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The Science of the Total Environment 314 –316 (2003) 303–334 0048-9697/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0048-9697(03)00062-7 A review of dissolved oxygen modelling techniques for lowland rivers B.A. Cox* Department of Geography, The University of Reading, Whiteknights, Reading RG6 6AB, UK Accepted 1 January 2003 Abstract This review introduces the methods used to simulate the processes affecting dissolved oxygen (DO) in lowland rivers. The important processes are described and this provides a modelling framework to describe those processes in the context of a mass-balance model. The process equations that are introduced all require (reaction) rate parameters and a variety of common procedures for identifying those parameters are reviewed. This is important because there is a wide range of estimation techniques for many of the parameters. These different techniques elicit different estimates of the parameter value and so there is the potential for a significant uncertainty in the model’s inputs and therefore in the output too. Finally, the data requirements for modelling DO in lowland rivers are summarised on the basis of modelling the processes described in this review using a mass-balance model. This is reviewed with regard to what data are available and from where they might be obtained. 2003 Elsevier Science B.V. All rights reserved. Keywords: Dissolved oxygen; Biochemical oxygen demand; Photosynthesis; Respiration; Mass-balance model; Rate parameter 1. Introduction A sufficient supply of dissolved oxygen (DO) is vital for all higher aquatic life. The problems associated with low concentrations of DO in rivers have been recognised for over a century. The impacts of low DO concentrations or, at the extreme, anaerobic conditions in a normally well- oxygenated river system, are an unbalanced eco- system with fish mortality, odours and other aesthetic nuisances. When DO concentrations are reduced, aquatic animals are forced to alter their breathing patterns or lower their level of activity. *Tel.: q44-118-9316553; fax: q44-118-9755865. E-mail address: [email protected] (B.A. Cox). Both of these actions will retard their development, and can cause reproductive problems (such as increased egg mortality and defects) and yor deformities. The variability in DO in rivers is caused by the influences of many factors, and the major influ- ences can be categorised as being either sources or sinks of DO in rivers. The major sources of DO include: Reaeration from the atmosphere; Enhanced aeration at weirs and other structures; Photosynthetic oxygen production; The introduction of DO from other sources such as tributaries.
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Page 1: A Review of Dissolved Oxygen Modelling Techniques for Lowland

The Science of the Total Environment 314–316(2003) 303–334

0048-9697/03/$ - see front matter� 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0048-9697(03)00062-7

A review of dissolved oxygen modelling techniques for lowlandrivers

B.A. Cox*

Department of Geography, The University of Reading, Whiteknights, Reading RG6 6AB, UK

Accepted 1 January 2003

Abstract

This review introduces the methods used to simulate the processes affecting dissolved oxygen(DO) in lowlandrivers. The important processes are described and this provides a modelling framework to describe those processes inthe context of a mass-balance model. The process equations that are introduced all require(reaction) rate parametersand a variety of common procedures for identifying those parameters are reviewed. This is important because thereis a wide range of estimation techniques for many of the parameters. These different techniques elicit differentestimates of the parameter value and so there is the potential for a significant uncertainty in the model’s inputs andtherefore in the output too. Finally, the data requirements for modelling DO in lowland rivers are summarised on thebasis of modelling the processes described in this review using a mass-balance model. This is reviewed with regardto what data are available and from where they might be obtained.� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Dissolved oxygen; Biochemical oxygen demand; Photosynthesis; Respiration; Mass-balance model; Rate parameter

1. Introduction

A sufficient supply of dissolved oxygen(DO)is vital for all higher aquatic life. The problemsassociated with low concentrations of DO in rivershave been recognised for over a century. Theimpacts of low DO concentrations or, at theextreme, anaerobic conditions in a normally well-oxygenated river system, are an unbalanced eco-system with fish mortality, odours and otheraesthetic nuisances. When DO concentrations arereduced, aquatic animals are forced to alter theirbreathing patterns or lower their level of activity.

*Tel.: q44-118-9316553; fax:q44-118-9755865.E-mail address: [email protected](B.A. Cox).

Both of these actions will retard their development,and can cause reproductive problems(such asincreased egg mortality and defects) andyordeformities.

The variability in DO in rivers is caused by theinfluences of many factors, and the major influ-ences can be categorised as being either sourcesor sinks of DO in rivers. The major sources ofDO include:

● Reaeration from the atmosphere;● Enhanced aeration at weirs and other structures;● Photosynthetic oxygen production;● The introduction of DO from other sources such

as tributaries.

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304 B.A. Cox / The Science of the Total Environment 314 –316 (2003) 303–334

The main causes of oxygen depletion, or sinks,are:

● The oxidation of organic material and otherreduced matter in the water column;

● Degassing of oxygen in supersaturated water;● Respiration by aquatic plants;● The oxygen demand exerted by river bed

sediments.

If the volume of water in a water body isV andthe concentration of DO isC, then the variationof DO in that water body may be described as:Accumulations Mass iny Mass outq sourcesy sinks, or in simple mathematical terms as:

dMsMyM q(PyR)qC yBODySODi o Rdt

yC "DS (1)D

where t is the time,M is the mass flux of DOi

entering the water body,M is the mass fluxo

leaving, C represents the aeration and reaerationR

processes, BOD is the biochemical oxygen demandrepresenting the oxidation of organic material,SOD is the sediment oxygen demand,C repre-D

sents degassing of oxygen andDS representschanges in the water body due to transport fromexternal sources.

In this review, the major processes are examinedin more detail and are expressed in mathematicalterminology in the form of differential equations.The list of processes described above is limitedand one could attempt to construct a mathematicalmodel based on descriptions of the movement ofeach chemical element in the system. Such a modelwould explicitly simulate all of the chemical reac-tions and biological processes affecting each ele-ment—be they dissolved, adsorbed on particles, ora part of some plant or animal. However, such amodel would be prohibitively complex and few ofthe required data would be available as they wouldnot be measured, or perhaps not even be measur-able. As a result, most mechanistic or process-based models are actually semi-empirical althoughthey do represent individual processes. For exam-ple, many models of DO usually employ modifiedor extended versions of the classical equations ofStreeter and Phelps(1925) for the BOD and DO

profiles along natural streams. BOD is a mixtureof many different elements and so if one usesBOD the model strictly cannot close the mass-balance. However, this deficiency is generallyassumed acceptable given the alternative of anover-parameterised, overly complex model thatmay be impossible to implement.

2. Influences on DO concentrations in lowlandrivers

In the UK, stream DO studies have been under-taken since the late nineteenth century Theriault(1927). By the turn of the century, the science ofDO measurement and interpretation had progressedrapidly following research in the UK by the RoyalCommission on Sewage Disposal while in theUnited States, studies on the Ohio River wereconducted between 1914 and 1916. These inves-tigations provided the basis of the classic mathe-matical modelling of DO by Streeter and Phelps(1925). This model incorporates the two primarymechanisms governing the fate of DO in riversreceiving sewage, the decomposition of organicmatter and atmospheric aerationyreaeration.According to their theory, biochemical oxidationis the only sink and atmospheric reaeration is theonly source of oxygen(i.e. photosynthesis andrespiration are ignored). The simplest manifesta-tion of the Streeter–Phelps model is for a reach insteady-state(i.e. time invariant) characterised byplug flow with constant hydrology and geometry.Mass-balances may be written as:

dL dDU syK LU sK LyK D (2)r d adx dx

whereU is average stream velocity,L is the amountof oxidisable organic material as oxygen equiva-lents (i.e. the BOD), x is the distance along thereach moving downstream andD is the DO deficit(the difference between the DO concentration ifsaturated and the actual concentration). The otherterms are the rates of processes affecting the DO:K is the surface reaeration rate;K is the totala r

removal rate of organic matter;K is the decom-d

position (i.e. oxidation) rate in the stream; andK sK qK , whereK is the net rate of sedimen-r d s s

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305B.A. Cox / The Science of the Total Environment 314 –316 (2003) 303–334

Fig. 1. How an open system affected by a BOD load is described by the Streeter–Phelps model. DO is the DO concentration,Cs

is the saturation concentration of DO andL is the BOD concentration.

tation and re-suspension of BOD. IfLsL and0

DsD at time zero then these equations can be0

solved for:

yK yU x( )rLsL e0

K Ld 0yK yU x yK yU x yK yU x( ) ( ) ( )a r aDsD e q e ye (3)Ž .0 K yKa r

These two equations constitute the classic‘Streeter–Phelps’ model of BOD and DO profilesalong a river.

A simple application of this model describes theclassic DO sag curve that occurs below sewagedischarges in streams as illustrated in Fig. 1. Thisapplication represents a stream that is originallyunpolluted and so has DO concentrations nearsaturation. A large BOD load is then added, forexample, from untreated sewage, and this elevatesthe levels of dissolved and solid organic matter.As oxygen levels drop, atmospheric reaerationtakes place to compensate for the oxygen deficit.Initially, the reaeration is dwarfed by the oxidationof the BOD as the organic matter is consumed.However, with time, the amount of organic matternot assimilated decreases and so the rate of oxygenloss decreases. At some point, the oxygen deple-tion and reaeration will balance and at this point,the lowest or ‘critical’ level of oxygen is reached.Beyond this point, reaeration dominates and so

oxygen levels begin to rise again towards thesaturation concentration.

Traditionally wastewater engineers have beencontent with this description and have concentratedon how best to estimate the rate parametersK ,aK and K so that the position and extent of thed s

DO minimum for a given polluting load can bepredicted. However, while the equations of Streeterand Phelps(1925) have been found to be adequateto give a schematised representation of the varia-tion of DO concentration downstream of a pointdischarge, subsequent studies have pointed out thatother sources and sinks need to be considered(Dobbins, 1964; Camp, 1963; Owens et al., 1964;Edwards and Owens, 1965). Thus, the processessuggested for inclusion in a model of DO are:

● the removal of BOD by sedimentation oradsorption;

● the addition of BOD by the re-suspension ofbottom sediments or by the diffusion of partiallydecomposed organic matter from the bed sedi-ments into the water above;

● the addition of BOD by local runoff;● the removal of oxygen from the water by the

action of gases in the sediments;● the removal of oxygen by the respiration of

plankton and fixed plants;● the addition of oxygen by the photosynthesis of

plankton and fixed plants;

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● the addition of oxygen by atmosphericreaeration;

● the redistribution of BOD and DO by the effectof dispersion, particularly when the pollutingload varies suddenly;

● the removal of oxygen by nitrifying bacteria.

