OCEN 475/677

Environmental Fluid MechanicsProblems

CIVIL College Station, Texas 77843-31363136 TAMU (979) 845-4517 FAX (979) 862-8162

1 Concentration, Diffusion, and the Diffusion Equation

2

1.1 Environmental Fluid Mechanics in the News

Find an example of a news article that deals with transport of a passive substance (dissolved or

suspended substance or chemical) in a situation that demonstrates the application of concepts

from environmental fluid mechanics. You may choose a newspaper or web article, and the event

should have occurred in the last 10 years. Attach a copy of the article and write a short paragraph

describing it’s relationship to environmental fluid mechanics. Be sure to use proper grammar and

your best writing skills when writing your summary paragraph.

3

1.2 Definitions

Write a short, qualitative definition of the following terms:

a. Concentration

b. Mass fraction

c. Diffusion

d. Chemical transport

e. Fick’s law

f. Mass flux

4

1.3 Concentrations in Water

A student adds 1.0 ml of a stock Rhodamine WT solution to 100.0 ml of water at 20◦C. Rhodamine

WT solution is a mixture that contains a 20% mass fraction of pure Rhodamine WT in water.

Assuming the solution is dilute so that we can neglect the equation of state of the solution, compute

the concentration of Rhodamine WT in the mixture in units of mg/l, mg/kg, ppm, and ppb. Be

sure to report the correct number of significant figures for each calculation.

5

1.4 Concentrations in Air

Air is a mixture of gasses containing a mass fraction of 21.7% oxygen gas (O2). For air with a

density of 1.227 kg/m3, compute the concentration of oxygen in the units of mg/l, mg/kg, mol/l,

and ppm.

6

1.5 Diffusive and Mass Fluxes

Aluminum dissovles into a well from a plate at the bottom of the well. No other source of aluminum

exists, so that aluminum can only enter through the bottom. The concentration of alluminum near

the bottom of the well is given by CAl = −(7/3)z, where z is the height above the bottom of the

well in meters and CAl is the concentration of aluminum in ppb.

a. Compute the diffusive flux of aluminum in µg/(m2s) at z = 0 assuming a one-dimensional model

(this assumes that the aluminum concentration does not depend on lateral position in the well).

b. Compute the mass flux of aluminum in µg/s into the well at z = 0 if the plate diameter is 1.5

meters.

For both calculations, use the molecular diffusion coefficient for aluminum in water (choose a

diffusion coefficient for a similar weight metal if you cannot find the diffusion coefficient of Al).

7

1.6 Chemical Fluxes

Fick’s law describes the diffusive flux of passive scalars. A vector for the combined diffusive and

advective flux of a passive tracer in one-dimension is given by

qx = uC −D∂C∂x

(1)

Compute the net flux of the chemicals in the following examples using the appropriate form of this

equation.

a. CO2 exits an automobile exhaust pipe, where the velocity of gas is 30 cm/s and the CO2

concentration inside the pipe is 0.05 g/l. Neglecting diffusion, what is the net flux of CO2 in

g/(cm2·hr)? If the diameter of the exhaust pipe is 3 cm, what is the associated mass flow rate

in g/hr?

b. O2 is consumed at the sediment water interface. The oxygen concentration profile at the sedi-

ment/water interface is linear with a slope of 0.4 mg/(l·cm), higher concentration existing above

the sediments in the water column. Ground water with O2 concentration of 2.1 mg/l also seeps

into the lake at a rate of 1.5 cm/hr. What is the net oxygen flux at the lake/sediment inter-

face in mg/(cm2·day)? The molecular diffusion coefficient for oxygen is D = 0.15 · 10−4 cm2/s.

Remember to specify a magnitude and direction for the flux vector.

8

1.7 Definitions

Write precise definitions of the following terms:

a. molarity

b. molality

c. mole fraction

d. molar volume

e. molecular weight

Please be sure to cite your sources for the definitions you provide.

9

1.8 Concentration Measures

Consider methane (CH4) dissolved in pure water. The saturation concentration at 10◦C is 0.0017

in units of molarity. Convert this concentration to the following alternative units:

a. mg/l

b. mol/l

c. mol/kg

d. mg/kg

e. (G) mole fraction assuming a dilute solution.

(U) For each of the units above, specify the type of concentration measure (i.e., concentration,

mass fraction, etc.).

10

1.9 Gas Mixtures

A gas bubble of effective diameter 1 cm has a mole fraction of 0.8 CH4, 0.15 C2H6, and 0.05 C3H8.

It is submerged 30 m in seawater (temperature 20◦C and density 1025.7 kg/m3). Calculate the

concentration of ethane (C2H6) in this bubble in units of:

a. mol/l

b. mg/l

c. ppm

d. ppb

You may assume the gases in the bubble behave like an ideal gas.

11

1.10 Fick’s Law

During a recent anoxic event in a lake, phosphorus in the form of phosphate PO3−4 was released

from the sediments. A concentration profile in mg/m3 above the sediments was measured and fit

by the equation

C = 0.008(z + 0.02)−2 (2)

where z = 0 at the bottom of the lake and z is positive upward in units of meters.

a. Compute the diffusive flux of phosphorus (µg/(m2s) of P) at the sediment water interface using

Fick’s Law.

b. (G)Assume the lake has a uniform depth of 10 m over a region 1 km2 and that the concentration

profile is the same everywhere. Compute the mass loading of phosphate to the lake in g/d.

Assuming the wind mixes this mass loading uniformly throughout the lake, compute the change

in the phosphate concentration in mg/l resulting from this loading.

12

1.11 Concentration Measures

A student adds 1 ml of 18 mol/l stock solution of hydrochloric acid (HCl) to 10 l of water. The

resultant solution is dilute. The molecular weight of chloride is Cl = 35.4527 g/mol.

a. What is the molar concentration (mol/l) of chloride (Cl) in the mixture?

b. What is the normal concentration (mg/l) of chloride in the mixture?

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2 Integral Solutions to the Diffusion Equation

14

2.1 Integral Transport Equation

Consider a continuously stirred tank reactor (CSRT) in a water treatment facility with initial

concentration of phosphate C0 = 0 and having a volume V = 3 m3. Phosphate is added to the

reactor through an inflow having Cin = 5 mg/l and Qin = 0.1 m3/s. The outflow from the reactor

is also Qout = 0.1 m3/s so that the volume of the reactor remains constant.

(a.) Solve for the concentration C(t) in the CSTR over time and plot the result.

(b.) What is the steady-state concentration in the reactor?

(c.) If you want the steady-state concentration in the reactor to increase by a factor of 2, what

options do you have to achieve this goal?

(d.) (G) Resolve part a., this time assuming the outflow is twice the inflow. Plot the solution for

C(t) and determine when the tank will run dry. Is there a steady state solution for this case?

If not, why?

15

2.2 Marina Design

You are working on the design of a marina that is to have a surface area of 1 km by 2 km in an

area that has a design tidal excursion of ±0.75 m for an average 25 hour tide. There is a freshwater

input from a nearby stream that averages Qf = 0.01 m3/s. Determine the following by satisfying

each of the three criteria specified in the design manual, namely V ∗t ≥ 13Vm, E ≥ 0.5, and complete

flushing withing 2 to 4 days.

