Math Problem Statement

A company has been discharging a non-reactive pollutant at a concentration of 15 g/m 3 to a lagoon for several years. The wastewater flow rate is 0.1 megaliters (ML) per day and the lagoon detention time is 10 days. The lagoon is assumed to be completely mixed. The overflow from the lagoon discharges into an adjacent river. Determine: a. The steady state concentration of the non-reactive pollutant in the effluent from the lagoon that overflows into the river. b. The concentration of the non-reactive pollutant in the effluent leaving the lagoon 10 days after the influent concentration is suddenly increased to 150 g/m 3 .

Solution

We are dealing with a pollutant discharge into a completely mixed lagoon, and we are tasked with determining:

  • (a) The steady-state concentration of the non-reactive pollutant in the lagoon effluent.
  • (b) The concentration of the pollutant after a sudden increase in influent concentration after 10 days.

Given:

  • Initial influent concentration, C0=15g/m3C_0 = 15 \, \text{g/m}^3
  • Wastewater flow rate, Q=0.1ML/day=100m3/dayQ = 0.1 \, \text{ML/day} = 100 \, \text{m}^3/\text{day}
  • Lagoon detention time, Td=10daysT_d = 10 \, \text{days}
  • Lagoon is completely mixed.

Assumptions:

  1. The pollutant is non-reactive, so it does not decay or react in the lagoon.
  2. At steady state, the concentration in the effluent equals the concentration in the lagoon due to complete mixing.
  3. After the sudden increase in influent concentration, the system behaves according to the first-order differential equation for the completely mixed lagoon.

Solution:

Part (a) Steady-State Concentration

At steady state, the concentration of the pollutant in the lagoon effluent, CeC_e, is the same as the influent concentration C0C_0. This is because the lagoon is completely mixed and no reaction or decay occurs.

Thus, the steady-state concentration of the non-reactive pollutant in the effluent is: Ce=C0=15g/m3C_e = C_0 = 15 \, \text{g/m}^3

Part (b) Concentration 10 Days After the Influent Increase

Now, we need to calculate the concentration in the lagoon after 10 days, following a sudden increase in influent concentration from C0=15g/m3C_0 = 15 \, \text{g/m}^3 to C1=150g/m3C_1 = 150 \, \text{g/m}^3.

For a completely mixed lagoon, the concentration C(t)C(t) at time tt after the change is governed by the first-order mixing equation: C(t)=C1+(C0C1)et/TdC(t) = C_1 + (C_0 - C_1) e^{-t/T_d} Where:

  • C(t)C(t) is the concentration in the lagoon at time tt,
  • C0=15g/m3C_0 = 15 \, \text{g/m}^3 is the initial concentration,
  • C1=150g/m3C_1 = 150 \, \text{g/m}^3 is the new influent concentration,
  • Td=10daysT_d = 10 \, \text{days} is the detention time,
  • t=10dayst = 10 \, \text{days} is the time after the concentration change.

Substituting the known values: C(10)=150+(15150)e10/10C(10) = 150 + (15 - 150) e^{-10/10} C(10)=150+(135)e1C(10) = 150 + (-135) e^{-1} Using e10.3679e^{-1} \approx 0.3679: C(10)=150135×0.3679C(10) = 150 - 135 \times 0.3679 C(10)15049.67C(10) \approx 150 - 49.67 C(10)100.33g/m3C(10) \approx 100.33 \, \text{g/m}^3

Thus, 10 days after the sudden increase in influent concentration, the concentration of the non-reactive pollutant in the effluent will be approximately 100.33g/m3100.33 \, \text{g/m}^3.


Let me know if you would like further details or clarifications!

Related Questions:

  1. How is the detention time of a lagoon calculated?
  2. What factors affect the steady-state concentration in completely mixed lagoons?
  3. How would the results change if the pollutant was reactive or decayed over time?
  4. Can we use the same equations for partially mixed lagoons?
  5. What would happen to the effluent concentration if the flow rate doubled?

Tip:

For systems like lagoons, the concept of "complete mixing" simplifies calculations because it assumes uniform concentration throughout, but real systems might exhibit partial mixing, which would require more complex models.

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Math Problem Analysis

Mathematical Concepts

Steady-state concentration
First-order differential equations
Complete mixing in lagoons
Pollutant concentration modeling

Formulas

C_e = C_0
C(t) = C_1 + (C_0 - C_1)e^{-t/T_d}

Theorems

First-order mixing equation for completely mixed lagoons

Suitable Grade Level

University Level (Environmental Engineering, Fluid Dynamics)