Of these processes, the dominant ones whichrelate to the oxygen balance of a river are(Bennettand Rathburn, 1972)

● the oxygen demand of the carbonaceous andnitrogenous wastes in the water;

● the oxygen demand of the bottom deposits;● any immediate chemical oxygen demand(COD);

● the oxygen required for plant respiration;● the oxygen produced by plant photosynthesis;● the oxygen gained from atmospheric reaeration.

3. Process descriptions

The six dominant processes given by Bennettand Rathburn(1972), as listed above, must beformulated as mathematical equations if they areto be used to simulate DO in a river using amathematical model. This procedure is familiar toall water quality models and many models sharesimilar mathematical descriptions of the physical,chemical and biological processes in a river.

3.1. BOD oxidation, sedimentation and re-suspension

The BOD is the amount of oxygen required bymicro-organisms(to respire) as they consumeorganic matter. The amount of organic waste in awater body is measured by its demand on thewater’s oxygen resources. In rivers that passthrough populated areas, the majority of BOD inrivers will generally come from effluent dischargesof organic matter (for example, from sewage treat-ment works (STW)), but it can also be sourcedfrom dead algae and other organisms. Althoughgenerally small, the input of BOD to a river reachfrom dead organisms can be very significant afteralgal blooms or large fish kills causing furtherproblems downstream of the event itself. BODconcentrations vary with time and location along

a river because as oxygen is used in oxidising thewaste, the amount of waste material decreases andso the BOD drops. BOD can also be removedfrom the water by sedimentation or it can be addedto the water by scouring and re-suspension ofbottom deposits. All of these factors must beconsidered in order to model the BOD distributionalong a river.

The rate at which biochemical oxidation takesplace is usually assumed a first-order process, i.e.the rate of oxidation is proportional to the amountof organic matter remaining in the water. The netrate of sedimentation and scour is also generallyassumed proportional to the amount of BOD pres-ent. Therefore, an equation for the distribution ofBOD can be written as(Thomann and Mueller,1987):

≠L ≠L¯ w xqu sK Alg yK LyK L (4)x ad d s

≠t ≠x

where L is the mean BOD concentration in thewater column;x is the distance downstream, isuxthe mean longitudinal velocity;wAlgx is the algalbiomass contribution;K is the rate constant forad

the BOD addition due to algal death;K is thed

rate constant for the biochemical oxidation; andK is the rate constant for the net rate of settlings

and re-suspension of BOD. In terms of DO con-centrations then, the rate of change of DO due toBOD oxidation alone could be expressed as a first-order process in a well-mixed system as:

dCsyK L (5)ddt

where C is the DO concentration, and which isobtained by the solution of Eq.(5) above oncethe rateK has been estimated. Because the oxi-d

dation is carried out by organisms, the rate willalso depend on nutrient availability, but this israrely considered in water quality models.

The BOD is usually measured using a 5-daytest (BOD ), but this may not account for the5

whole of the oxygen demand that the organicmatter in a sample can exert given sufficient time(i.e. the ultimate BOD or BOD). The BOD datau

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307B.A. Cox / The Science of the Total Environment 314 –316 (2003) 303–334

Table 1Typical values of BOD decomposition rates for various levelsof treatment(Chapra, 1997)

Treatment K (day ) at 20 8Cy1d BOD yBOD5 u

Untreated 0.35(0.20–0.50) 0.83Primary 0.20(0.10–0.30) 0.63Activated sludge 0.075(0.05–0.10) 0.31

used in the DO models will invariably be theBOD (because of the availability of these data).5

However, it might be of merit to estimate BODu

and use this when modelling DO if the time-scalesbeing simulated are sufficiently long. If the first-order decomposition model holds, an extrapolationcan be made to estimate BOD by using the first-u

order decay(or oxidation) rate K (Thomann andd

Mueller, 1987):

BOD5BOD s (6)u y5Kd1ye

Using the assumption that the BOD and5

BOD can be conceptualised as ‘fast’ and ‘slow’u

components of BOD, Chapra(1997) providesestimates of the relative proportions of fast andslow BOD for STW effluent(Table 1).

This estimate can be further refined by usingthe concept of a fast and slow BOD such that eachcomponent has its own reaction rate, such that theultimate BOD would be expressed by:

BOD5BOD s (7)u y5K y5Kfbod sbodw x1y (1ya)e qae

where,a is the proportion of ‘slow’ BOD,Kfbod

is the reaction rate constant for the oxidation of‘fast’ BOD andK is the same reaction rate, butsbod

for ‘slow’ BOD.

3.2. Sediment oxygen demand

Sediment(or benthic) oxygen demands(SOD)result from organic matter being deposited andincorporated in the channel bed. The sources ofSOD may be allochthonous(external) such as leaflitter and humic substances, or autochthonous(internal) such as the settling of BOD. It may also

represent the demand by resident invertebrates forrespiration. There are two aspects to this sedimentdemand: oxygen diffusing into the sediments thenbeing consumed, and reduced organic matter enter-ing the water column where it is oxidised.

The main influencing factors on the effect ofthe SOD are: the concentration of DO in theoverlying water; temperature; the characteristics ofthe bed(physical, chemical and biological); thevelocity of the water in the stream and the char-acteristics of the interstitial(or pore) water, nutnutrient availability may also be a factor. Com-monly used methods for modelling SOD use azero-order or constant source term, but this is notideal because it treats SOD as a model input ratherthan as a simulated part of the in-stream processes(Chapra, 1997). Alternative models have beendeveloped that relate the SOD to the sedimentedBOD and divide the sediments into anaerobic andaerobic layers(Di Toro et al., 1990) so that theoxidation of methane and ammonium can be sim-ulated. However, these methods are complex andcan involve many parameters that may not bemeasured and so must be evaluated by calibratingthe model to fit observed data.

A simpler approach is to assume that the SODuptake is related to the transfer of oxygen betweenthe overlying water and the sediments, i.e.:

dC dC K KV syKA C´ sy Cfy C (8)s B Edt dt HV

C FD GAs

whereK is the transfer coefficient(m day ) andy1

C is the DO concentration in the water(mgO l ), V is the water volume andA is the bedy1

2 s

area in contact.VyA can be approximated by thes

river depth, but it actually represents the availabi-lity of oxygen to the bed.

Variations of the reaction rate(K) with temper-ature (T) are commonly predicted using the fol-lowing Eq. (9):

TyT( )refKsK u (9)ref

where the most frequently usedT is 20 8C. Thisref

form is based on the Arrhenius equation, which

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provides the temperature dependence of a reactionbased on its activation energy. Assuming that watertemperatures will only vary over a limited range(say 0–408C), u is defined as a constant:

Elnusy (10)

RT Tref

whereT is the temperature in8C, R is the universalgas constant(8.314 J K mol ) and E is they1 y1

activation energy of the reaction.The temperature can also be specified in degrees

Kelvin if consistent use is made(it is the differencefrom the reference temperature that is important).Here, degrees Celsius are used. Zison et al.(1978)has reported a range of 1.04–1.13 foru and avalue of 1.065 is commonly employed. However,below 10 8C, the rate will decline faster thanindicated by this equation and from 0 to 58C itwill approach zero.

The influence of the oxygen in the overlyingsediments is clear. If the DO concentration in thewater goes to zero, the SOD will cease. Converse-ly, above a certain concentration it is usuallyassumed that the SOD is independent of theoxygen concentration in the overlying water. Baity(1938) found this to be the case when DO con-centrations were above just 2 mg O l . This cany1

2

be represented by a saturation relationship,

CK s K (11)SODC SOD( ) K qChs

where K is the new oxygen-dependent SODSOD

rate,C is the concentration of DO in the overlyingwater andK is the half-saturation value for thishs

dependence. Thomann and Mueller(1987) usedthe data of Fillos and Molof(1972) to estimateK and obtained a value of 0.7 mg O l , whichy1

hs 2

corresponds to independence at oxygen concentra-tions greater than approximately 3 mg O l . Ay1

2

simple SOD model would therefore be as follows:

Ty20( )

(12)dC K 1.065SODsdt H

3.3. Nitrification and the nitrogenous biochemicaloxygen demand

Nitrogenous matter in waste generally consistsof organic compounds such as proteins, urea,ammonia and nitrate together with intermediatedecomposition products of the proteins such asamino acids, amides and amines in varyingamounts. With time, the proteins are broken downby hydrolysis in a series of steps into amino acidsin a process called deamination. In this process,ammonia (NH ) is released and this can then3

combine with hydrogen ions in the water to formthe ammonium ion(NH ). Thus, ammonia inq

4

natural waters is invariably the result of either thedirect discharge of the material in wastewaters orthe decomposition of organic matter(Thomannand Mueller, 1987).

The ammonium in water can be oxidised underaerobic conditions to nitrite(NO ) by bacteria ofy

2

the genusNitrosomonas as described by the fol-lowing equation(Gaudy and Gaudy, 1980):

Nitrosomonasq q y2NH q3O ™ 4H q2H Oq2NO (13)4 2 2 2

From the stoichiometry of this reaction, 3.43 gof oxygen will be utilised in the oxidation of eachgram of nitrogen to nitrite and the nitrite formedin this reaction can then be oxidised further tonitrate(NO ) by bacteria of the genusNitrobac-2y

3

ter (Gaudy and Gaudy, 1980):

Nitrobactery 2y2NO qO ™ 2NO (14)2 2 3

This reaction consumes an extra 1.14 g ofoxygen per gram of nitrite-nitrogen in its oxidationto nitrate. Thus, the total oxygen requirement forthe process of nitrifying ammonium to nitrate is4.57 g for each gram of ammonium-nitrogen.However, some of this ammonium may be useddirectly in cell production, and so the actualoxygen utilisation may be closer to 4.2 g oxygenper gram of ammonium oxidised(Gaudy andGaudy, 1980) provided:

● there are adequate numbers of the nitrifyingbacteria available;

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309B.A. Cox / The Science of the Total Environment 314 –316 (2003) 303–334

● there is an alkaline environment to neutralisethe resulting acids;

● there is sufficient phosphate;● there is sufficient oxygen, i.e. at least 1–2 mg

O l (Chapra, 1997).y12

The source of nitrifying bacteria is usually thesurrounding soils, where one would expect to findthem in large numbers. In streams and rivers, thepresence of bacteria will generally depend on theavailable substrate and the nature and treatment ofthe wastewaters.