(a.) What is the mean water depth that should be maintained in the marina?

(b.) What is the recommended inlet width to achieve good mixing?

(c.) If you need to accomodate boats with greater draft than the mean water depth, what design

strategies do you suggest? What complications would they bring to the problem?

(d.) (G) At what freshwater flowrate Qf does n = 3 satisfy the design criteria?

16

2.3 Well-mixed Pond

A farmer owns a pond with an algae problem. Input to the pond is through an irrigation canal

with flow rate Q = 0.001 m3/s. Usesage of the pond water is at the same rate so that the pond

volume remains constant at 3,600 m3. To control algal growth, the farmer adds a 100 ppm copper

sulfate solution to the irrigation canal inflow at the rate of 0.5 l/hr.

Assume the pond is continuously stirred, has one inlet and one outlet, and that the initial

concentration in the pond is zero.

1. What is the concentration of copper sulfate in the stock solution in mg/l? Assume the density

of water is 998.2 kg/m3.

2. What will be the steady-state concentration of copper sulfate in the pond in mg/l?

3. The optimal pond concentration is 2.5 mg/m3. When will the pond reach that concentration?

4. If the farmer stops copper sulfate injection when the concentration reaches 3.0 mg/m3, how

long thereafter will it take for the pond concentration to drop below 2.0 mg/m3?

5. (G) If water removal from the pond is split evenly between usage (outflow from the pond)

and evaporation (which does not remove copper sulfate), copper sulfate will build up in the

pond at a different rate. Write down the differential equation with initial condition to be

solved in this case. Clearly define your variables.

17

2.4 Marina Design Evaluation

The University of Texas Marine Sciences Institute maintains a marina at the coordinates 27◦50′16.58′′ North

latitude and 97◦3′11.86′′ West longitude in Port Aransas, Texas. Tidal amplitude data are available

from

http://nearshore.utmsi.utexas.edu/tides.htm

Nautical charts are available here

http://www.nauticalcharts.noaa.gov/mcd/NOAAChartViewer.html

Evaluate this marina based on the three design criteria specified in the design manual, namely

V ∗t ≥ 13Vm, E ≥ 0.5, and complete flushing withing 2 to 4 days.

As a minimum, accomplish the following:

a. What is the mean water depth that should be maintained in the marina? Based on the nautical

chart, do you think this depth is likely in this marina?

b. What is the inlet Strouhal number KW ? What level of mixing does this predict (high, medium,

or low)?

c. If there were a spill in the marina, how long would it take for the concentration of contaminant

to decrease to 25% of its initial value?

d. Based on your analysis, is this a good marina design? Why or why not?

e. (G) Repeat this analysis for the public marina located at 27◦50′25.34′′ N and 97◦3′40.25′′ W

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2.5 Integral Version of the Diffusion Equation

A lake may be approximated over large times as a 4 m deep, well-mixed container 15 km by 15 km

in area. A river discharges into the lake at a rate of 1 m3/s, and the net outflow from the lake is

also 1 m3/s. A paper mill on the upstream river plans to begin discharge of a mill by-product at a

rate of 1 l/s of 500 mg/l solution. Assume that the mill by-product does not undergo any chemical

or biological transformations in the lake and that the wind keeps the lake well-mixed.

a. Derive an equation for the concentration of mill by-product in the lake over time as a function

of the given variables.

b. Plot the concentration in the lake over time as the lake progresses from clean conditions to the

steady-state by-product concentration.

c. What is the steady-state concentration of mill by-product in the lake and when does the lake

concentration get “close” to this value?

d. When does the concentration in the lake first reach 50% of the steady-state concentration?

e. Given the times listed above, do you think conditions in the lake remain constant for long enough

that steady state conditions will ever apply? If not, how can you apply this model to simulate

the lake?

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3 Differential Solutions to the Transport Equation

20

3.1 Instantaneous Point Source in a Pipe

Consider the pipe section depicted in Figure 1. A student injects 5 ml of 20% (by mass) Rhodamine-

WT solution (specific gravity 1.15) instantaneously and uniformly over the pipe cross-section (A =

0.8 cm3) at the point x = 0 and the time t = 0. The pipe is filled with stagnant water. Assume

the molecular diffusion coefficient is Dm = 0.13 · 10−4 cm2/s.

a. What is the concentration at x = 0 at the time t = 0?

b. What is the standard deviation of the concentration distribution 1 s after injection?

c. Plot the maximum concentration in the pipe, Cmax(t), as a function of time over the interval

t = [0, 24 h].

d. How long does it take until the concentration over the region x = ±1 m can be treated as

uniform? Define a uniform concentration distribution as one where the minimum concentration

within a region is no less than 95% of the maximum concentration within that same region.

A M

-x x

Figure 1: Definitions sketch for one-dimensional pure diffusion in an infinite pipe.

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3.2 Measuring Diffusion Coefficients 1

A chemist is trying to calculate the diffusion coefficient for a new chemical. In his experiments,

he measured the concentration as a function of time at a point 5 cm away from a virtual point

source diffusing in three dimensions. Select a set of coordinates such that, when plotting the data

in Table 1, D is the slope of a best-fit line through the data. Determine the value of D from this

data.

Based on this coordinate transformation, what is more important to measure precisely, concen-

tration or time? What recommendation would you give to this scientist to improve the accuracy

of his estimate for the diffusion coefficient?

Table 1: Measured concentration and time for a point source diffusing in three-dimensions.

Time Concentration

(days) (µg/cm3 ±0.03)

0.5 0.02

1.0 0.50

1.5 2.08

2.0 3.66

2.5 4.81

3.0 5.50

3.5 5.80

4.0 5.91

4.5 5.81

5.0 5.70

5.5 5.54

6.0 5.28

6.5 5.05

7.0 4.87

7.5 4.65

8.0 4.40

8.5 4.24

9.0 4.00

9.5 3.84

10.0 3.66

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3.3 Measuring Diffusion Coefficients 2

As part of a water quality study, you have been asked to assess the diffusion of a new fluorescent dye.

To accomplish this, you do a dye study in a laboratory tank (depth h = 40 cm). You release the

dye at a depth of 20 cm (spread evenly over the area of the tank) and monitor its development over

time. Vertical profiles of dye concentration in the tank are shown in Figure 2;the x-axis represents

the reading on your fluorometer and the y-axis represents the depth.

a. Estimate the molecular diffusion coefficient of the dye, Dm, based on the evolution of the dye

cloud.

b. Predict at what time the vertical distribution of the dye will be affected by the boundaries of

the tank.

0 0.02 0.04 0.06 0.08

0

5

10

15

20

25

30

35

40

Concentration [g/cm3]

Dep

th [

cm]

Profile after 14 days

0 0.01 0.02 0.03 0.04 0.05

0

5

10

15

20

25

30

35

40

Profile after 35 days

Concentration [g/cm3]

Dep

th [

cm]

Figure 2: Concentration profiles of fluorescent dye for two different measurement times.

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3.4 Mass fluxes

A one-dimensional concentration profile near a fixed-concentration sewage discharge is

C(x) = C0

(1− erf

(x√4Dt

))(3)

where C0 is the initial concentration near the source, erf is the error function, t is the time since

the start of release, x is the distance downstream, and D is the diffusion coefficient. Compute the

net mass flux vector qx at x = 10 m and t = 6 hours. Use D = 1 · 10−3 m2/s.

