## Problems

2.1 A small stream with a flow rate of 0.1 m3/s empties into a river that has a flow rate of 2 m3/s. The stream is affected by mining operations and is contaminated with arsenic at a concentration of 50 mg/L. The river is not affected by mining and has an arsenic concentration of 0.03 mg/L upstream from the small stream. What is the arsenic concentration in the river downstream from the stream?

2.2 Consider an electroplating facility that discharges to a river liquid wastes containing chromium. The effluent flow rate is 0.05 m3/s and the flow rate of the river is 5 m3/s. If the concentration of chromium is not allowed to exceed 100 |g/L, what is the maximum allowable concentration of chromium in the effluent?

2.3 Consider a contaminant having a concentration C0 at t = 0 which undergoes first-order degradation. Generate a table of C/C0 as a function of time expressed as the number of half-lives up to a maximum of 10.

2.4 What is the concentration in air (mg/m3) corresponding to 75 ppm (by volume) of H2S?

2.5 Cobalt-60 is a radioactive form of cobalt that has a half-life of 5.2 yr. It is produced in nuclear reactors as the result of neutron activation of 59Co and 60Ni, and it is a major constituent of low-level radioactive waste. The current inventory of 60Co in a trench at a low-level waste disposal facility is 1012 Bq.

(a) What is the mass of 60Co in the trench?

(b) What will the inventory be 100 years from now (in Bq)?

2.6 In the Chernobyl nuclear accident, approximately 106 Ci (3.7 x 1016 Bq) of 137Cs was released to the atmosphere. Calculate the mass that was released. The half-life of 137Cs is 30 yr and its atomic weight is 137 g/mol.

2.7 The concentration of ethylene dibromide (EDB) in an aquifer is 100 mg/L. Biodegradation of the contaminant can be approximated as a first-order process with a rate constant of 0.02 yr-1. How long will it take for the concentration to decrease to 20 mg/L?

2.8 The pesticide carbaryl (the active ingredient in Sevin) undergoes photolysis when exposed to sunlight. Presented in the table are data from laboratory tests in which a simulated surface water was contaminated with carbaryl and exposed to the sun. Estimate a first-order rate constant for photolysis from these data.

Time |
Carbaryl |
Time |
Carbaryl |

(days) |
(mg/L) |
(days) |
(mg/L) |

0 |
148 |
20 |
52 |

2 |
137 |
50 |
15 |

5 |
113 |
100 |
1.1 |

10 |
94 |

2.9 Consider the problem in Example 2.3. What would the removal rate constant need to be to keep the steady-state concentration below 10 mg/m3?

2.10 Solve the differential equation in Example 2.6 using the Laplace transform technique (see Appendix A).

2.11 Consider the constant-source first-order removal model. Show that the steady-state concentration is M/Vk in two different but equivalent ways:

(a) Set dC (t)/dt = 0 in Eq. 2.12 and solve for Css.

2.12 In some states the legal intoxication limit is 0.8 g of ethanol per liter of body fluid. Use the constant-source first-order removal model to calculate the alcohol content in a person who over the course of 2 hours consumes four cans (355 mL/can) of beer that has an ethanol content of 6 g per 100 mL. The removal rate constant is 0.3 h-1 and the person's volume of body fluids is 40 L. Does the concentration exceed the limit? If so, how long will it take for the concentration to decrease to the limit?

2.13 Consider the constant-source first-order removal model when the initial concentration is nonzero [i.e., C (0) = C0].

(a) Using Laplace transforms, solve the differential equation for C(t).

(b) Find the steady-state concentration, Css.

(c) Sketch C(t) vs. t for (i) C0 > Css and (ii) C0 < Css.

(d) Determine the time required for the concentration to come within 1% of the steady-state concentration.

2.14 A risk assessment is to be performed for a proposed incinerator. One of the pathways to be analyzed is atmospheric transport of lead (Pb) and subsequent uptake by crops at a large truck farm located near the proposed site, as illustrated in Figure 2.13. The following expression for the lead flux vector (including both the advective and dispersive components) is obtained by fitting a curve to the predictions of an atmospheric dispersion model:

jrb(x,y,0) = 30(x - y2)e-2% + 3(2x - y2)e~xiy + 5(y2 - x)e-(x/2)iz

where jPb has units of mg/km2 • s and x and y are in kilometers. (Unit conversions are contained within each term.) The equation above is valid for |y| < -Jx. Using the flux vector, find the lead deposition rate (in mg/s) on the farm.

2.15 The two countries Smokylvania and Envirostan share a border, and they have a long-standing history of conflict, resulting from fundamental differences in societal values. Smokylvania is a heavily industrialized country whose people value material goods and a high standard of living. Envirostan is a pastoral country whose people value a clean environment and a high quality of life. Envirostan is downwind from Smokylvania, and its government wants the government of Smokylvania to impose stricter controls on atmospheric releases from its industrial facilities. The government of Smo-kylvania does not want to burden its industries with environmental controls the government considers to be unnecessary.

A risk assessment is to be performed to determine if emissions from Smokylvania are affecting the environment in Envirostan. One of the contaminants of potential concern is SO2. The problem is to determine the emission rate (kg/s) of SO2 from Smokylvania into Envirostan. The wind velocity vector and SO2 concentration along the 100-km border between the two countries are

The x-axis lies along the border between the two countries and the z-axis is the vertical direction. The y-axis is positive in Envirostan. Unit conversions are contained within each term of the expressions.

2.16 Consider an ecosystem covering an area of 1 km2 consisting of the compartments specified in Table 2.3. Consider 1 kg of a contaminant that is introduced into the ecosystem. The partition coefficients for the contaminants are as follows:

TABLE 2.3 Compartment Data for Problem 2.16 | ||

Compartment |
Dimensions |
Mass Density |

Bottom sediment (b) Fish (f) |
1000 m x 1000 m x 1000 m 1000 m x 800 m x 0.125 m 1000 m x 200 m x5 m 1000 m x 200 m x 0.05 m 1 ppm by volume in water |
1.3 kg/m3 1500 kg/m3 1000 kg/m3 1500 kg/m3 500 kg/m3 |

TABLE 2.4 Results of Calculations for Problem 2.16 | ||

Compartment |
Concentration [mg(c)/m3 or mg(c)/kg or mg(c)/L] |
Fractional Inventory |

Air Soil

Water

Sediment

Fish

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