Environmental Modeling

Chapter 5:Fate and Transport Concepts for lake

Systems

Copyright © 2006 by DBS

Quote“[Mathematics] The handmaiden of the Sciences”-Eric Temple

Bell

Concepts

• Descriptive aspects of lakes• General mathematical concepts• Pulse and step models• Chemical aspects• Limitations• Professional modeling• Remediation

Case Study: Lake Onondaga

• Contaminated with raw sewage, salt (sodium and calcium chloride) from soda ash industry

• 1946 mercuric waste discharged from production of chlorine via mercury cell process

http://www.onlakepartners.org

Case Study: Lake Onondaga

• Placed on Superfund list in 1995

• Most contaminated lake in US

– Nutrients PO43-, NO3

-, NH4+, bacteria, turbidity, salinity,

mercury, excess sedimentation

• Metals do not degrade, est. 165,000 lbs of Hg

• Much of which becomes methylated and bioconcentrated

Case Study: Lake Onondaga

Introduction

• Inland surface water 2% Earth’s surface < 1% fresh water

Types of Lakes20% Earth’s fresh water

12% Earth’s fresh water

Types of Lakes

The 15 largest lakes in the world (insert is outline of Great Britain) all drawn to same scale. The numbers indicate the rank in area, while the figures in brackets denote surface area in square kilometers (after Ruttner, 1963; in Burgess and Morris, 1987; updated to 1996 by ESIG/NCAR).

Largest by volume, 20% Earth’s freshwater

Types of Lakes

Large lakes are impressive but not representative of lakes in general which tend to be a lot smaller

Types of Lakes

> 50 lakesMore common lakes

Types of Lakes

• Formed via glacial, volcanic and tectonic activity (also landslides, dissolution of limestone and human-made reservoirs)

• Table 5.3• Physical geography of lake and watershed is important for

modeling– Lakes on flat land are shallow and subject to mixing by wind– Deeper lakes stratify during summer which inhibits mixing

Types of Lakes

Input Sources

• Point – well-defined source

e.g. end of a pipe, smokestack, drain

• Non-point - less well defined, cannot be pinpointed

e.g. outboard motor from a boat

Input Sources

Aerial sources – short and long range

In-Lake sources – bioturbation, storms, dredging

Input Sources

• Eutrophication – input of plant nutrients leads to uncontrolled growth of algal blooms

• low degree of mixing, limited O2 input

Stratification of Lakes

Monomictic – one thermal stratification eventDimictic (2 thermal stratification events)

Highly mixed

Summer

Frozen lake in Winter

Stratification of LakesSummer

Sun heats surface

Heating continues

Overturns in autumn

Becomes anoxic as O2 in hypolimnion is consumed by MO’s

NO3- + 10H+ + 8e- → NH4

+ + 3H2O

Stratification of lakesCarbon substrate (CHO) is oxidized as terminal e- acceptors are reduced

• Startification complicates modeling– Hypolimnion and epilimnion do not mix during summer – pollutant level

is underestimated if total volume is used– Lake is split into boxes based on epi-hypo volumes– Fate of pollutants in anoxic hypolimnion requires separate consideration

• Surface - pollutants can be oxidized by MO’s and photochemistry• Bottom - other types of transformation reaction due to anoxic conditions

– Pollutant may be present from previous event– Pollutant may be released slowly from sediments

e.g. mercury may be biomethylated• Fall overturn usually renews the lake with nutrients, may also expose biota to

chemically changed pollutants

End

• Review

Conceptual Model DevelopmentWhat do you need to build a lake model?

Start with mass balance:

Accumulation = inputs - outputs + sources - sinks

What terms or parameters do we need?

Terms for Modeling

V = the volume of the lake (m3)

Qi = the inlet flow from the main inlet to the lake (m3/yr)

Qe = the outlet, or effluent, flow rate from the lake (m3/yr) (We usually assume that Qi is equal to Qe and we represent both by simply Q)

Ci = the average pollutant concentration in the inlet to the lake (kg/m3) (this value is zero in many cases)

Ce = the average pollutant concentration in the lake (kg/m3) and the concentration in the effluent from the lake

k = the first-order rate constant for removal of pollutant from the lake (year-1)

W = the total mass flux of pollutant in the lake, which is equal to the sum (QiCi + QeCe)

Retention Time, Effective Mixing Volumes

• How long will water (and pollutant) stay in the system?

• Hydraulic retention time (to)

to = V/Q eqn. 5-1

• V may change based on stratification • Mixing is difficult to estimate

– Entire volume may not mix, use fluorescent dye or radioactive tracer

Assumes lake is completely mixed

Chemical Reactions

• Kinetic Degradation (in situ determinations)

Ct = Co e(-kt) eqn. 5-2 Photochemical, biological, chemical and nuclear

Sedimentation

• Removal via sorption to particles and settling

• Particle size decreases, surface area increases, % OM increases – sorption increases

Clays have smallest settling velocities – deposit in deepest and calmest regions of lake

Classification Particle Diameter (μm) Settling Velocity (cm s-1)

Clay < 2 10-8 to 2 x 10-4

Silt 2.0-20 2 x 10-4 to 2 x 10-2

Fine sand 20-200 2 x 10-2 to 2

Coarse sand 200-2000 2 to 20

gravel > 2000 > 20

From Stokes’ law

Sedimentation

• Highest concentration of polluted sediments found in deepest region of lakes– Polutant focusing