24

3.5 River Spill

An railroad accident results in the instantaneous release of 500 kg of organic nitrogen fertilizer into

the Brazos river (flow rate Q = 150 m3/s and cross section A = 750 m2). The accident occurs

35 km upstream of an irrigation intake. Assume the spill mixes rapidly over the depth and is

well-distributed in the lateral direction. The effective diffusion coefficient is 5 m2/s. Answer the

following:

a. When will the center of mass of the spill pass the irrigation intake?

b. Is the river diffusion or advection dominant?

c. What will be the maximum concentration measured at the inlet in kg/m3?

d. What will be the width of the contaminant cloud when it passes the inlet, and how long will it

take for the cloud to pass? Assume the cloud width is ±2σ.

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3.6 Alternative River Release

If the accident in Problem 3.5 occurs at the river edge (i.e., not well-mixed across the river):

a. Sketch the locations of the first 5 most important image sources to model this spill?

b. If the lateral diffusion coefficient is 0.01 m2/s, when does the spill first start to sense the opposite

bank?

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3.7 Two-dimensional Instantaneous Point-Source Solution

A boating accident results in a spill of 10 kg of MTBE (a gasoline additive) which mixes rapidly

over the depth in a shallow lake and dissolves into the local ambient water. Analyze the mixing

of this spill for a lake having a depth of 4 m, a weak current toward the east (x-direction) at

5 cm/s, and mixing coefficients Dx = 0.5 m2/s and Dy = 0.1 m2/s.

a. Plot the concentration distribution along the x- and y-axes, where the spill location is at the

point (0, 0), for t = 1, 3, and 10 days following the spill.

b. Plot the concentration at a drinking-water intake 600 m south and 10 m west of the spill site

over time.

c. When does the maximum concentration occur at the drinking water intake?

d. What is the Peclet number for mixing in the x-direction? Is this flow diffusion or advection

dominant?

e. A fish spawning bed is located 1 km east and 350 m south of the spill site. Over what period

of time should the site be monitored for MTBE contamination as a result of this spill?

For these problems, you may ignore the lateral boundaries of the lake which are far away from

the spill site.

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3.8 Image Source Solutions

Approximate a marina as a rectangular box of length 400 m and width 100 m with uniform depth

of 4 m. Wind and waves work to keep the marina well-mixed so that the longitudinal diffusion

coefficient (along the length of the marina) is 5 m2/s and the lateral diffusion coefficient (along

with width) is 2 m2/s; the effective vertical diffusion coefficient is 1.5 m2/s.

A spill occurs in the middle of the marina at the surface of the water. Determine the following:

a. How long after the spill may the spill be considered well-mixed in the vertical? After this time,

the spill may be considered as a two-dimensional point source in the plan-view of the marina.

b. When does the spill cloud grow large enough that the sides of the marina must be taken into

account in the solution? I.e., when do you need to include images sources to obtain a reliable

solution?

c. Draw a sketch of the marina and show the locations of the necessary image sources to account for

two-dimensional mixing of the point source in the marina. Write down the summation solution

for concentration based on these image sources.

d. When can the spill be considered well-mixed throughout the marina? Based on this calculation,

how accurate is it to use the CSTR solutions for marina design applied to this marina?

28

3.9 Spill Analysis

A railroad accident results in the instantaneous release of 500 kg of organic nitrogen fertilizer into

the Brazos River (flow rate Q = 120 m3/s and cross-sectional area A = 520 m2) at the edge of the

river. Use simple characteristic scales of the problem to answer the following key questions. Let

the diffusion coefficients be 5 m2/s in the longitudinal, 0.2 m2/s in the lateral, and 0.003 m2/s in

the vertical directions, respectively.

a. How far downstream is the release well-mixed in the vertical?

b. How far downstream does the spill first touch the opposite stream bank? When does this occur?

What is the maximum concentration in the river?

c. An irrigation intake is 25 km downstream on the same side of the river as the spill. What is the

maximum concentration at the intake? When does it occur? For how long is the intake affected

by the spill?

29

3.10 Mixing of Joining Rivers

One river (left) with a high concentration of sediment joins another river (right) with a negligible

sediment concentration. The width of the low concentration river near their union is 40 m while

the high concentration river is 80 m wide. Assume the river width and depth do not change much

after the union, and both rivers are shallow and have the same velocity. At one particular day the

mean velocity downstream of the union is 1 m/s and the diffusion coefficient is 0.1 m2/s.

a. Estimate the time required tmix and the distance downstream xmix until the low sediment

concentration river is considered to be well-mixed with the sediment from the high concentration

river. Use a relative concentration of 95% as the criteria for the well-mixed condition.

b. If there is a water intake located on the low-sediment side of the river at 3 km downstream from

the river union, do you expect the water taken from the intake to contain a significant amount

of sediment? Justify your answer.

30

3.11 Image Source Solutions

Approximate a marina as a rectangular box of length 400 m and width 100 m with uniform depth

of 4 m. Wind and waves work to keep the marina well-mixed so that the longitudinal diffusion

coefficient (along the length of the marina) is 5 m2/s and the lateral diffusion coefficient (along

with width) is 2 m2/s; the effective vertical diffusion coefficient is 1.5 m2/s.

A spill occurs in the middle of the marina at the surface of the water. Determine the following:

a. How long after the spill may the spill be considered well-mixed in the vertical? After this time,

the spill may be considered as a two-dimensional point source in the plan-view of the marina.

b. When does the spill cloud grow large enough that the sides of the marina must be taken into

account in the solution? I.e., when do you need to include images sources to obtain a reliable

solution?

c. Draw a sketch of the marina and show the locations of the necessary image sources to account for

two-dimensional mixing of the point source in the marina. Write down the summation solution

for concentration based on these image sources.

d. When can the spill be considered well-mixed throughout the marina? Based on this calculation,

how accurate is it to use the CSTR solutions for marina design applied to this marina?

31

3.12 Spill Analysis

A railroad accident results in the instantaneous release of 500 kg of organic nitrogen fertilizer into

the Brazos River (flow rate Q = 120 m3/s and cross-sectional area A = 520 m2) at the edge of the

river. Use simple characteristic scales of the problem to answer the following key questions. Let

the diffusion coefficients be 5 m2/s in the longitudinal, 0.2 m2/s in the lateral, and 0.003 m2/s in

the vertical directions, respectively.

a. How far downstream is the release well-mixed in the vertical?

b. How far downstream does the spill first touch the opposite stream bank? When does this occur?

What is the maximum concentration in the river?

c. How far downstream can the spill be considered well-mixed in the lateral direction?

d. An irrigation intake is 25 km downstream on the same side of the river as the spill. What is the

maximum concentration at the intake? When does it occur? For how long is the intake affected

by the spill?

32

3.13 Dye Study

In order to evaluate the diffusion coefficient in Wolf Pen Creek, a group of Aggie Engineers conducts

a dye study. They inject 20 g of pure Rhodamine WT uniformly across the depth and width of the

creek at George Bush Dr. They measure the dye concentration 1 km downstream at Texas Ave.

and another 2 km downstream (3 km total distance from the injection) at Hwy. 6.