In order to model the uptake of oxygen, thereare two main approaches available. One approachuses a nitrogenous BOD(NBOD) to characterisethe process, while the other approach mechanisti-cally models the organic nitrogen, ammonia andnitrate explicitly. The NBOD method uses anoverall oxidation rate of the organic and ammoni-acal nitrogen together(called the total Kjeldahlnitrogen or TKN). Since no attempt is made tosimulate the actual reactions using the NBODmethod, this is a semi-empirical approach, but byusing an oxygen utilisation of 4.57 g O g N , they1

NBOD (L ) can be estimated by:N

L s4.57TKN (15)N

and therefore:dC

syK L (16)n NdtwhereK is the rate of NBOD oxidation.n

Clearly, such an approach is rather simplisticand it will not be able to accurately simulate thetimings of the real reactions when it uses oneprocess to describe a set of separate reactions. Themechanistic approach which models organic nitro-gen (N ), ammonium(NH ), nitrite (NO ) ando 4 2

nitrate (NO ) involves the following equations3

(Chapra, 1997):dNosyKoadtw xd NH4

w xsK N yK NHoa o ai 4dtw xd NO2

w x w xsK NH yK NOai 4 in 2dtw xd NO3

w xsK NO (17)in 2dt

where K is the rate of the reaction convertingoa

organic nitrogen to ammonium-nitrogen,K is theai

reaction rate of the first stage of nitrification(thatoxidises ammonium to nitrite) and K is thein

reaction rate of the second stage of nitrificationwhere nitrite is converted to nitrate. Under lowDO conditions(and in the anaerobic conditions ofmuddy sediments), a process of denitrification mayalso occur. This denitrification process involves abacterial reduction of nitrate to nitrogen gas, butthe oxygen produced is used by bacteria in respi-ration and so this will not act as an oxygen source.

An approach such as that described by Eq.(17)for nitrogen has been shown to be reasonable(DiToro, 1976) and, using this method, the rate ofconsumption of DO due to the nitrogenous oxygendemand can be expressed as:

dCw x w xs3.43K NH q1.14K NO (18)ai 4 in 2dt

where C is the concentration of DO. Whilst thismethod is more elegant than the NBOD method,three parameters rather than one are required. Themethod also requires greater amounts of informa-tion on input concentrations of all four nitrogenspecies. This means that the semi-empirical meth-od is more suitable in situations where there areinsufficient data available to use the mechanisticmodel.

Curtis et al.(1974) studied nitrification in therivers of the Trent Basin in the UK and found thatthe growth rates of bothNitrosomas andNitrobac-ter bacteria were essentially identical. However,laboratory experiments carried out by Alexander(1965) showed that theNitrobacter were five timesas efficient as theNitrosomonas in transformingnitrite and ammonium, respectively. This wouldsuggest that the ammonium concentration is thecontrolling factor for the rate and that the processmight be modelled based only on the temperature,and the ammonium andNitrosomonas concentra-tions (Knowles and Wakeford, 1978).

Denitrification is a biological process, whichtakes place under anaerobic conditions in the river,for example, in mud and bacterial films on thesurfaces of stones gravel and leaves. During thisprocess, nitrate is transformed into nitrogen gas by

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(predominantlyPseudomonas) bacteria, which canescape into the atmosphere due to its low solubilityin water. The oxygen produced in this reduction isused by the denitrifying bacteria because of theanaerobic conditions and so does not have aninfluence on the DO concentrations, but denitrifi-cation can be an important process for nitrogenremoval in rivers. Thus, in order to have a propernitrogen balance, this process must be includedwherever it is known to occur.

3.4. Chemical oxygen demand

Many types of industrial effluents contain sub-stances such as iron sulphite and aldehyde that areoxidised relatively rapidly by DO in a water body.These substances all exert an oxygen demand onthe water’s DO content and this demand is equiv-alent to a ‘fast’ COD. In some rivers, this demandcan be very important, but in others where indus-trial effluent discharges are relatively minor, thisis not the case. It should also be noted that thetechnique for measuring the COD uses high tem-peratures and strong oxidising agents and so doesnot reflect natural conditions. Thus, even wheredata are available, they may not actually reflect asignificant sink of oxygen in the naturalenvironment.

3.5. Atmospheric aerationyreaeration

Reaeration is the process of absorption of atmos-pheric oxygen into the water and is regarded asone of the most important factors controlling thewaste assimilation capacity of a river becausephotosynthesis is the only other source of oxygenreplenishment and this is limited to daylight hoursonly. In order to understand the basic mechanismof the transfer of oxygen to any water body fromthe atmosphere, one can consider a conceptualtwo-phase system such as a beaker of water.

The container of water is open to the atmosphereand can therefore interact with it. If the water isallowed to come to equilibrium with the atmos-phere above it, the concentration of DO that willbe reached will be fixed for a given temperatureand pressure(for that body of water). This isknown as the oxygen saturation concentration and

is described by Henry’s law, which states that, themass of any gas that will dissolve into a givenvolume of liquid, at a constant temperature, isdirectly proportional to the pressure that the gasexerts above that liquid. Thus,

psH C (19)e s

wherep is the partial pressure of O(mmHg), C2 s

is the saturation concentration of DO in the liquid(mg O l ) and H is Henry’s constant (mmHg/y1

2 e

(mg O l )). The saturation concentration of DOy12

will be dependent on the air pressure and thetemperature and salinity of the water. In lowlandfreshwater rivers, the most important effect onoxygen saturation is the temperature and variousmethods are available for calculating its value. Themost frequently used equation in water qualitymodelling is that developed by Elmore and Hayes(1960) for distilled water:

C s14.652y(0.41022T)s2 y5 3q 0.007991T y 7.7774=10 T (20)Ž . Ž .

whereT is the temperature in degrees Celsius.If the liquid in the conceptual system has its

DO concentration decreased very rapidly, say byan input of a reduced pollutant or BOD, the drivingforce for the restoration of the equilibrium DOcondition is the difference between the actual andsaturated concentrations. The rate of change ofDO with respect to time can be approximated asbeing proportional to the oxygen deficit, with theexchange taking place across the interfacial areaof the water’s surface. The proportionality constantis called the reaeration constantK and the deri-a

vation of this exchange is based on a ‘two-film’theory where it is assumed that a gaseous film isat the atmosphere side of the air–water interfaceand a liquid film present on the side of the water.In order to transfer oxygen from the atmosphereto the water, it is necessary for a parcel of waterto travel from the bulk liquid to the interface.Oxygen is then able to diffuse to the water throughthe gaseous and liquid films. In each film, resis-tances are encountered and the relatively highHenry’s constant for oxygen reflects a high ‘par-titioning’ of oxygen into the gaseous phase relative

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311B.A. Cox / The Science of the Total Environment 314 –316 (2003) 303–334

to the liquid phase. Therefore, the gas film hasrelatively low resistance compared to the liquidfilm and the liquid film is said to be the limitingcontrol for the exchange. Assuming complete mix-ing, the flux of oxygen through the controllingliquid film equals the time rate of change of DOin the vessel:

1 dm V dCNs s sK C yC (21)Ž .L sA dt A dt

i.e.

dCV sK A C yC (22)Ž .L sdt

where,N represents the rate of mass transfer perunit time per unit area, dmydt is the time rate ofthe mass transfer,A is the area through whichdiffusion occurs,V is the volume of the liquid andK is the DO interfacial transfer coefficientL

(m day ) and C is the actual DO concentrationy1

in the water(mg O l ). This can also be writteny12

as

dCsK C yC (23)Ž .a sdt

where K is the volumetric reaeration coefficienta

(day ), which reflects the proportionality alreadyy1

discussed, and is given by

K ALK s (24)a V

or

KLK s (25)a H

whereH is the ratio of the surface area to volume(m) usually approximated as the depth.

The source term for reaeration in a DO mass-balance equation such as Eq.(2) may be repre-sented as:

K C yC (26)Ž .a s

and the reaeration rate coefficient(K ) will dependa

on:

● internal mixing and turbulence due to velocitygradients and fluctuation;

● temperature;● wind mixing;● waterfalls, dams and rapids;● surface films

because each of these processes will affect theturbulence in the surface layer of the water andthus control the rate of oxygen absorption. How-ever, it is more common for the effects of water-falls and other flow structures to be dealt withseparately such that they act as a direct source ofDO into the system.

3.6. Plant respiration and photosynthesis

The presence of aquatic plant in a water bodywill have a marked effect on the DO resourcesand the variability of DO throughout a day or fromday to day(Thomann and Mueller, 1987). Phyto-plankton, macrophytes and periphyton all contrib-ute to this effect and are important because oftheir ability to photosynthesise. During photosyn-thesis, the chlorophyll-containing plants use solarenergy to convert carbon dioxide into carbohy-drates and release oxygen in the process. Thissubjects the water to pure oxygen unlike atmos-pheric reaeration at the surface that occurs with anatmosphere of only approximately 21% oxygen.Since saturation values refer to the standard atmos-phere, photosynthesis can result in supersaturatedvalues and DO concentrations of 150–200% ofthe air saturation are common in highly productivestreams(Thomann and Mueller, 1987).

Depending on the depth of the stream, two typesof plant will tend to grow. In deeper rivers, plantswithin the water column(phytoplankton) dominateand their distribution declines with depth in rela-tion to decreasing light penetration. In the shallow-er streams where light can reach the bottom,bottom plants tend to dominate. These plants caninclude rooted and attached macrophytes andmicrofloral growths called periphyton. Unfortu-nately, frequent measuring of the plant activitydirectly in a stream is not a trivial task, particularly

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Fig. 2. A typical plot of DO concentrations measured by an automatic water quality monitor at Mildenhall on the River Kennet(Williams et al., 2000; Neal et al., 2001).

for shallow streams dominated by periphyton andmacrophytes.

