentration 1 (case 2) — Product production at steady state, single substrate limiting (8)

A environmental engineering problem uses Steady-state mass balance. Given flow 1 (Q1) = 8.5000 MGD; flow 2 (Q2) = 9.0000 MGD; concentration 2 (C2) = 36.5000 mg/L; blended concentration (C) = 19.8000 mg/L, determine the concentration 1 (C1) in mg/L.

Given

$flow 1 (Q_{1}) = 8.5000 MGD$
$flow 2 (Q_{2}) = 9.0000 MGD$
$concentration 2 (C_{2}) = 36.5000 mg/L$
$blended concentration (C) = 19.8000 mg/L$

Find

concentration 1 (C1), in mg/L

Start with the thinking

The governing relation printed in this handbook section is Steady-state mass balance.
Everything except C1 is given, so isolate C1 symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
Step 2 — Rearrange the relation so that C1 stands alone on the left-hand side.
Step 3 — List the givens: flow 1 (Q1) = 8.5000 MGD, flow 2 (Q2) = 9.0000 MGD, concentration 2 (C2) = 36.5000 mg/L, blended concentration (C) = 19.8000 mg/L.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$C_{1} = 2.1176\ \text{mg/L}$
Step 6 — Check: returning C1 = 2.1176 mg/L to
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
reproduces the given quantities, and both sides carry the same units.

Answer:

C_{1} = 2.1176\ \text{mg/L}

Why the other options are there

4.2353 — kept a factor of two that cancels in the correct rearrangement.
1.0588 — dropped that same factor in the other direction.
2.3294 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 9

First-order BOD decay — solve for ultimate BOD (case 2) — Product production at steady state, single substrate limiting (9)

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1400 1/day; time (t) = 8.5000 day; remaining BOD (Lt) = 359.0 mg/L, determine the ultimate BOD (L0) in mg/L.

Given

$rate constant (k) = 0.1400 1/day$
$time (t) = 8.5000 day$
$remaining BOD (Lt) = 359.0 mg/L$

Find

ultimate BOD (L0), in mg/L

Start with the thinking

The governing relation printed in this handbook section is First-order BOD decay.
Everything except L0 is given, so isolate L0 symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$L_t = L_0 e^{-k t}$
Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.
Step 3
$List the givens: rate constant (k) = 0.1400 1/day, time (t) = 8.5000 day, remaining BOD (Lt) = 359.0 mg/L$
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$L_{0} = 1180\ \text{mg/L}$
Step 6 — Check: returning L0 = 1,180 mg/L to
$L_t = L_0 e^{-k t}$
reproduces the given quantities, and both sides carry the same units.

Answer:

L_{0} = 1180\ \text{mg/L}

Why the other options are there

2,360 — kept a factor of two that cancels in the correct rearrangement.
590.0 — dropped that same factor in the other direction.
1,298 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Example 10

Steady-state mass balance — solve for blended concentration (case 3) — Product production at steady state, single substrate limiting (10)

A environmental engineering problem uses Steady-state mass balance. Given flow 1 (Q1) = 19.5000 MGD; concentration 1 (C1) = 4.5000 mg/L; flow 2 (Q2) = 10.5000 MGD; concentration 2 (C2) = 48.0000 mg/L, determine the blended concentration (C) in mg/L.

Given

$flow 1 (Q_{1}) = 19.5000 MGD$
$concentration 1 (C_{1}) = 4.5000 mg/L$
$flow 2 (Q_{2}) = 10.5000 MGD$
$concentration 2 (C_{2}) = 48.0000 mg/L$

Find

blended concentration (C), in mg/L

Start with the thinking

The governing relation printed in this handbook section is Steady-state mass balance.
Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

Step 1 — State the governing relation:
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
Step 2 — Rearrange the relation so that C stands alone on the left-hand side.
Step 3 — List the givens: flow 1 (Q1) = 19.5000 MGD, concentration 1 (C1) = 4.5000 mg/L, flow 2 (Q2) = 10.5000 MGD, concentration 2 (C2) = 48.0000 mg/L.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$C = 19.7250\ \text{mg/L}$
Step 6 — Check: returning C = 19.7250 mg/L to
$C = (Q_1 C_1 + Q_2 C_2) / (Q_1 + Q_2)$
reproduces the given quantities, and both sides carry the same units.

Answer:

C = 19.7250\ \text{mg/L}

Why the other options are there

39.4500 — kept a factor of two that cancels in the correct rearrangement.
9.8625 — dropped that same factor in the other direction.
21.6975 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Product production at steady state, single substrate limiting

Product production at steady state, single substrate limiting

Core formulas for this FE topic

Worked exam-style examples

Product production at steady state, single substrate limiting

Core formulas for this FE topic

X1 = YP / S _ S0 − Si jNeeds technical review

kd=microbialdeathrateorendogenousdecayrateconstant(time−1)kd = microbial death rate or endogenous decay rate constant (time-1)kd=microbialdeathrateorendogenousdecayrateconstant(time−1)

km=maximumgrowthrateconstant(time−1)km = maximum growth rate constant (time-1)km=maximumgrowthrateconstant(time−1)

Ks=saturationconstantorhalf−velocityconstant=concentrationatµmax⁡/2Ks = saturation constant or half-velocity constant = concentration at µ\max /2Ks=saturationconstantorhalf−velocityconstant=concentrationatµmax/2

S=concentrationofsubstrateinsolution(mass/unitvolume)S = concentration of substrate in solution (mass/unit volume)S=concentrationofsubstrateinsolution(mass/unitvolume)

Y=yieldcoefficient[(mass/Lproduct)/(mass/Lfoodused)]Y = yield coefficient [(mass/L product)/(mass/L food used)]Y=yieldcoefficient[(mass/Lproduct)/(mass/Lfoodused)]

µ=specificgrowthrate(time−1)µ = specific growth rate (time-1)µ=specificgrowthrate(time−1)

µmax⁡=maximumspecificgrowthrate(time−1)=Ykmµ\max = maximum specific growth rate (time-1) = Ykmµmax=maximumspecificgrowthrate(time−1)=Ykm

Worked exam-style examples