The creek is 0.25 m deep, 5 m wide, and has a mean velocity of 0.15 m/s. Assume that the

longitudinal (along-stream) effective diffusion coefficient is D = 10 m2/s,

a. When will the maximum concentration be measured at Texas Ave.?

b. What is the maximum concentration that will be measured at Texas Ave.?

c. What is the width of the dye cloud when it passes Hwy. 6?

d. If the students want to document the shape of the dye cloud as it passes Hwy. 6, over what time

period should they take measurements?

e. (G) If the dye passage is to be measured at both stations (Texas Ave. and Hwy. 6), do the

students need two instruments to measure dye concentration, or can they just use one?

33

3.14 Spill

A boating accident 6 km offshore (south east) of Port Aransas results in an oil spill of 100 kg

of gasoline. Wind and waves mix the gasoline through the top 1 m of the ocean surface and

the gasoline spreads laterally within that layer with effective diffusion coefficients in the offshore

direction of 5 m2/s and in the cross-shore direction of 8 m2/s. A weak longshore current produces

a 5 cm/s drift of the cloud toward the south west.

a. Write down the governing equation to solve for gasoline concentration resulting from this spill.

Use symbols (e.g. M , Dx, etc.) in the equation. Define your variables and sketch the problem

and coordinate system. Neglect lateral boundaries.

b. How long after the spill will you need to include image sources in the solution?

c. What is the maximum gasoline concentration 12 hrs following the spill?

d. (G) The Coast Guard is required to monitor the spill as long as the concentration at one standard

deviation away from the center of mass exceeds 10 µg/l. How long must they monitor the spill?

34

3.15 Tanker Accident

A tanker runs aground on the Mississippi River, where the width of the river 1.4 km, the depth is

10 m, and the flow rate is 6,300 m3/s. The accident results in a continuous release of hydrochloric

acid (HCl) at the left bank of the river at a rate of 100 l/min of 18 mol/l solution.

a. Compute the mass flux at the source m in kg/s of chloride. The molecular weight of chloride is

Cl = 35.4527 g/mol.

b. A long time after the start of the accident, the concentration field near the spill will be at steady

state. Sketch the lateral concentration distribution qualitatively (without numbers) for several

downstream cross sections up to the point where the toxic cloud begins to interact with the

opposite bank.

c. What is the downstream location where the cloud touches the opposite bank where you would

need to add image sources to account for the far bank?

35

Station 1:

Dye injected @x = 0 mQd = 100 cm3/sCd = 50 mg/l

x = 70 mCd = 10 ug/lCl = 0.5 ug/l

x = 170 mCd = 8 ug/lCl = 0.9 ug/l

Station 2:

Station 3:

River cross-section:W = 1 md = 0.5 m

Figure 3: Dye study to determine the source of Lindane contamination in a small stream.

3.16 Continuous-Release Dye study

A small stream has been found to be contaminated with Lindane, a pesticide known to cause

convulsions and liver damage. Groundwater wells in the same region have also been found to

contain Lindane, and so you suspect that the river contamination is due to groundwater inflow.

To test your theory, you conduct a dye study using a continuous release of dye. Based on the

information given in Figure 3, what is the groundwater volume flux and the concentration of

Lindane in the groundwater between Stations 2 and 3? The variables in the figure are Qd the

volume flow rate of the dye at the injection, Cd the concentration of the dye at the injection and

at downstream stations, Cl the concentration of Lindane in the river at each station, W the width

of the river, and d the depth in the river. (Hint: this is a steady-state problem, so you do not

need to use diffusion coefficients to solve the problem other than to determine whether the dye is

well-mixed when it reaches Station 2. Recall that the mass flow rate is

m = CQ (4)

and must remain constant along the river in a steady state problem unless there are inflows or

outflows of mass or volume.)

(G) Due to problems with the pump, the dye flow rate has an error of Qd = 100 ± 5 cm3/s.

Assume this is the only error in your measurement and report your measurement uncertainty.

36

4 Turbulent Diffusion

37

4.1 Turbulent diffusion in the ocean

Ten surface drogues are released into a coastal region at local coordinates (x, y) = (0, 0). The

drogues move passively with the surface currents and are tracked using radio signals. Their locations

at the end of t1 = 1 and t2 = 20 days are given in the following table.

Table 2: Drogue position data.

Drogue x(t1) y(t1) x(t2) y(t2)

Number [km] [km] [km] [km]

1 2.5 0.2 5.3 8.1

2 4.6 1.4 2.3 1.0

3 2.3 -1.2 6.6 3.9

4 3.1 -0.4 6.7 -2.8

5 1.5 0.8 0.5 4.2

6 1.4 2.1 10.1 3.6

7 4.7 2.1 6.6 -1.4

8 2.7 0.2 6.0 -2.9

9 1.5 2.6 3.2 2.1

10 4.9 2.3 -4.0 1.7

1. Estimate the advection velocity and the lateral coefficients of diffusion (Dx and Dy) for this

coastal region.

2. Using the radio links, the positions of all ten drogues can be collected within ten minutes. Sup-

pose the radio link were to break down and the positions were instead determined through

visual observation. Even using a helicopter, it requires nearly four hours to locate all ten

drogues. How does this change the accuracy of your data? Can you still consider the mea-

surements to be synoptic?

3. Later, a freight ship is caught in a winter storm off the coast where this drogue study was

conducted. High winds and rough seas cause several shipping containers to be washed over-

board. One of the containers breaks open, releasing its contents: 29,000 children’s bathtub

toys. Estimate how long it will take for the toys to begin to wash up on shore assuming

the same transport characteristics as during the drogue study and that the spill occurs 1 km

off the coast. (This really happened in the Pacific Ocean, and the trajectory of the bathtub

toys, plastic turtles and ducks, were subsequently used to gain information about the current

system.)

4. Writing assignment (pay attention to grammer, spelling, and style): Read the article from

EOS (Vol. 88, No. 1, 2 January 2007) on Tub Toys in the Pacific Subarctic Gyre referenced

38

in class. Write a one paragraph summary (typed) of the article and explain how it relates to

this work-out problem and to the determination of mixing and transport rates in the ocean.

Provide proper referencing and do not plagiarize any of the article.

39

4.2 Mixing of joining rivers

Referring to the picture below, one river (left) with a high concentration of sediment joins another

river (right) with a negligible sediment concentration. The width of the low concentration river

near their union is 40 m while the high concentration river is 80 m wide. Assume the river width

and depth do not change much after the union, and both rivers are shallow and have the same

velocity. At one particular day the mean velocity downstream of the union is 1 m/s and the lateral

diffusion coefficient is 0.1 m2/s.

• Estimate the time required tmix and the distance downstream xmix until the low sediment con-

centration river is considered to be well-mixed with the sediment from the high concentration

river. Use a relative concentration of 95% as the criteria for the well-mixed condition.

• If there is a water intake located on the low-sediment side of the river at 3 km downstream

from the river union, do you expect the water taken from the intake to contain a significant

amount of sediment? Justify your answer.

40

4.3 Turbulent diffusion

A tanker runs aground on the Mississippi River, where the width of the river B is 1.4 km, the

depth is 10 m, and the flow rate is 6,300 m3/s. The accident results in a continuous release of

hydrochloric acid (HCl) at the left bank of the river at a rate of 100 l/min of 18 mol/l solution.