Because this process requires solar energy, itwill only occur during daylight hours. However,the plants and algae will consume oxygen as theyrespire and this will occur continuously. In a mass-balance DO model, therefore, the oxygen termsfor representing photosynthesis(P) and respiration(R) will be PyR, and for these two processesonly:

dCsPyR (27)

dt

The photosynthetic production of oxygendepends on several factors including the lightintensity, the optical density of the water, watertemperature, carbon dioxide and nutrient availabi-lity, and the quantity(and type) of plants(Edwardsand Owens, 1965). The rate of respiration is oftenassumed constant, but clearly will also be depend-ent on oxygen and nutrient availability, and thequantity (and type) of plants. The combination ofthese two processes can produce strong diurnal(asshown in Fig. 2) and seasonal effects on the DOconcentrations.

Photosynthesis therefore begins at dawn andends at dusk and the total photosynthesis of thewater column roughly follows the approximatelysinusoidal variation of the solar irradiance duringthe day(Kirk, 1994). However, there can be photo-

inhibition effects close to the surface that willreduce the photosynthetic rate per unit volume. Ifdinoflagellates are the dominant organisms, theremight also be a reduction in the photosyntheticrate during the middle part of the day due to theirdownward migration to areas of lower light inten-sity (Kirk, 1994).

In temperate and ArcticyAntarctic latitudes,there is a marked seasonal variation in aquaticphotosynthesis. This is caused by and is the causeof the seasonal variation in plant biomass. Biomassand photosynthesis rates are both low in the winterdue to the decrease in irradiance and temperature.In deeper waters, flow circulation may also takephytoplankton to depths where the light intensityis insufficient for photosynthesis to take place. Inspring, irradiance and temperature increase andthis, combined with the availability of nutrients,leads to a ‘bloom’ in phytoplankton and increasedphotosynthesis. The behaviour during the rest ofthe seasons tends to be complex and highly vari-able (Kirk, 1994), affected by zooplankton graz-ing, fluctuations in nutrient availability, or majorchanges in species composition due to changes inthe water quality. For example, higher pH valuesin late summer tend to favour blue–green algae(or cyanobacteria) rather than diatoms.

Thus, overall, photosynthesis will tend to dom-inate during the growing season whilst decompo-sition and respiration may dominate outside of thisgrowing season. During the growing season, the

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swings in stream oxygen concentration due todiurnal light variations can be enough to leave thewater supersaturated in the afternoon and yetseverely depleted just before dawn(Chapra, 1997).It is therefore crucial that, particularly in smallstreams, the effects of plants are not ignored inany river oxygen modelling that is carried out,although very shallow streams might show restrict-ed effects due to interactions with the atmosphere(Kirk, 1994). This applies especially with regardto predicting the effects of waste input to the riversystem, for example, in simulating the impact of anew STW. In general, values of the average dailyareal production rate range from approximately 0.6to 6 g (O )m d for moderately productivey2 y1

2

streams. Highly productive streams can range from6 to 40 g(O )m d (Chapra, 1997).y2 y1

2

4. Parameter estimation for modelling DO inlowland rivers

The processes affecting DO in rivers that havebeen described in Section 3 provide a modellingframework to simulate those processes in a mass-balance model. The process equations describedthere all require(reaction) rate parameters and socommon procedures for identifying those parame-ters are reviewed here. Any of the parameters canbe evaluated by a process of calibration where theparameter values are adjusted until the simulateddata fit observed data. However, in multi-parame-ter systems it may not be true that the parametersare independent, and so it will not be possible toensure that a unique set of parameter values isobtained. A variety of statistical methods forparameter estimation also exist, including theextended Kalman filter(Jazwinski, 1970; White-head et al., 1981), instrumental variable estimation(Young and Whitehead, 1977), and general sensi-tivity analysis (Spear and Hornberger, 1980).However, in this section, alternative methods forobtaining estimates of the rate parameters aredescribed that are based on field measurementsand theoretical considerations, or empiricalrelationships.

4.1. BOD oxidation, sedimentation and re-suspen-sion rates

The usual method for estimating the BOD rateis by laboratory experiment and involves a similartechnique to that used to measure BOD. In theassay, a sewage sample is added to series of‘closed-batch reactors’ or bottles containing waterwith a known DO content. Small amounts ofbacteria are then added to each bottle and they aresealed and placed in an incubator. Values ofKr

can vary from 0.01 to 0.5 day depending on they1

type of waste and the degree of stabilisationalthough this could rise up to 2.0 day immedi-y1

ately downstream of effluent discharges(Williams,1993).

Another method of determining the rate constantis to use measurements of BOD at successivestations on a river. This process can be describedby solving the BOD Eq.(28) here limited to aconstant flow within a channel with a constant-geometry:

≠L ≠LsyU yK L (28)r

≠t ≠x

whereU is the mean reach velocity andK is ther

total removal rate(day ) of BOD, which isy1

composed of oxidation and settling, i.e.

K sK qK (29)r d s

Given the constant-flow assumption, at steady-state, this becomes

≠LU syK L (30)r

≠x

and, if complete mixing is assumed at the topstation, an initial concentration can be calculatedas the flow-weighted average of the upstreamconcentration in the river(subscript r) and anyother input(subscript i):

Q LqQ Li i r rL s (31)0 QqQi r

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Fig. 3. A plot of BOD downstream from a point source oforganic pollution(after Chapra, 1997).

Using this initial condition one can solve Eq.(28) for:

yK yU x( )rLsL e (32)0

However, one can also use this solution toestimate the removal rate,K by taking logarithms,r

i.e.:

xlnLslnL y K (33)0 rU

and this has the form of a straight line. Thus, ifBOD measurements at successive stations are plot-ted on a log-scale againstxyU (which is alsoknown as the time constant,t), the gradient willbe equal to the combined rateK if the BODr

diminishes as a fixed first-order decay. Fig. 3shows a typical pattern that might be observed ina stream receiving sewage. In this case, a singlestraight line is not obtained. Instead, the rate ishigher immediately below the discharge. Thesehigher rates are usually caused by the fast degra-

dation of readily decomposable organics and thesettling of sewage particulates.

The coefficientK for the rate of settling ands

re-suspension of BOD needs to be determinedfrom BOD profile measurements even though set-tling and re-suspension are physical processes,governed by the flow conditions and particle size.One might assume that the removal by settling canbe estimated given a depth(H) and settling veloc-ity (v ), i.e.s

vsK s (34)s H

using typical settling velocities such as 0.1–0.5m s . However, particulate matter can flocculatey1

into different sizes depending on the particle pop-ulation in the receiving water(Kranck, 1974) andthis will have a marked effect. Under steady-stateconditions one might assume that the amount ofsettling and scour should be equal, but even inthis scenario the BOD concentrations in the mate-rial settling and that being scoured might not beequal. Because of these factors, the coefficientKs

is usually just an average for the river reach beingstudied and Dobbins(1964) provides a methodfor its calculation provided thatK has alreadyd

been determined, for example, from laboratorymeasurements. This technique is similar to thegraphical method shown in Fig. 3, but it alsoaccounts for addition of BOD along the reach.

4.2. SOD oxidation rate

SOD in rivers is usually measured either bymodelling the observed oxygen levels or by directmeasurement. The modelling approach is extreme-ly common and requires a DO model to bedeveloped for the water body where all of the ratesexcept for the SOD have been determined. TheSOD rate can then be estimated by adjusting itsvalue until the simulated DO values match thoseobserved. Although widespread, this calibrationmethod is flawed, in principle, because it assumesthat all of the other model parameters are knownwith some certainty, when this is rarely the case.The direct measurement of SOD is made byenclosing the sediments and some overlying water

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Table 2SOD values(from Thomann, 1972; Rast and Lee, 1978)

Bottom type and location K yg O m day20 y2 y1SOD 2

Average value Range

Sphaerotilus (10 g-dry wt. m )y2 7 –

Municipal sewage sludge:Outfall vicinity 4 2–10Further downstream(or ‘aged’) 1.5 1–2Estuarine mud 1.5 1–2Sandy bottom 0.5 0.2–1Mineral soils 0.07 0.05–1Areal hypolimnetic oxygen demand(AHOD) in lakes – 0.06–2

in a chamber and measuring the concentration ofDO in the water over a period of time. This canbe carried out in the field or laboratory but hasthe same flaws as the similar BOD and light–darkbottle tests also described in this review. Studiesby Rolley and Owens(1967) showed that thisparameter varied considerably from river to river;and that the rates found ranged from 0.144 to 9.84g O m day where there was an overlying DOy2 y1

2

concentration of 7 mg O l at 158C. However,y12

they do not explain this variation. Typical valuesare listed in Table 2 and, in general, values fromapproximately 1 to 10 g O m d are probablyy2 y1

2

indicative of enriched sediments(Chapra, 1997).The organismSphaerotilus sp. (sometimes called‘sewage fungus’ despite it being an attached fila-mentous higher bacterium) is included to indicatethat the processing of organic matter in sedimentsis not the sole domain of simple bacteria. In fact,freshwater(or Zebra) mussels(Dreissena poly-morpha) can also have a significant effect(Efflerand Siegfried, 1994).

4.3. Nitrification rates

If the oxidation of nitrogen compounds is wellpredicted by an NBOD model, the NBOD rate canbe determined in the same manner as described inSection 4.1 for the carbonaceous BOD rate. If thisis done, the values found for the rate parameterK are approximately the same as for the CBODn

(Thomann and Mueller, 1987). Thus, values ofbetween 0.1 and 0.5 day would be reasonabley1

in deeper rivers, with smaller streams perhaps

displaying rates as high as 1 day at 208Cy1

(Thomann and Mueller, 1987). The effect of tem-perature on the rate coefficient is given by:

Ty20( )K sK 1.08 (35)n T n 20( ) ( )

for 10(T(30, whereT is the temperature in8C,but values for the temperature coefficient(1.08)have been reported to be in the range 1.0548–1.0997(Zison et al., 1978). The temperature lim-itations are given, because the bacteria are notactively multiplying below 108C and above 308Ctheir activity is severely limited. Therefore, outsideof this range,K may be assumed zero.n

If, instead, one assumes that the nitrificationrate is limited by the oxidation of ammonium byNitrosomonas bacteria(Alexander, 1965; Knowlesand Wakeford, 1978), then, provided there aresufficient bacteria, the rate can be described by afirst-order decay rate such as:

w xd NH4Ty20( )w xsK NH 1.08 (36)an 4dt

where wNH x is the concentration of ammonium4

and K is the ammonium nitrification rate at 20an

8C found by Knowles and Wakeford(1978) to bebetween 0.01 and 0.5 day .y1

The process of denitrification may also beassumed to be at a first-order decay rate propor-tional to the nitrate concentration in the water.Other effects will be due to both the temperatureand the bed area since the denitrifying bacteria

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will be predominantly on the river bed. If theeffects of the bed area and substrate type arecombined into the rate parameter and the temper-ature dependency is taken to be that of Toms etal. (1975) then:

w xd NO3

w xs(yK yH) NO (37)den 3dt

and

0.0293TK sK (1.06981)=10 (38)den den,20

where wNO x is the nitrate concentration,K is3 den

the denitrification rate coefficient,K is theden,20

rate coefficient at 208C and T is the watertemperature in degrees Celsius.

