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---------------------------------------------------------------------------------------------------------------- CIVL 4100B Environmental Systems Analysis, Fall 2014, HW 5 ---------------------------------------------------------------------------------------------------------------- Due: 28 November 2014, Friday, during lecture A river has 5 reaches of certain lengths. On average, the temperature of the water in the river is 20oC. As shown in the figure below, Polluter A is located at the top of Reach 1, Polluter B at the top of Reach 3 and Polluter C at the top of Reach 5. At present, due to the lack of regulations, the polluters are discharging untreated BOD into the river. The polluters are also withdrawing water from the river, just before their respective points of discharge, each at the same flow rate as its discharge. At the top of Reach 4 is the town of Eee that withdraws water from the river at a rate of 10 m3/s to supplement its supply of drinking water. At the top of Reach 2, a tributary enters the river. Just upstream of the point of entry, the stream flow of the tributary is 20 m3/s, BOD concentration 10 mg/L and DO deficit 3 mg/L. Also, just upstream of Reach 1 where Polluter A withdraws water, the flow rate of the river is 200 m3/s, BOD concentration 5 mg/L and DO deficit 2 mg/L.

Background BOD both in the river and from the tributary decays with a rate constant of 0.2 day-1. BOD from the Polluter A decays with a rate constant of 0.5 day-1, and from Polluters B and C with a rate constant of 0.3 day-1. The table below gives the average water velocities and depths for the five reaches, and for each, the number of uniform segments that the reach is to be further divided into when modeling it (as instructed below).

BOD concentration = 10 mg/L DO deficit = 3 mg/L Stream flow = 20 m3/s

Reach 1 30 km

Reach 2 25 km

Reach 3 10 km Reach 4

5 km

Reach 5 20 km

BOD concentration = 5 mg/L DO deficit = 2 mg/L Stream flow = 200 m3/s

Withdrawal = 10 m3/s

Eee

Polluter A

Polluter B

Polluter C

2

Reach number

Number of segments

Average water velocity (m/s)

Average depth of water (m)

1 15 0.5 2

2 15 0.25 1

3 10 0.3 1.5

4 5 0.1 1.7

5 15 0.2 1

The costs of building and operating wastewater treatment plants to reduce the BOD concentrations of the wastewater discharges from the three polluters vary depending on their percentages of BOD reduction. The costs are as defined by the piecewise linear functions below. For all i = A, B, C:

∑=

=4

1pipipi xcC … (1)

Ci is the cost of building and operating a wastewater treatment plant to treat Polluter i’s discharge. cip is the cost gradient of piece p of the piecewise linear function for Polluter i. xip is the fraction of BOD removed that is associated with piece p of Polluter i’s cost function. For all i = A, B, C:

= ∑

=

4

1pip

uii x -1LL … (2)

Li is the BOD concentration of the discharge from Polluter i after treatment, and Li

u the BOD concentration of the raw discharge from Polluter i before treatment. In the piecewise linear functions as given by Equations (1) and (2), the first (and least steepest), second, third and fourth pieces represent the costs of primary, secondary, tertiary and advanced tertiary treatment, respectively. Primary treatment provides up to 35% BOD removal, secondary treatment up to 80%, tertiary treatment up to 90% and advanced tertiary treatment up to 99%. The following table gives the flow rates, untreated BOD concentrations and DO deficits of the discharges from the three polluters to the river. The second table gives the cost gradients of the piecewise linear functions to estimate the costs of wastewater treatment.

Polluter, i Flow rate of discharge to river (m3/s)

BOD concentration of untreated discharge

to river, Liu (mg/L)

DO deficit of discharge to river (mg/L)

A 8 500 3 B 10 300 4 C 3 200 2

Polluter, i

Cost gradient, cip (million $/yr)

Primary treatment

Secondary treatment

Tertiary treatment

Advanced Tertiary

treatment A 0.25 0.5 1 3 B 0.35 0.7 1.5 4 C 0.3 0.6 1.2 3.5

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(a) By expanding the Streeter-Phelps equation for multiple reaches, discharges and BOD types,

develop a spreadsheet model of the river to predict the concentrations of BOD and DO along the river. Using the results of the model, plot a graph showing the BOD and DO concentrations along the entire river for the status quo where there is no wastewater treatment to reduce the discharges of BOD from the polluters to the river.

(b) Using linear programming, find the combination of xip that gives the least overall cost of

improving the water quality along the river such that DO concentration is at least 5 mg/L at all points, and total BOD concentration is no greater than 10 mg/L where the town of Eee withdraws water for its drinking water supply. Take note of the optimum.

(c) Modify the least cost model in part (b) such that it is required that the three polluters reduce the

BOD concentrations of their discharges by the same percentage. Solve the modified model, and take note of the new optimum.

(d) Repeat part (c), but now, instead of uniform percent reductions, the three polluters are required

to have uniform BOD concentrations in their discharges.

(e) With your results from parts (b) to (d), fill out the table below.

Item (b)

Least cost

(c) Uniform % reductions

(d) Uniform

concentrations

Total BOD discharged (g/s)

Total treatment cost (million $/yr)

Minimum DO concentration in river (mg/L)

BOD concentration where Eee is (mg/L)

% reduction in BOD discharged

--- Discharger A

--- Discharger B

--- Discharger C

BOD concentration in discharge

--- Discharger A

--- Discharger B

--- Discharger C (f) Repeat part (c). Instead of uniform percent reductions, the three polluters are now required to

pay a BOD discharge tax. For each g/s of BOD discharged, the polluters are required to pay a certain amount of tax. Obtain solutions for the following tax rates, in $/yr per g/s: 200, 250, and 300.

(g) Repeat part (c). Instead of uniform percent reductions, the three polluters are now paid a

subsidy reducing their BOD discharges. For each g/s of reduction in BOD discharge, the polluters are paid a certain amount of subsidy. Obtain solutions for the following subsidy rates, in $/yr per g/s: 200, 250, and 300.

(h) With your results from parts (f) to (g), fill out the table below. Based on your results, which tax

rate do you recommend if a tax program were to be implemented to achieve the water quality goals described in part (b)? Which subsidy rate do you recommend if a subsidy program were to be implemented?

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BOD discharge Tax

Tax rate ($/yr per g/s) 200 250 300





Total tax paid (million $/yr)

Total financial cost (million $/yr)

BOD treatment subsidy

Subsidy rate ($/yr per g/s) 200 250 300





Total subsidy received (million $/yr)


(i) Repeat part (c), but now, instead of uniform percent reductions, the three polluters are initially

allocated BOD discharge permits. They are then allowed to buy and sell the permits among themselves. The quantity of permits held by each polluter determines the amount of BOD discharge it is allowed. The polluters are initially assigned permits according to the optimal formula under the uniform percent reductions program in part (c). Assume zero transaction costs and fully rational behavior. With your results, fill out the table below. From the results, do you expect the trading to lead to better or worse water quality?

BOD discharge permit trading






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