1. Estimate the vertical and lateral turbulent diffusion coefficients.

2. How far downstream can the release be considered well-mixed in the vertical dimension?

3. Using Appendix B of the Socolofsky and Jirka textbook, write down the solution for the

concentration distribution in the river after the source can be considered two-dimensional

in plan view and before the spill interacts with the opposite bank. You may assume that

longitudinal diffusion is negligible (i.e., ∂C/∂x ≈ 0).

4. (U) Plot the concentration distribution C(y) at the downstream locations x = 12, 50, and

150 km. At what downstream location do you need to include image sources in your solution?

5. (G) If the pH depression at the edge of a region within B of the spill is required to be less

than 0.5 pH units, is it acceptable to allow the tanker to drain at the rate specified above?

41

5 Dispersion

42

5.1 Dye study planning

To estimate the mixing characteristics of a small stream, an engineer injects 5 g of dye instanta-

neously and uniformly over the river cross section (A = 5 m2) at the point x = 0. A measurement

station is located 1 km downstream and records a river flow rate of Q = 0.5 m3/s. In order to design

the experiment, the scientist assumed that the longitudinal mixing coefficient is Dx = 0.1 m2/s. If

the fluorometer has an accuracy of ±0.2 µg/l, did the engineer plan to inject sufficient dye to get

an accurate concentration distribution at the measuring station?

43

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9x 104

Stage in ft

Dis

char

ge in

cfs

Figure 4: Discharge versus stage for the Brazos River at SH 21 near Bryan, Texas.

5.2 River flow data and dispersion

The USGS reports daily streamflow measurements across the United States using a network of river

flow gauges. Each gauge consists of a stilling well in which the river flow stage (depth relative to

a known datum) is measured. The depth measurement is converted to river flow using a carefully

calibrated stage-discharge relationship. Measurements are made on an hourly basis, and the daily

mean flow is reported in a National database of stream flow data available at

http://waterdata.usgs.gov/nwis

A detailed summary of the gauge development process is also available online at

http://ga.water.usgs.gov/edu/measureflow.html

As an example, Figure 4 shows a plot of daily flow versus river stage for the Brazos River measured

at SH 21 near Bryan, Texas.

To get an understanding of this data and river mixing coefficients, complete the following:

1. By comparing the USGS flow gauge network with maps in Google Earth or other resources,

select a river in your home state. This exercise is in preparation for a dye study, so keep in

mind that you will have to have access to the river in order to inject the tracer and measure

44

its concentration at two or more stations downstream. The measurement stations have to be

far enough downstream that the dye has mixed across the vertical and lateral boundaries,

but close enough to the injection that the dye is clearly measureable. The river section you

select should have at least one USGS flow gauge that reports 50 years or more of stage and

discharge. Turn in the latitude and longitude of your selected flow gauge, the gauge I.D., and

a Google Earth map of the gauging station (include at least 1 km of river reach in your map).

2. Download the USGS flow data for your selected river segement and get the data in an appro-

priate form to analyze using Excel, Matlab, or a program of your choice. Plot the discharge

as a function of the river stage for the period of record. Be sure to label your axes. Turn in

a copy of this plot.

3. The stage measurement is not a direct measure of the river flow depth. From your plot in the

previous section, estimate the river stage at zero flow. Formulate an equation for the river

flow depth given the stage measurement. Explain how the real average flow depth may be

different from this depth measurement. Turn in a summary of your analysis.

4. From all available data, estimate the parameters necessary to compute the longitudinal dis-

persion coefficient using the equations from Fischer et al. (1979) and Deng et al. (2001) at

the USGS flow gauge. You may use Google Earth satellite images to estimate flow width.

You may need to obtain USGS topographical maps to estimate the river slope. These are

available here:

http://nationalmap.gov/ustopo/index.html

Turn in a summary of your proposed methods.

5. Create a plot of the dispersion coefficient using the two methods (Fischer and Deng et al.)

as a function of time for last year (calendary year 2010). Turn in a copy of this plot. Also

include a discussion of your estimate of the error in these results.

45

5.3 Research Literature

Read the articles

• Deng et al. (2001) “Longitudinal dispersion coefficient in straight rivers,” J. Hydraulic Engi-

neering 127(11): 919–927.

• Ho et al. (2002) “Ditermination of longitudinal dispersion coefficient and net advection in

the tidal Hudson River with a large-scale, high resolution SF6 tracer release experiment,”

Environmental Science and Technology 36(15): 3234-3241.

Compare and contrast these articles in a typed essay (maximum 1 page, 12 point font, single spaced,

1 inch margins).

46

5.4 Dispersion Coefficients by Numerical Integration

Using the velocity profile data in Table 3, estimate the longitudinal dispersion coefficient by per-

forming a numerical integration of

DL = − 1

A

∫ W

0u′h

∫ y

0

1

Dyh

∫ y

0u′hdydydy (5)

where h is the water depth, u′ is the shear deviation velocity, Dy is the lateral turbulent diffusion

coefficient, and z is the vertical coordinate. You should obtain a value of DL = 1.5 m2/s. How

does this value compare with estimate using the equations from Fischer et al. (1979) and Deng et

al. (2001)?

Table 3: Stream velocity data for calculating a longitudinal dispersion coefficient.

Station Distance from Total depth Measurement Velocity

number bank d depth, z/d u

[cm] [cm] [–] [cm/s]

1 0.0 0 0 0.0

2 30.0 14 0.6 3.0

3 58.4 42 0.2 6.0

0.8 6.4

4 81.3 41 0.2 16.8

0.8 17.6

5 104.1 43 0.2 13.4

0.8 13.6

6 137.2 41 0.2 13.6

0.8 14.2

7 170.2 34 0.2 9.0

0.8 9.6

8 203.2 30 0.2 5.0

0.8 5.4

9 236.2 15 0.2 1.0

0.8 1.4

10 269.2 15 0.2 0.8

0.8 1.2

11 315.0 14 0.6 0.0

12 360.7 0 0 0.0

47

5.5 River tracer Study

Engineers conduct a tracer study in a local stream to determine the aeration rate at the air/water

interface and the longitudinal dispersion coefficient using SF6 tracer. The river is 200 m wide, 3 m

deep, and flows at 0.4 m/s. The shear velocity should be taken as u∗ = 0.1u. Two measurement

stations are located at 50 and 200 km downstream of the release; breakthrough curves of dye

concentration at these stations are shown in Figure 5.

1. Estimate the dispersion coefficient using the models of Fischer et al. (1979) and Deng et al.

(2001).

2. If the release is an instantaneous point source in the center of the river, how far downstream

will the tracer become well-mixed across the stream cross section?

3. If the dispersion coefficient is DL = 700 m2/s, what is the expected standard deviation σ of

the tracer cloud at the first measurement station?

4. Using DL = 700 m2/s and assuming SF6 is conservative (no off-gasing), what mass of tracer

must be injected so that the peak concentration at station 1 is 1 · 10−7 kg/m3?