4.4. Atmospheric reaeration rates

The transfer of oxygen between the water andthe atmosphere, as a function of internal mixingand turbulence, has been the subject of much studyand investigation. This is because, in most models,the addition of oxygen to the water from theatmosphere is the most significant source of DO.Hornberger and Kelly(1975) expressed that ‘esti-mates of the(rate of) exchange of oxygen betweenthe atmosphere and water are critical in severalareas of water quality control and monitoring,’ andthat these rates are needed for both predicting theeffects of waste disposal on a water body and formeasuring the rates of community photosynthesisand respiration.

Three general techniques are used for ‘measur-ing’ the reaeration coefficientK of a stream ora

river. These are(a) the DO balance,(b) thedisturbed-equilibrium technique and(c) the tracertechnique. The DO balance method consists ofmeasuring the various sources and sinks of DOand determining the amount of reaeration neededto balance the equation. The disturbed-equilibriummethod relies on producing an artificial DO deficit(by adding a reduced chemical such as sodiumsulphite to the stream) and then measuring thesubsequent recovery rate by measuring the DOconcentration upstream and downstream of the

reach. An inert radioactive gas such as Krypton isused as a tracer for oxygen and correlates the rateof desorption of the tracer gas with the rate ofadsorption of oxygen.

The latter methods are used most commonly inthe literature. However, the DO balance methodcan be very useful since these methods do notrequire specific experiments to be carried out. Allthat is required is the information provided byautomatic DO monitors—provided that the dataare of a sufficient frequency and quality. Since thelatter methods are relatively self-explanatory; onlythe DO balance methods for measuring the rateparameter are described here.

4.4.1. DO balance methodsTechniques have been developed that use the

diurnal variation in DO concentrations to estimatereaeration, photosynthesis and respiration rates.These techniques are based on one of two familiarequations, depending on whether it is for day-timeor night-time measurements:night-time:

dCsK C yC yR (39)Ž .a sdt

day-time:

dCsK C yC qPyR (40)Ž .a sdt

whereR is the respiratory oxygen uptake rate andP is the photosynthetic oxygen production rate.Solutions to these equations based on continuousmeasurements of DO have been provided by Odum(1956), Hornberger and Kelly(1975) and Chapraand Di Toro(1991).

The oldest, simplest and most widely usedmethod is that of Odum(1956). It estimatesKa

from the rates of DO change and saturation deficitsjust before dawn and just after sunset:

B E B EDC DCSSC F C FyD G D GDt Dt SR

K s (41)a w zx |C yC y C yCŽ . Ž .SS SRs sy ~

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Here the subscripts ‘SS’ and ‘SR’ denote thetimes just after sunset and just before sunrise,respectively;C is the concentration of DO andCs

is the DO saturation concentration. No specificlimitations are given on this method, but it willnot perform well if the oxygen deficit(C yC)s

does not change significantly over the ‘dark’period.

The method of Odum(1956) only considers theconditions at the beginning and end of the darkperiod, but more complex methods can be devisedthat make use of the data from the whole of thedark period or the whole of the light period.Hornberger and Kelly(1975) created two methodsof estimating the reaeration rate. The first andsimpler method, like that of Odum(1956) usesonly that part of the DO curve where there is noinfluence by sunlight and, therefore, no influenceby photosynthesis. This method derives least-squares estimates ofR and K by using repeateda

attempts to predict the post-sunset DO concentra-tions at the end of sequential time intervals fromthe concentrations at the beginning of each inter-val. As with Odum(1956), the starting equationsare Eqs.(39) and (40) given at the start of thissection and, assuming that the respiration rate hasa constant valueR , integrating from timet to0 j

t yields:jq1

R0 yKd¯C yCs C yC y 1ye (42)Ž . Ž .jq1 j s,j j Ka

where

dst yt (43)jq1 j

and is the mean saturation concentration ofCs,j

DO over that interval, calculated from the watertemperature. The values ofR andK may then be0

estimated by choosing values so that the estimateof C best fits (by least-squares methods) thejq1

measured data. Simplifying the notation by letting:

dsC yCj jq1 j

¯a sC yCj sj jyK daj s1ye1

R0j s (44)2 Ka

where the subscript ‘j’ refers to sequential timeintervals during the night, it can be seen that thevalue of K that gives the best fit will be obtainedby minimising a function,J for the dark hourswhere:

2o oJ j ,j s d ya j qj j (45)Ž . Ž .1 2 j j 1 1 28j

and the superscript ‘o’ refers to observed data. Adirect minimisation of this function yields:

2B Ea dy a a dj j j j jC F8 8 8 8D Gj j j j

j sy (46)2

N a dy a dj j j j8 8 8j j j

and

dj8j

j s (47)1

a yNjj 28j

Therefore:

1K sy log 1yj (48)Ž .a 1

d

and

R sK j (49)0 a 2

This method is very similar to that of Odum(1956), but it uses all of the dark-hours data, and,as with the Odum method, this method will notwork well if (C yC) does not vary much durings

the night. The second method by Hornberger andKelly (1975) includes the light hours(or day-time)measurements and so involves solar radiationto estimate the photosynthetic rate, and thereforeto estimate the reaeration rate. As before, thisinvolves fitting parameters to an exact solution ofthe linear DO mass-balance equation Eq.(40) andthen using these parameters to predict the DO atthe end of sequential time intervals from the

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concentrations at the beginning of each interval.To do this, (PyR) is assumed to be linearlyrelated to the light intensity and that:

PsaI andRsR (50)0

whereI is the incident radiation anda is a constantof proportionality. Integrating Eq.(40) gives:

B Ea R0¯ ¯C FC yCs C yCq Iyjq1 j s,j j jD GK K

yKd1ye (51)Ž .

To ease notation, if the same values ofj and1

j as before are used along with2

aj s (52)3 K

the best-fit will be obtained by minimising thenew J function:

2o o oJ j ,j ,j s d ya j yI j j qj j (53)Ž . Ž .1 2 3 j j 1 j 1 3 1 28j

A direct minimisation yields:

gj sy3

b

j smj qb2 3

1j s (54)1 ˆmj qbŽ .3

where:

2 2ˆ ˆasym d Iqmm dq I qNm y2m Ij j j j8 8 8 8ˆ ˆˆbsymb a d dyb d Iqmb dj j j j j j8 8 8 8

q2 a Iy2m a q2Nmby2b Ij j j j8 8 82 2ˆ ˆgsyb a dq a qbb dy2b a qNb

(55)j j j j j8 8 8 8

and:

IyNmj8ms

dj8a yNbj8

bs (56)dj8

where

2d Ij j8 8Iyj8 d Ij j8ms

Ad a Ij j j8 8

a yj8 d Ij j8bs (57)A

and

Ny d Ij j8 8As (58)

d Ij j8

So oncej , j andj have been calculated from1 2 3

Eq. (54), K andR may be computed from:a 0

yK daj s1ye1

R0j s2 Ka

aj s (59)3 Ka

The equations are rather long, but do not requiresolving using iterative calculations like some otherday-time solutions. It should be noted that typo-graphical errors were found in three of the inter-mediate terms as presented on page 734 ofHornberger and Kelly(1975), but that the correctderivations are given here(G.M. Hornberger, per-sonal communication, 2001).

A more recent technique has been developed byChapra and Di Toro(1991) and is mainly usedfor estimating the rates of photosynthesis andrespiration in a water body based on the diurnalDO curves. However, it does produce an estimateof the reaeration rate in the process of doing so.

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The method is described in more detail in Section4.5. However, it is worth noting here that thismethod for computingK is particularly sensitivea

when the difference between the time of theminimum DO deficit and the time of the solarnoon (i.e. the time lag) is low. Because of this,some authors(e.g. Williams et al., 2000) haveattempted to use another estimate ofK in ordera

to calculate the photosynthesis and respirationrates.

A review of these productivity methods byKosinski (1984) found that all of the techniqueswere highly accurate, and that this was particularlytrue for the day-time methods. This review alsoconsidered the impacts of noise, data resolution,extreme low or high productivities, temperaturevariations, and fluctuating light conditions. Noisydata and long observation intervals were not aserious problem for most of the methods and couldbe improved with the use of a moving DO averageinstead of single DO data points(Kelly et al.,1974), but low productivity or high reaerationrates were found to cause serious impairments inall of the methods. Therefore, it is possible toobtain a value of the reaeration rate parameter byexperimentation or data analysis, but this is notacceptable from a modelling perspective, becauseit is known that the rate parameter will vary underdifferent stream conditions.

4.4.2. Semi-empirical methods based on meanhydraulic parameters

Because the reaeration rate parameter is soimportant to DO modelling, many attempts havebeen made to estimate this from other parametersthat can be simulated andyor from those that maybe assumed not to vary greatly under the expectedconditions. Such studies have generally used oneof the experimental ‘measurement’ techniques toestimate the reaeration rate in real streams orflumes in which many hydraulic parameters arealso measured. These data are then used either toform an empirical relationship or to test theoreti-calyconceptual models of atmospheric aeration andreaeration. Unfortunately, none of the presentlyavailable models of the oxygen absorption processin open-channel flows is sufficiently developed toaccurately predict the reaeration coefficient from

mean hydraulic parameters alone. As a result,water quality models, instead, generally rely onsemi-empirical and empirical regression equationsto estimate this coefficient. Numerous studies havebeen made and many prediction equations havebeen presented for the reaeration coefficient,K .aThe studies have ranged from theoretical investi-gations(O’Connor and Dobbins, 1956) to empir-ical field studies(Churchill et al., 1962; Owens etal., 1964; Edwards and Owens, 1965). For thefield studies, the general procedure was to measurethe reaeration coefficient indirectly by measuringchanges in DO under closely controlled conditions(i.e. disturbed-equilibrium techniques). The vari-ous studies have covered a wide range of riversituations from shallow short-run streams in Eng-land, to deep, wide and slow moving rivers in theUS.