• Regions of higher energy flow that are well mixed contain larger particles – lower levels of pollutants

Sedimentation

• Equations 5.3 and 5.4 can be used to estimate amount removed via sedimentation

Sedimentation

where is the settling velocity (length/time)• g is the acceleration due to gravity (length/time2) s is the density of the spherical particle (mass/length3) f is the density of the fluid (mass/length3)• r is the spherical particle radius (length) is kinematic viscosity of the fluid (length2/time)

= (2/9) g (s/f - 1) r2

Sedimentation

where• rA is the rate of decrease in pollutant A concentration per unit

volume of water (mass/length3-time) (NOTE UNITS)• Kd is the distribution coefficient (or Kp is the partition

coefficient) is the particle settling velocity (length/time)• S is the suspended solids concentration (mass/length3)• H is the water depth (length)• C is the pollutant concentration in the water (mass/length3)

rA = - Kd S

H (Kd S + 1) C

End

• Review

Modeling ApproachesModeling using Differential Equations

• Continuous (step) model• W(t) is not zero• Net pollutant concentration in lake is result of (1) decrease due to

flushing via effluent river and first-order decay, (2) pollutant increase due to contant imput from source

V dC + CV(1/t0 + k) = W(t) dt

C(t) = W (1-e-βt) + C0e-βt

βV

Where β = 1/t0 + k and C0 = background conc. pollutant in the lake

Using Laplace transform

Two Basic Mathematical Models for lakesStep Input of Pollutants

TYPO!

Assuming zero background,Coe-βt = 0Continuous input

Two Basic Mathematical Models for lakesStep Input of Pollutants

System reaches equilibrium, concentration approaches C = W / βV

Example Problem

A lake in a rural community has an average surface area of 5000 m2 and a mean depth of 50 m. A stream exits the lake with an annual average flow rate of 45,000 m3/yr. Aerial application of an insecticide to the area introduces the compound to the lake. The average annual loading of this pollutant to the lake is 50 kg/day. Assuming a first-order removal of the insecticide from the lake (half-life = 43.8 days) and that the background concentration of the insecticide in the lake is negligible, answer the following:

What is the retention time of the water in the lake?

What is the equilibrium concentration of insecticide in the lake?

What is the concentration after 0.0100 years?

Example Problem

Sensitivity Analysis

• Goal

C(t) = W

V 1 - e t

Two Basic Mathematical Models for lakesPulse Input of Pollutants

Instantaneous input C(t) = C0e-(1/t0+k)t

Two Basic Mathematical Models for lakesPulse Input of Pollutants

Example Problem

Consider the problem used in the step input example. Monitor the fate of the insecticide in the system if the continuous input is ceased after 1 year.

Develop a formula to express the concentration of insecticide as a function of time (pulse input)

Calculate how long it will take for the insecticide concentration to reach 0.100 mg/L (detection limit).

Example Problem

Sensitivity Analysis

Limitations to our Models

• Different input function• Variable or incomplete mixing• No internal sources (from pollutated sediments)• Multi-component reservoirs• Sedimentation of pollutant-laden particles

End

• Review

Remediation (Clean Up)

• Source elimination!• Father Time• in situ versus ex situ (not really an option)• Cap sediments• Dredge sediments

– Mostly for estuaries and shipping channels– There are various forms but most release considerable

sediment and water containing desorbed pollutions back into the lake system

• Short term events may use chemical additives (activated carbon?)

http://www.dec.state.ny.us/website/der/projects/ondlake/

End

• Review

Further ReadingJournals and Reports

• Dunnivant, F.M., Coates, J.T., and Elzerman, A.W. (2005) Labile and non-labile desorption rate constants for 20 PCB cogeners from lake sediment suspensions. Chemosphere, Vol. 61, No. 3, pp. 332-340.

• Eisenreich, S.J., Looney, B.B., and Thornton, J.D. (1981) Airborne organic contaminants in the Great Lakes ecosystem. Environmental Science and Technology., Vol. 15, No. 1, pp. 30-38.

• Engstrom, D.R., Swain, E.B., Henning, T.A., Brigham, M.E., and Brezonik, P.L. (1994) Atmospheric mercury deposition to lakes and watersheds: A quantitative reconstruction from multiple sediment cores, In: Environmental Chemistry of Lakes and Rerservoirs, Advances in Chemistry Series 237, Baker, L.A. ed., American Chemical Society, Washington DC, Chapter 2, pp. 33-66, 1994.

• Holloway, T., Freore, A., and Hastings, M.G. (2003) Intercontinental Transport of air pollution: Will emerging science lead to a new hemispheric treaty? Environmental Science and Technology, Vol. 37, No. 20, pp. 4535-4542.

• Lapple, C.E. (1961) The little things in life. Stanford Research Institute Journal (third quarter), Vol. 5, pp. 95-102.

• Spiefhoff, H.S., and Hemond, H.R. (1996) History of toxic metal discharge to surface waters of the Aberjona watershed. Environmental Science and Technology, Vol. 30 No. 1, pp.121-128.

• Wetzel, R.G. (1990) Land-water interfaces: Metabolic and limnological regulators. Verh. Int. Verein. Limno., Vol. 24, pp. 6-24.

Books

• Ruttner, F. (1963) Fundamentals of Limnology, University of Toronto Press, Toronto.

• Wetzel, R.G. (2001) Limnology: River and Lake Ecosystems, Academic Press, New York.