5. From the data in the figure, estimate the standard deviation of the tracer cloud at each

station. From these data, estimate the dispersion coefficient in m2/s from the equation

σ22 − σ21∆t

= 2DL (6)

6. If 2 kg of SF6 are injected, what is the apparent gas transfer velocity kl in m/s?

48

0 10 20 30 40 50 60 701

0

1

2

3

4

5

6

7x 10 8

Time (hrs)

C (k

g/m

3 )

80 100 120 140 160 180 2002

0

2

4

6

8

10x 10 9

Time (hrs)

C (k

g/m

3 )

Figure 5: Concentration of SF6 at measurement stations 1 (top panel) and 2 (bottom panel).

49

5.6 Dye Study

An engineer conducts a dye study in a local stream of width 10 m, depth 0.3 m, flow velocity

0.3 m/s, and shear velocity of 0.02 m/s. The study uses Rhodamine WT dye and a fluorometer

with a measurement range of 0 to 250 ppb with detection limit of 0.01 ppb.

a. How far downstream as a minimum must the first measurement station be? Assume an injection

in the center of the stream. State your assumptions and specify some kind of factor of safety.

b. (U) What mass of Rhodamine WT should be injected to achieve a good measurement at a

measurement station 6 km downstream of the injection?

c. (G) What mass of Rhodamine WT should be injected to achieve good measurements at stations

located 6 and 15 km downstream of the injection?

50

6 Chemical Reactions

51

6.1 Reaction Order

A Chemical reaction is of order 1.5.

1. What are the units of the rate constant?

2. What is the solution to the rate equations (i.e. what is C(t))?

3. Write an expression for the half-life and e-folding time.

52

6.2 Reaction kinetics

In the atmosphere, N2O5 will transform to NO2 due to the chemical reaction:

2N2O5 → 4NO2 + O2 (7)

Experimental data show the concentration of N2O5 varies as a function of time. Giving the mea-

sured data in the following table, determine

1. The order of the reaction.

2. The reaction constant.

Table 4: Chemical reaction rate data for N2O5 in the atmosphere.

Time N2O5 Concentration

[min] [mg/l]

0 0.833

10 0.620

20 0.465

30 0.357

40 0.265

50 0.196

60 0.144

70 0.110

53

6.3 Water Treatment Reactors

A sedimentation pond is used to remove heavy metals. Untreated water flows into the pond where

it is rigorously mixed (treat as a continuously stirred tank reactor) and brought into contact with

other chemicals that remove the metals. A single outlet is installed for the pond. Assume the

inflow and outflow rates are identical, and assume metals are removed in a fist-order reaction. The

tank volume is 15 m3 and the flow through rate is 50 l/s.

1. What is the required die-off rate such that 90% of the metals in the input stream are reomved

in the pond?

2. How much higher can the flow through rate be if the reaction rate constant is doubled?

3. Does the pond efficiency depend on the inflow concentration? Why or why not?

54

6.4 Radioactive Cleanup

Cooling water at a nuclear power plant is contaminated by 131I. The plant manager decides to

retain the water in a pond on site to allow the radiation to decay to a safe level before discharging

into neighboring coastal waters.

Inflow to the pond is 10 l/s at a concentration of 15,000 Bq/l. The safe discharge concentration

is 150 Bq/l. The pond operates at steady state and constant volume. The half-life of 131I is 8 days.

1. What is the decay constant k in s−1?

2. What is the required pond volume? Assume that mechanical mixers keep the pond continu-

ously mixed.

3. If the inflow concentration doubles, what will be the new volume necessary to maintain the

safe discharge concentration?

55

6.5 VOC Spill

A traffic accident could result in 25 kg of TCE being released into the Brazos River, where the

width is 75 m, the depth is 1.3 m, the flow velocity is 0.07 m/s, and the channel slope is 0.000015.

Assume the first-order rate constant for TCE volatilization is 0.32 day−1. If the water quality

standard is 3 ppb, will the standard be exceeded at an irrigation works 8 km downstream of the

spilll?

Your client would like you to estimate the maximum TCE concentration at an irrigation works

8 km downstream for this type of spill. If your uncertainty in the dispersion coefficient is δDL =

2DL, what is the uncertainty in your answer to your client?

56

6.6 Bacteria Contamination

A wastewater treatment plant (WWTP) discharges bacteria into a nearby stream at a rate of

4 · 108 #/s. The discharge is through a multiport diffuser on the bottom of the stream such that

the effluent is evenly distributed across the stream width. The stream is 10 m wide, 0.15 m deep,

has a flow rate of 0.5 m3/s, and the shear velocity is 0.038 m/s.

1. How far downstream of the discharge can the effluent be considered well-mixed in the vertical

direction?

2. Once the effluent is vertically well-mixed, what is the bacteria concentration C0 in the stream

in #/m3?

3. If the concentration in the stream 20 km downstream of the discharge is at the safe swimming

level of 2 · 107 #/m3, what is the effective die-off constant k for the bacteria in this stream

in day−1?

4. If bacteria are also lost to sediments at the bed at a rate of 2.5 day−1, what is the die-off rate

of bacteria in the water column?

57

6.7 Mississippi River Plume

The Mississippi River is a major sediment source to the Northwest Gulf of Mexico. The river plume

travels toward the Mexico-Texas boarder and stays attached to the coast due to geophysical fluid

flow patterns in the Gulf. Most of the sediment settles out of the water column by the time the

plume is south of Corpus Christi, about 750 km from the source.

To understand the dominant time-scales and processes in this flow, consider the following sim-

plified model. Set the coast as a wall at y = 0 extending in the x = ±∞ directions and let the Gulf

circulation result in a uniform current of 15 cm/s in the negative x-direction. Let the Mississippi

River be a continuous line source at (0, 0) with a discharge of 10,000 m3/s with a concentration

of 675 mg/l particulate matter. Ignore the velocity field caused by the river flow, and assume the

river plume extends over a 30 m deep mixed layer on the continental shelf (assume this to be the

constant water depth on the shelf).

1. Determine an appropriate lateral turbulent mixing coefficient Dt,y so that the plume is about

75 km wide by the time it has traveled 750 km.

2. If sediment is a conservative tracer, what is the maximum concentration measured at 750 km

downstream. Remember to use appropriate image sources to produce the coastline.

3. If the actual sediment concentration is 0.5% of the value above for a conservative tracer, what

is the effective die-off rate of sediment in the Mississippi River plume?

4. (G) Can you relate the die-off rate obtained above to an effective sediment settling velocity?

How does this velocity compare to sediment grain sizes you expect in the Mississippi river

plume? Do you think mechanical settling is the dominant mechanism to remove sediment

from the Mississippi River plume?

58

6.8 Combined Sewer Overflow

During a storm, raw sewage and rainfall runoff spill out of a sewer system and into a residential

stream through a single sewer pipe. The spill occurs over a period of 1 hour, with a flow rate of

0.1 m3/s and bacterial concentration of 6 · 108 #/100 ml. The river has a depth of 2.5 m, width of

350 m, average flow velocity of 0.8 m/s, and shear velocity of 0.035 m/s.

a. If you need to predict the concentration 40 km downstream at a local swimming area, can you

treat the overflow as an instantaneous point source? Justify your answer quantitatively.

b. What is an estimate of the dispersion coefficient using the formula by Fischer et al. (1979)?

c. If DL = 1000 m2/s, what is the maximum concentration that occurs at the swimming area?