All of the empirical, semi-empirical and concep-tual studies are based on a general differentialequation of the oxygen balance in a river:

dDsK LyK D (60)d adt

and, ignoring BOD oxidation,

dDsyK D (61)adt

whereD is the DO deficit,L is the BOD concen-tration,K is the BOD oxidation rate andK is thed a

reaeration rate. When integrated this becomes:

yK taD sD e (62)t 0

which is usually presented in the literature as:

yk taD sD 10 (63)t 0

where D is the deficit at time zero andk is a0 a

different reaeration coefficient, related toK bya

K s2.303k . This crucial factor is sometimes over-a a

looked in the literature. It needs to be rememberedthat the rate required for a model isK (Eqs. (2)a

and(22)), which is equivalent toK in the original2

Streeter–Phelps equations, and notk . In thisa

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section, all parameter estimations quoted will befor the reaeration coefficientK in units of dayy1

a

based on the original work.In addition, most empirical methods are based

on the Ohio River findings of Streeter and Phelps(1925), where the reaeration coefficientk is influ-a

enced by the hydraulic and physical characteristicsof the river channel. The relationship was formu-lated empirically as:

n1EUk s (64)a 2H

where U is the flow velocity (m s ), H is they1

depth (m) and E and n represent the(undeter-1

mined) empirical constants dependent on the phys-ical and hydraulic conditions of the channel. Inthe original description,n was related to the1

function ‘mean velocity increase per 5-footincrease in river stage,’ and the constantE, wasrelated to the slope of the channel and to an‘irregularity factor,’ which is a measure of therelative roughness of the channel bed.

Unfortunately, with each experiment and theo-risation that followed the work by Streeter andPhelps(1925), came a different method to estimatethe rate parameter. This is highlighted in the workby Parkhurst and Pomeroy(1972) who say that‘the literature shows a multiplicity of reaerationformulas with little consensus’, and that ‘theseformulas cannot all be right’. Many of the empir-ical prediction equations appear to only fit the fewsets of data from which they were derived, whilethe conceptual models suffer because they allcontain coefficients that depend on a number ofdifferent flow variables. Nonsensically, the com-bination of coefficients and variables chosen for agiven prediction equivalent may not even maintaindimensional consistency. Lau(1972) found fromhis laboratory experiments that the dimensionlessvariable is a function of both the friction¯K hyua

factor f and the Reynolds number whereyuhyy

is the kinematic velocity of the water—showingthat the simple empirical prediction equations thatrelateK to andh cannot be satisfactory. How-ua

ever, in a number of the semi-empirical cases thecoefficient may incorporate one or more physicalparameters.

Wilson and Macleod(1974) tested 16 predictionequations against published data covering a widerange of hydraulic variables and found that eventhe ones with the best correlations gave veryunreliable prediction rates. They suggested that thepoor performance of these equations might beattributable to one or more missing variables(based on dimensional analysis). Of the equationstested by Wilson and MacLeod, those of Dobbins(1964) and of Parkhurst and Pomeroy(1972) gavethe best-fit overall to all of the data investigated.However, Lau(1972) warns that the variation ofk (and thereforeK ) cannot be covered by a singlea a

equation. Therefore, the selection of an equationfor K is usually based on the user’s judgement bya

considering the desired complexity, hydraulic sit-uation and available data. However, as a part ofthis work it has been found that many reviews onthis topic have reproduced one or more equationscontaining errors in that they are not found to beconsistent with the original published equations.Some of the confusion arises because, as alreadynoted, theoretical considerations require that theparameter is actually logarithmic and the base ofthe logarithm is not always reported. In addition,the units of time can be confused between days,hours and seconds, for example. Because of theconfusion that exists in the literature, this reviewhas re-examined the original texts for each of themajor studies in this field. This was done to ensurethat the equations presented here are correct andthat consistent use of units is made. Table 4summarises the results of this extensive review ofwork covering a period of more than 40 years andthe terms used in the equations are described inTable 3.

4.4.3. SummaryWithin the large range of equations presented

here, some patterns have been found. For example,those equations incorporating only simple hydrau-lic parameters of stream velocity and depth exhibita tendency to predict higher values ofK thana

those observed, except of course for the data setused to formulate that equation(Wilson and Mac-leod, 1974). It has also been found that equationsbased on flume data tend to predict higher ratesthan those based on river data for similar condi-

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Table 3Description of the terms used in Table 4

B 0.976q0.0137(30-T)1.5 CA 1.0qF2

C4 0.9qF E Energy dissipation per unit mass of flowsSUgF Froude numbersUy(6gH) g Acceleration due to gravityH Mean depth Ad 9.68q0.054 (T-20)Q Mean flow rate S SlopeT Temperature U Mean velocityu* Shear velocitys6(HSg) W Width

tions (Bennett and Rathburn, 1972). Of the simpleempirical equations mentioned in this paper, thebest results probably come from the modifiedequation by Churchill et al.(1962) provided byIsaacs and Gaudy(1968). However, none of theequations incorporating only the simple hydraulicparameters of stream velocity and depth allowaccurate predictions ofK for all flow situationsa

(Wilson and Macleod, 1974). Given that the var-iation in the predicted coefficients by empiricalmeans appears to be larger than could be attributedto experimental error, it is suggested that variablesmay have been omitted from the equations(Rath-burn, 1977).

The equations involving energy terms performsomewhat better than those that do not, especiallywith regard to the differences between river andlaboratory situations(Wilson and Macleod, 1974).Of these methods, those of Dobbins(1964, 1965)and Parkhurst and Pomeroy(1972) give the mostgenerally reliable predictions over the whole rangeof data, with the Parkhurst and Pomeroy equationproducing the lowest standard error of prediction(Wilson and Macleod, 1974). The reviews byRathburn(1977) and Bennett and Rathburn(1972)also rate the equation by Dobbins to be the mostreliable. However, even these ‘best’ predictionsare found to be often lacking when compared withobservations and in many cases the observed ratescould be several times larger or smaller than thosepredicted. Wilson and Macleod(1974) suggestthat this might be due to uncertainties in theoxygen balance determination. However, it seemslikely that the discrepancies arise because of errorsin the theory. For example, the equations mightlack one or more of the physical variables affectingthe reaeration rate.

Using dimensional analysis techniques, the esti-mates by both Dobbins(1964, 1965) and Parkhurstand Pomeroy(1972) contain incomplete sets ofvariables. The missing variables could for exam-ple, relate to surface-active contaminants. Thesecompounds can lead to a reduction in the observedreaeration rate and yet not one of these estimationmethods could predict this effect. Examining par-tial correlation coefficients, it has been found thatof the variables usually incorporated in the rateestimates, the most important was depth, thenaverage velocity and then slope and that the widthwas negligible(Bennett and Rathburn, 1972). Ifonly the flume data were examined, the order ofimportance was slightly different with slope, width,depth and average velocity in descending order ofimportance.

There is therefore, no consensus by which reaer-ation rates can be estimated with certainty on ariver without measuring directly(say by a tracerexperiment). The productivity methods wouldappear to be more accurate than those using arelationship based on hydraulic parameters(Kosin-ski, 1984), but they are far more data intensiveand require data of a high quality. Furthermore,productivity methods are not suitable for use inpredictive models since they require the data thatsuch a model would be designed to simulate!However, these methods could be useful as ameans of improving the simulation of primaryproduction in rivers, where one has access to(evenbrief) records of high-intensity measurements atimportant times of the year such as spring andsummer(Williams et al., 2000). If temporal vari-ability is very low, then it would also be possibleto use the estimates from the more complex pro-ductivity methods in a model based on a single

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Table 4A summary of the available reaeration coefficient prediction equations from the literature

Type of system Equation Range of hydraulic variables used to generate or verify theequation

Number Reference

Discharge Meanvelocity

Meandepth

Energyslope

Ka 20( )

of data

(m s )3 y1 (m s )y1 (m) (my1000m)

(day )y1

points

Conceptual model

0.25S Ty20( )K s11.057 1.016a 1.25H – 0.06–1.280.27–11.28 0.027–3.6 0.04–11.1 43

O’Connor &Dobbins (1956)

0.5U Ty20K s3.952 1.016a 1.5H

Large rivers

0.969U Ty20( )K s5.0140 1.0241a 1.673H 27–489 0.56–1.52 0.65–3.480.126–2.35 0.52–12.8

30 means of509measurements

Churchill et al.(1962)

Recirculating flume

0.408E 20yT( )K s68.4 1.016a 0.66H – 0.07–0.65 0.02–0.06 0.75–41 24–265 58Krenkel andOrlob (1962)

Conceptual and flume

K sa0.375 0.125B EC A E 2.7510BEA d

C F1.7535 coth1.5 0.5D GC H C4 4 – – – –Approximately0.1–100 –

Dobbins(1964,1965)

Small streams

0.73U Ty20( )K s6.9152 1.0241a 1.75H 0.08–1.03 0.04–0.56 0.12–0.740.156–10.6 0.71–113 32

Owens et al.(1964)

Large rivers and flume

U Ty20( )K s5.1340 1.0182a 1.33H71.2–15705 0.55–1.62 1.16–15.5 – 0.17–2.46 )14

Langbein andDurum (1967)

Recirculating cylindri-cal flume

B EU Ty20( )C FK s4.7531 1.0241a 1.5D GH – 0.17–0.50 0.15–0.46 – 3.08–29.8 52Isaacs andGaudy(1968)

Recirculating 0.61 mflume

UK s4.05a 1.5H – 0.17–0.34 0.05–0.15 0.43–1.0 31–74 12

Eloubaldy(1969)*

uUK s154a H –

Recirculating cylindri-cal flume

UK s3.60a 1.5H – 0.09–0.51 0.30–0.45 – 1.3–11.7 48

Isaacs et al.(1969)*

Recirculating 0.20 mflume

0.85B EUC FK s10.9aD GH – 0.20–0.58 0.05–0.15 – 20–43 18

Negulescu andRojanski(1969)*

Flumes and large rivers

B EuU0.5 Ty20( )C FK s24.86081qF 1.016Ž .aD GH – 0.06–1.52

0.01–11.28

0.03–20.38 15.32–168 131

Thackston andKrenkel (1969)

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Table 4(Continued)

Type of system Equation Range of hydraulic variables used to generate or verify theequation

Number Reference

Discharge Meanvelocity

Meandepth

Energyslope

Ka 20( )

of data

(m s )3 y1 (m s )y1 (m) (my1000m)

(day )y1

points

Small and large rivers

0.607UK s5.5773a 1.689H 0.08–489 0.04–1.52 0.12–3.48

0.126–10.16 0.52–113 62

Bennett andRathburn(1972)

Small, mountainstreams K s2.30(1955.2Ey1657Sq20.87)a

0.003–0.007 0.03–0.56 0.02–0.27

0.100–370 4.4–8160 97 Holtje(1972)*

1.483K s4440Ea

Flumes and large rivers

3B EU uUC FK s2506.7aD GH U – 0.07–1.52 0.01–3.48 0.65–41 0.52–265 140 Lau(1972)

Recirculating 0.76 mflume

0.703B EUC FK s4.52a 1.5D GH

0.001–0.003 0.02–0.14 0.03–0.19 – 1.7–23 239

Padden andGloyna(1972)*

Sewers

K sa2 0.3751q0.17F (SU)Ž .

Ty20( )23.0400 1.0212H

0.01–0.249 0.45–3.60 0.05–0.48 0.71–68.4 2.21–155 74

Parkhurst andPomeroy(1972)

Medium to large rivers

0.6U Ty20( )K s4.1528 1.016a 1.4H 27–489 0.06–1.520.27–11.28 0.027–3.6 0.04–12.8 73 Bansal(1973)

Recirculating 0.359 mflume

uUK s123a H – 0.29–0.88 0.03–0.11 – 24–113 19

Alonso et al.(1975)*

Small to medium sizedstreams 0.5K s0.8880.30q5570.8SŽ .a

0.013–11.6 – – 0–7.90 0.34–20.7 20 Foree(1976)*

Small streams K s22700SUa 0.01–1.05 0.08–0.38 – 0.2–13.3 1.8–49 59 Grant(1976)*Non-tidal streamsincluding pools andwaterfalls K s31200SUa 0.008–30 -0.67 –

0.095–56.8 0.08–305 605

Tsiovoglou andNeal (1976)*

Small streams K s3170Sa

0.008–0.23 – – – – 28

Tsiovoglou andNeal (1976)*

Small, lowland streams

0.76 2.66 1.13U (1qF) SK s23000a 0.60H 0.02–0.35 0.06–0.32 0.13–0.50

0.390–2.03 1.0–15.0 29

Thyssen andJeppesen(1980)*

Shallow macrophyterich streams

0.734 0.938784U Sk sa 0.42H 0.02–2.9 0.06–0.52 0.12–1.37 0.3–7.4 0.2–97.7 144

Thyssen et al.(1987)

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Table 4(Continued)

Type of system Equation Range of hydraulic variables used to generate or verify theequation

Number Reference

Discharge Meanvelocity

Meandepth

Energyslope

Ka 20( )

of data

(m s )3 y1 (m s )y1 (m) (my1000m)

(day )y1

points

Large streams

0.524(US)K s517 U-a 0.242Q0.556, pool and riffle

0.0028–210

0.003–1.83

0.0457–3.05 0.01–60 – 99

Melching andFlores(1999)

0.528(US)K s596 U-a 0.136Q0.556, pool and riffle

0.313(US)K s88 U-a 0.353H0.556, pool and riffle

0.333(US)K s142 U-a 0.66 0.243H W0.556, channel control

References with asterisks have information taken from the review by Thyssen et al.(1987), not from the original source. All terms are as used in Table 3.

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Table 5Flow structure parameters after Department of the Environment(1973)

Dam type b

Flat broad-crested regular step 0.70Flat broad-crested irregular step 0.80Flat broad-crested vertical face 0.80Flat broad-crested straight slope face 0.90Flat broad-crested curved face 0.75Round broad-crested curved face 0.60Sharp-crested straight slope face 1.05Sharp-crested vertical face 0.80Sluice gates with submerged discharge 0.05

set of continuous data. If this is not the case, thenit is likely that one will have to rely on thehydraulic methods where suitable data are morelikely to be available.

4.5. Enhanced aeration at flow structures

Structures present in the river such as weirs anddams can influence aeration and increase DOconcentrations by 1–3 mg O l over very shorty1

2

distances(Bowie et al., 1985). Numerous predic-tive equations have been developed to simulatethe effects of these structures and these arereviewed by Butts and Evans(1983). A report bythe Department of the Environment in the UK(1978) identified nine classes of structures, andquantified an aeration coefficient(b) for use inthe following formula:

C yCs urs s1q0.38abh(1y0.11h)C yCs d

=(1q0.046T) (65)

where C is the saturation concentration of DO,s

C is the concentration above the structure andu

C is the concentration below the structure. Thed

parameter ‘a’ is a ‘water quality factor’ and rangesfrom 0.65 for grossly polluted streams to 1.8 forclean streams,h is the head loss over the structurein meters andT is the water temperature in degreesCelsius. Values for the parameters ‘b’ are basedon the type of structure as described in Table 5.

4.6. Plant respiration and photosynthesis rates

Various methods have been described for deter-mining the gross primary production, total com-munity respiration rates in flowing water systems.These include light and dark bottle techniques,

C-bicarbonate uptake techniques(e.g. Kevern14

and Ball, 1965) and the methods based on DOevolution in the river that were described in theprevious section(e.g. Odum, 1956; Owens et al.,1969; Kelly et al., 1974; Hornberger and Kelly,1975; Chapra and Di Toro, 1991).

4.6.1. Light and dark bottle methodThe light and dark bottle technique is similar in

principle to the measurement of BOD, in that abottle environment is used to simulate the chemicalenvironment of the water body. The techniqueinvolves placing a sample of the water into twobottles. One bottle is clear or ‘light’ allowing bothphotosynthesis and respiration to occur. The otherbottle does not allow light to penetrate and is thus‘dark.’ This second bottle should exhibit respira-tion only. The DO concentration in each bottle isdetermined and then they are sealed and placed inthe river. Some time later the bottles are openedand the new DO concentration measured. Thedifference in the concentrations can then be usedto estimate the photosynthesis and respiration ratesin the following way.

If the experiment was conducted over a periodof time t, and the initial and final concentrationsin the light bottle are denotedC and C , then ifli lf

one assumes a zero-order(i.e. constant rate) reac-tion, the net photosynthetic rate can be determinedby:

C yClf liP s (66)net t

whereP is defined as:net

P sP yR (67)net b c

and the subscript b is used to distinguish that thisis a ‘bottle’ rate.R is the community respirationc

rate, i.e. it reflects plant respiration, but also anyother oxygen consuming reactions such as bacterialrespiration.

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Fig. 4. The variation in photosynthesis over the diurnal cycleas represented by a half-sinusoid.

The community respiration in the dark bottle isdetermined in a similar fashion:

C yCdi dfR s (68)c t

where the initial and final concentrations of DOin the dark bottle areC and C , respectively.Pdi df b

can then be calculated using Eqs.(66) and (67).The community respiration can be corrected toplant respiration(or R , because this is a bottleb

rate) by accounting for the BOD, i.e.:

R sR yK L (69)b c d fi

whereK is the BOD oxidation rate described ind

Section 4.1 andL is the initial filtered BOD.fi

This technique provides average rates over theperiod of the study, but whilst the respiration mightbe assumed constant it is clear that this is unlikelyto be the case for photosynthesis. If constancy isthe case, then some manipulation ofP is requiredb

in order to obtain maximum and average photo-synthesis rate. The main drive for this variation isthe variation in light intensity over a diurnal cycle,which can be idealised as a half-sinusoid(Chapra,1997). In most water-quality models, photosynthe-sis is assumed directly proportional to the availableenergy from sunlight(although this is not strictlytrue (Kirk, 1994), i.e.:

P AI (70)t t( ) ( )

whereP is the rate of primary oxygen productionand I(t) is the available light. If this holds true,the photosynthesis rate can also be represented bya half-sinusoid:

w zx |P sP sin v tyt t -t-tŽ .t m r r s( ) y ~

(71)

P s0 otherwiset( )

whereP is the maximum rate,v is the angularm

frequency,t is the time of sunrise,t is the timer s

of sunset,f is the photoperiod(i.e. the fraction ofthe day subject to sunlight) and T is the dailyp

period as illustrated in Fig. 4 below.

By integrating Eq.(71) between the beginning(t ) and end(t ) of the test and setting the result1 2

equal toP (t yt ), the maximum photosynthesisb 2 1

rate can be found. So:

w zt2 pP sin tyt dtsP tyt (72)x |Ž . Ž .m r b r| fTy ~t p1

and integrating gives:

P sP t ytŽ .m b 2 1

fTp (73)S Ww z w zT Tp pU Xp cos t yt ycos t ytx | x |Ž . Ž .1 r 2 rT TfT fTy ~ y ~V Yp p

The average daily rate can then be found bytaking an average over the whole day, i.e.:

Tp

| P dt 2ft( )0Ps sP (74)mT pp

However, the light–dark bottle method formeasuring gross primary production of the com-munity suffers from the same problems as BODbottle measurements. Thus, the rate found is sel-dom applicable in flowing waters because muchof the community is benthic and heterogeneousrather than planktonic. Furthermore, ‘any measure-

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ment made without the normal turbulent flow maybe questioned on the grounds that production is afunction of current flow,’ Odum(1956).

4.6.2. Dissolved oxygen balanceAn alternative method for estimating rates of

photosynthesis and respiration is based on a linearDO mass-balance equation and high-quality con-tinuous DO data as described in the previoussection(e.g. Odum, 1956). Such methods are lessexpensive and therefore ideal for small systems,but they do have limitations and do require high-quality inputs in order to get good results. In theyears since Odum’s work others such as O’Connorand Di Toro (1970), Schurr and Ruchti (1975),Erdmann (1979a,b) have extended and refined thisidea and more recently Chapra and Di Toro(1991)have given a graphical method. The methods ofSchurr and Ruchti(1975) and Chapra and Di Toro(1991) are described here, but the methods ofOdum (1956) and Hornberger and Kelly(1975)described in the previous section may also be usedfor the purpose of estimating community photo-synthesis and respiration rates.