Assume a bacterial decay rate of 1.3 d−1 and that the bacteria cloud is well-mixed laterally

across the river by the time it reaches the swimming area.

d. (G) The recreational swimming water qualtiy standard for fecal coliform is ≤ 2000 #/100 ml.

Over what time period after the start of the spill should the swimming area be closed? As-

sume die-off is negligible just as the bacteria cloud passes the swimming area (frozen cloud

assumption).

59

6.9 Combined Sewer Storage Tank

During a storm, a combined sewage storage tank (volume of 750 m3) operates as a continuously

stirred tank reactor. The inflow rate is 0.1 m3/s and outflow occurs at the same rate as inflow

whenever the tank overflows; otherwise, there is no outflow.

a. During a long storm, the tank overflows for a long period such that the tank operation reaches

steady state. What is the inflow concentration if the outflow concentration is 6 · 108 #/100 ml

and the die-off rate is 1.3 d−1.

b. (U) At the end of the storm, the inflow and outflow stop and the tank is full with initial

concentration of 6 ·108 #/100 ml. When is the concentration reduced to the recreational boating

standard of 2 · 104 #/100 ml? Assume there is no inflow or outflow and the die-off rate remains

1.3 d−1.

c. (G) At the end of the storm, the sewer company pumps out the tank at a constant flow rate

of 0.005 m3/s. If the initial tank concentration is 6 · 108 #/100 ml, solve for the concentration

in the ouflow over time assuming the tank remains well-mixed, starts full and the die-off rate

remains 1.3 d−1.

60

6.10 Reactive Spill

A traffic accident on a bridge results in the sudden release of 200 kg of Pyrethrin (a botanical

insecticide) into the Brazos river (flow rate 30 m3/s, depth 3 m, width 300 m, and slope 0.00001)

at one of its banks.

a. How far downstream can the spill be considered well-mixed across the width and the depth?

b. What is the maximum concentration of the insecticide 100 km downstream if it is degraded with

a die-off rate of 0.03 d−1?

c. (G) If each mole of Pyrethrin degraded consumes 2 moles of oxygen, what would be the oxygen

deficit in mg/l at 100 km downstream? The molecular weight of Pyrethrin is 57.0287 g/mol. Do

you think that this oxygen anomaly is measureable? Why or why not?

61

7 Atmospheric Mixing

62

7.1 Boundary influence

The effects of a solid boundary are only felt after a plume grows large enough to touch the boundary.

Assuming a total plume depth of 4σz, find the distance downstream of the release point to where

a continuous source release at a height h above a solid boundary first touches the boundary.

63

7.2 Smoke-stack exhaust

A power company releases 1 kg/s of CO2 from a height of 30 m into a wind with average velocity

4 m/s. The sky is partly cloudy and the terrain surrounding the release is pasture land. Estimate

the maximum concentration at ground level downstream of the release. How do the results change

if the release point is lowered by 15 m?

64

7.3 Urban roughness

The results presented in this chapter were for surfaces with uniform roughness and elevations much

greater than the roughness height z0. How do you expect relationships for turbulence intensity to

change near the street level in an urban setting (where the roughness is largely due to buildings

and houses)?

65

7.4 Concentration of radioactive material

Compute the activity in Bq/m3 of 1 µg/m3 of Cs-137 in air. The half-life of Cs-137 is 30.17 years.

(G) Estimate the equivalent dose in Sv for human lung exposure to Cs-137 at the above atmo-

spheric concentration. Is this a dangerous level of Cs-137 for response workers who may breath this

air?

66

7.5 Fukushima radionuclide cloud

Read the article Chino et al. (2012), “Preliminary estimation of release amounts of 131I and 137Cs

accidentally discharged from the Fukushima Daiichi nuclear power plant into the atmosphere,” J.

Nuc. Sci. & Tech. , Vol. 48, No. 7, pp. 1120–1134.

a. In your own words, summarize the method used in this article to determine the source strength

of the accidental release. Based on the data in Table 2, over what time period can this accident

be treated as a continuous release, and what is the release rate over that time?

b. The authors assume a radioactivity ratio of 131I:132I:134Cs:137Cs to be 1:1:1:1. Using this as-

sumption, what is the mass flux of each isotope on 3/20/11 when the 131I release rate was

3.8·1014 bq/hr?

c. Use the methods in Chapter 6 of the Socolofsky & Jirka textbook to calculate the concentra-

tion of 131I in Bq/m3 on March 16th at JAEA, Tokai-mura. State all assumptions used and

clearly explain your solution. Compare your results with Figure 2 in the paper. How does your

calculation perform?

d. (G) Also calculate the concentrations of other radionuclides and the total activity in Bq/m3

at JAEA, Tokai-mura. Which radioactive component contributes the greatest activity? Which

component is the most toxic?

67

8 Boundary Exchange

68

8.1 BOD test

To determine the biodegradation rate coefficient of a particular waste, the waste is placed in solution

in a closed bottle, where the oxygen concentration in monitored over time. Table 5 gives the results

of a typical test. Based on all the data in the table, estimate the value of the rate coefficient, k.

Table 5: Measurements of time and oxygen concentration for a BOD test.

Time [O2]

[days] [mg/l]

0 9.00

2 5.21

4 3.81

6 3.30

8 3.11

10 3.04

12 3.01

14 3.01

16 3.00

18 3.00

20 3.00

69

8.2 Gas exchange rates

A river has the following characteristics: water depth h = 1.0 m, width B = 100 m, bottom slope

S = 9 · 10−4, and Manning coefficient n = 0.01. The turbulent intensity is 15%.

• Use the Lewis-Whitman model to estimate the transfer velocity kl. A typical layer thickness

δ is of order O(0.1mm) to O(1mm).

• Use the small-eddy film renewal model to estimate kl.

• Use the large-eddy film renewal model to estimate kl.

• Compare and discuss the three kl’s you obtained.

• Calculate the flux of dissolved oxygen (DO) into the river using the most reasonable kl (use

your own judgment with justification) among the three above. The saturated concentration

[DO]sat is 8.0 mg/L while the measured [DO] near the bottom of the river is 4.0 mg/L.

• What is the total mass transfer of DO into the river for every kilometer of length?

70

8.3 PCB contamination

An industrial plant releases PCBs (polychlorinated biphenols) through a diffuser into a river. The

river moves swiftly, with a modest sediment load. PCB is volatile (will off-gas into the atmosphere)

and can be adsorbed by the sediment in the river. Describe the network of complex interactions

that must be investigated to predict the fate of PCBs from this disposal source.

71

8.4 Sediment source of phosphorus

Phosphorus, in the form of phosphate (PO3−4 ), is often a limiting nutrient for algae production in

lakes. Because the uncontrolled growth of algae is undesirable, the discharge of phosphorus into

the environment should be minimized (this is why you cannot buy laundry detergent any more that

contains phosphorus). The U.S. Environmental Protection Agency recommends a limit of 0.05 mg/l

PO3−4 for streams that flow into freshwater lakes.

An old chemical plant recently shut off their phosphorus discharge; however, high concentra-

tions of phosphorus are still being measured downstream of the chemical plant. After further

investigation, the following facts were collected:

• The concentration of phosphate in the stream water upstream of the plant is C0 = 0.003 mg/l.

• The sediments in the stream are saturated with sorbed phosphorus for a distance of 2 km

downstream of the plant.

• The phosphate concentration at the sediment bed is kept constant by desorption at a value

of Cb = 0.1 mg/l.

• The design conditions in the stream are h = 2 m deep and u = 0.2 m/s.

• The phosphate transfer velocity at the sediment bed is given approximately by the relationship

(film-renewal model):

kl = 0.002u3/4

h1/4in m/s. (8)

Based on the above data: What is the concentration of phosphate in the stream just after passing

the region of contaminated sediments? If the stream carries a suspended sediment load, how would

that affect the concentration of phosphate in the stream?

72

8.5 Bubble Dissolution

Consider an air bubble dissolving in the upper water column of the ocean. Turbulence keeps the

bubbles suspended at 3 m depth.

a. Write down the conservation of mass equation for the bubble and list all of the parameters that

must be estimated in order to compute the dissolution. Assume the bubble is fixed in the water

column (depth z is constant), but that the surrounding fluid is turbulent.

b. (G) Solve for the bubble mass as a function of time. The initial bubble diameter is 1.5 cm. You

may make the following simplifying assumptions:

• Gas properties may be computed using the ideal gas law. The initial composition of the

bubble is 20% oxygen and 80% nitrogen. You may neglect gas stripping of any other

dissolved gases present in the upper ocean. The dissolved gas concentrations far from the

bubble interface are 9 mg/l oxygen and 12 mg/l nitrogen. The temperature is 20◦C, the

density of seawater is constant at 1025 kg/m3 and the salinity is 34.5 ppt.

• The mass transfer coefficient is constant at 1.0 · 10−4 m/s.

• The Henry’s law constants at standard conditions (atmospheric pressure and 298.15 K) are

5.7 · 10−7 and 2.2 · 10−7 kg/(m3·Pa) for oxygen and nitrogen, respectively. The enthalpies

of solution at the same standard conditions are ∆solH/R = 1650 K for oxygen and 1300 K

for nitrogen gas.

c. (G) Discuss how the actual dissolution would be different than you calculate above. What

parameters are not constant? Will the bubble likely dissolve faster or slower than predicted?

Why?

73

8.6 Aeration

A wastewater discharge in a river results in an oxygen deficit of 5 mg/l below saturation at a certain

water quality station (flow rate 1 m3/s, depth 0.5 m, width 10 m, and slope 0.00001).

a. Using the O’Connor and Dobbins formula (K = 3.93√V /(d3/2), where V is in m/s, d is in m,

and K is in day−1), estimate the rearation constant at the air/water interface.

b. If the measured gas transfer velocity is kl = 6 · 10−6 m/s, how far downstream will half of the

oxygen depression be recovered?

c. Solve for the oxygen deficit as a function of distance along the river using kl = 6 · 10−6 m/s.

d. (G) from your solution above, how far downstream does the oxygen deficit become negligible?

74

9 Jets, Plumes, and Thermals

75

9.1 Single-port Outfall

A single-port outfall discharges treated municipal wastewater into an unstratified stagnant ocean

at a depth of 15 m. The diameter of the outfall port is 0.7 m, the discharge velocity is 3 m/s, the

density of the wastewater is 998 kg/m3, and the density of the ambient seawater is 1024 kg/m3.

Estimate the plume dilution.

Repeat your calculations for when the current is 15 cm/s. Also, determine the approximate

current speed at which the plume changes from the BDNF to the BDFF regime.

76

9.2 Line-source Diffuser

Consider an outfall that is required to discharge treated domestic wastewater at 3 m3/s in 20 m of

water, where the 10-percentile current is 5 cm/s.

1. What would the wastewater dilution be if the wastewater is simply discharged out of the end

of the 900 mm diameter outfall pipe?

2. What would the wastewater dilution be if the wastewater were discharged through closely-

spaced ports along a 12 m long diffuser? Assume typical values for the wastewater and

seawater densities.

Compute the dilution at the end of the near field.

77

9.3 Plume Far Field

A 50 m long multiport diffuser discharges effluent at a rate of 6.5 m3/s and at a depth of 32 m.

If the ambient current is 20 cm/s, estimate the distance downstream of the diffuser to where the

dilution is equal to 150. Assume that the initial wastefield thickness is 30% of the depth.

78

9.4 Deepwater Horizon Accident 1

Use the equations derived in class for single-phase wastewater discharges to predict the trap height

of an oil and gas mixture resulting from a subsea accidental oil well blowout. Also calculate the

dilution at the trap height. Use the following values relevant to the Deepwater Horizon accident:

• The orifice diameter of the discharge pipe is 20 cm and discharges at 1500 m depth in the

ocean. The density of the ambient sea water is 1090 kg/m3.

• Flow rate of oil and gas is 0.1 m3/s for each. Density of oil is 845 kg/m3 and density of gas

is 100 kg/m3. Neglect the two-phase nature of the oil and gas and treat them instead as a

single-phase plume

• Current speed is 0.08 m/s. Stratification frequency is 0.002 s−1.

• (G) If you account for the multiphase nature of the plume, do you think the trap height will

be higher or lower? Why?

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9.5 (G) Deepwater Horizon Accident 2

Read the following two journal articles about the Deepwater Horizon accident

• Ryerson, et al. (2011). “Atmospheric emissions from the Deepwater Horizon spill constrain

air-water partitioning, hydrocarbon fate, and leak rate.” Geophys. Res. Lett., Vol. 38, L07803.

• Kessler, et al. (2011). “A Persistent Oxygen Anomaly Reveals the Fate of Spilled Methane

in the Deep Gulf of Mexico.” Science, Vol. 331, Issue 6015, pp. 312-315.

In a maximum one-page typed essay (single space, 1 in margins, 12 pt Times New Roman font)

discuss the following in a well-organized, inter-related set of paragraphs. You may also include any

other things you think are interesting or relevant to the topics in this class.

• Discuss the mass transfer processes at the bubble/water and oil/water interface that are

relevant to the discussion in the Kessler et al. paper.

• Discuss the mass transfer processes at the atmosphere/ocean interface that are relevant to

the discussion in the Ryerson et al. paper.

• In what ways do the measurements by Ryerson et al. corroborate the results in Kessler et

al.? How does Ryerson et al. use the solubility of each individual compound to interpret his

results?

• If you were building a model to simulate a deepwater oil and gas spill, what information in

these two papers would be important for the model development (algorithms) and the model

validation (proving the model is correct)?

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9.6 Wastewater Outfalls

The discharge from a chemical processing plant is planned to be discharged into Corpus Christi

Bay. The single discharge port of the diffuser is 10 cm in diameter, will be at 4 m depth, and will

flow at 10 l/s. The bay is unstratified with a density of 1020 kg/m3, and the waste stream has a

density of 998 kg/m3. The ambient currents are forced by the wind and average 15 cm/s.

a. Calculate the discharge, momentum and buoyancy length scales.

b. (G) given the values of these scales, explain the general physical behavior of the plume (i.e.,

what will it look like and why?).

c. Calculate the dilution at the end of the near field.

d. What options are available to increase the dilution?

e. (G) the ambient currents vary during the day from peaks around 20 cm/s and lows that are

stagnant. What is the range of dilutions (maximum and minimum) that may occur for these

currents?

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