Schurr and Ruchti(1975) employed two differ-ent methods in their study. The first, a solution ofrate equations is similar to the methods alreadydescribed and the second is a cross-correlationcomputational technique on DO records. Bothmethods are explained in this investigation andapplied to several Swiss rivers to obtain estimatesof the ‘oxygen exchange constant,’ and the ratesof photosynthesis and respiration. The basic prin-ciple for both methods comes from an idea ofEinstein(1920):

‘When a chemical reaction in the steady-state issubjected to a cosinusoidal(in time) externalperturbation of any of the thermodynamic varia-bles, the periodic response of the reactants to theperturbation will lag behind in phase(or time) byan amount depending upon the finite speed of thechemical process itself, and will exhibit an ampli-tude of oscillation reflecting the fundamental sus-ceptibility or inducibility of the system to theperturbation.’

In this situation, the chemical process is reaer-ation, the steady-state condition is that maintainedin the presence of constant respiration, and the

cosinusoidal perturbation is brought about byphotosynthesis.

The simple model used by Schurr and Ruchti(1975) is:

≠C § ™syKCqKC qaI(t)yb (75)A≠t

where is the outgassing constant or exchange™K

constant, is the invasion constant,C is the™K A

(constant) concentration of oxygen in the atmos-phere,b is the constant total respiration rate,I(t)is the instantaneous light intensity at timet on thesurface of the water andaI(t) is the instantaneousrate of photosynthesis at timet. It is assumed thatthe light intensity is given by a constant plus a24-h cosinusoidal function, i.e.:

I0I(t)s (1qcosVt) (76)2

where Vs2py24, and t is in hours. Separatingboth C and I into a mean plus a time-dependentpart, produces the following pair of equations:

¯C(t)sCqDC(t)I I0 0¯I(t)sIqDI(t)s q cosVt (77)2 2

and substituting the initial simple model equationreveals:

™ aI y2 aI y20 0KC b bACs y q sC y q§ § § § §sK K K K K≠(DC) aI§ 0qKDCs cosVt (78)

dt 2

Solving this resulting pair of equations usingstandard methods such as Fourier transforms gives:

DC(t)sA cosV(tyT) (79)

where the time delay,T is the difference betweenthe time of the DO maximum and the time of

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maximum sunlight(or solar noon) given by:

B E1 Vy1C FTs tan (80)§D GV K

and the amplitude of the response,A is one halfof the range of the DO variations, given by:

aI y20As (81)

B E§C F2 2 0.5D GV qK

Thus, measurement of the three parametersT, Aand O in conjunction with the oxygen saturationconcentration will provide estimates of the rate

parameters: aI y2 and b, i.e. the rates of§ ™K, K, 0

oxygen exchange, photosynthesis and respiration.The implications of using an ‘ideal’ sunlight

model are recognised by the authors(Schurr andRuchti, 1975) and so they also provide a methodthat incorporates a random opacity factor to rep-resent the ‘random’ fluctuations in cloud coverand atmospheric dust, etc. that one might experi-ence. However, it was found to be unsuccessful inthat the random noise can(and did) prevent theidentification of secondary maxima, which wouldnegate the use of the simple model. Instead, theauthors(Schurr and Ruchti, 1975) suggest that thesecond method is used that effectively averagesout the random effects of the clouds. This cross-correlation method uses the cross-correlation func-tion between the ideal sunlight and the measured

DO concentrations to produce estimates of and§K

A (actually the amplitude of the cross-correlationfunction), and then uses the same set of equationsas before to obtain the remaining rates.

Both techniques were applied to data collectedfrom several rivers in Switzerland, where the datahad previously been manipulated so that missingdata points would not upset the cross-correlationcalculations. It was found that, where the modelwas successful, the two methods gave very similarestimated of the rates so long as the data onlyexhibited single maxima each day. However, itwas also found that both methods and the firstmethod in particular were very poor at producingreliable estimates when the DO concentrations in

the rivers were affected by other influences suchas power stations and dams.

The ‘Delta method’ of Chapra and Di Toro(1991) is similar to that of Schurr and Ruchti(1975) in that it too uses the time-lag and ampli-tude of the diurnal DO variations to estimatevalues of the reaeration rate(K ), average dailya

photosynthetic rate(P ) and daily average respi-av

ration rate (R). In this method, the rates arecalculated from a series of diurnal DO curvesusing a piecewise solution. This is based on amass-balance DO model provided by O’Connorand Di Toro (1970), but simplified for the casewhere the DO deficit,D does not vary spatiallyi.e.:

dDqK DsRyP (82)a t( )dt

where D is the oxygen deficit(mg O l ), t isy12

the time in days andP is the primary productiont( )

rate (mg O l day ). The components of they1 y12

diurnal DO cycle are illustrated in Fig. 5 whereDis equivalent to twice the amplitude in the Schurrand Ruchti(1975) technique andf is equivalentto the time-delay,T.

The rates are then calculated by carrying outthree sequential steps. Firstly, the reaeration rate(K ) is found from the time difference,f betweena

the minimum DO deficit and the solar noon. Fromthe above model(Eq. (82)), there is a uniquerelationship betweenf and K , such thatf isa

controlled solely by the reaeration rate. An analyt-ical expression of this relationship is difficult toobtain as it involves the solution of a transcenden-tal equation, but a numerical solution is relativelystraightforward and is presented by Chapra and DiToro (1991) as a series of curves relatingk to fa

for various day lengths(f). It is, however, possibleto recalculate these curves as ‘look-up’ tables usingthe latitude and longitude for the area of interest.

Once the reaeration rate has been specified, theaverage daily photosynthetic oxygen productionrateP is calculated using the value ofK and theav a

deficit range (D). As before, there is a uniquerelationship between these parameters and againthe analytical expression of this relationship isdifficult to obtain, but Chapra and Di Toro(1991)

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Fig. 5. A graphical description of the components of the diurnal DO cycle used in the Delta method to estimate the rates ofphotosynthesis, reaeration and respiration in a stream(after Chapra and Di Toro, 1991).

present a series of approximations for differentranges ofK , i.e.:a

2B ED f y1C FfTyfq0.2 K -1.0 dayaD GP Tav

2y0.5Ka1yeŽ .D y1s q0.05110.1(K (5.0 day

(83)

ayKaP 0.5K 1yeŽ .av a

D p y1s K )5.0 daya2 2P yav K q(2p)a

whereT is the period(i.e. 1 day) where, if f (thelength of the day) is given in hours,T will be 24h, but if f is given in days thenT will be 1 day.Finally, the respiration rate may be obtained fromthe calculated estimates ofP andK by:av a

¯RsP qK D (84)av a

where is the mean daily oxygen deficit(mgDO l ).y1

2

It should be noted that the calculations of theDelta method are highly dependent on an accurateestimate off—especially when the time-lag isshort (i.e. the stream has a high reaeration rate)and the relationship becomes extremely sensitive.In such situations, any errors inK will be passeda

on to the estimates of the other rates. It is sug-gested here that under those conditions an alter-native method is chosen for calculating thereaeration rate.

4.6.3. SummaryThe simple DO model used by the DO balance

methods, and their solutions, involve some signif-icant assumptions and limitations. Since photosyn-thesis is represented as a half-sinusoid, the solutionwill not account for shading by clouds, banks andvalleys or overhanging plants. Furthermore, respi-ration is assumed constant throughout the day andnight (this assumption is also in the other produc-tivity methods). An improvement is suggested herewhereby actual solar radiation data could be usedto account for overcast skies, but this may nothelp significantly if the actual radiation input doesnot follow the assumed half-sinusoid pattern.

The rates of photosynthesis and respiration arenot easily obtainable from theoretical considera-tions and that bottle measurements may not becharacteristic of the whole stream. It is thereforenot surprising that some authors have attempted toderive more simplistic empirical estimates. Inves-tigations by the Water Pollution Research Labora-tory (1968) of photosynthesis in freshwaterstreams found that, in streams not suffering fromnutrient limitation, the rate of oxygen production

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was related to the incident solar radiation and theconcentration of chlorophyll-a in the water. Aleast-squares analysis revealed that the followingequation fitted the observed data best:

0.28Ps(0.95q31.7Chl)I (85)

where P is the quantity of oxygen produced perunit area per day(g O m day ), Chl is they2 y1

2

concentration of chlorophyll-a (mg l ) and I isy1

the total light energy incident at the water surface(Cal cm day ). A study on the Rivers Thamesy2 y1

and Kennet near Reading by Kowalczewski andLack (1971) suggested that the net oxygen pro-duction rate is related to: temperature, total solarradiation, chlorophyll concentration and euphoticdepth (i.e. the depth to which sufficient light istransmitted). Regressions performed on the datafound that the net production in the River Thamescould be expressed by:

P sy0.05Iq0.02 ChlD y0.47 (86)Ž .net a

where P is the net production rate(gnet

O m day ), I is the total solar radiationy2 y12

(kCal m day ), Chl is the areal concentrationy2 y1a

of chlorophyll (mg Chl-a m ) and D is they2

euphotic depth(m).If a model requires the simulation of organic

carbon as well as oxygen, the average compositionof a plant will need to be considered since thiswill not be CH O, due to the presence of proteins,2

lipids and nucleic acids, etc. as well as carbohy-drate. Thus, to simulate the growth in say an algalpopulation in-stream, the ratio O :CO(known as2 2

the photosynthetic quotient) that should be utilisedwill not be exactly one. In most models, thenutrient levels are assumed to be sufficient enoughto not be a limiting factor and in many lowlandrivers with relatively high effluent discharge thisis reasonable. In such situations, the photosyntheticquotient depends only whether the primary nitro-gen source is ammonium(NH ) or nitrateq

4

(NO ). In either case, the summary reaction for2y3

photosynthesis(or indeed respiration) will be giv-en by the following equation for ammonium-basedsystems:

q 2y106COq16NH qHPO q108H O2 4 4 2qµ ∂° C H O N Pq107O q14H (87)106 263 110 16 2

or the following equation for nitrate based systems:

y 2y106COq16NO qHPO q122H O2 3 4 2qq18H

µ ∂° C H O N Pq138O (88)106 263 110 16 2

If the photosynthetic quotient were one, themass conversion from oxygen to carbon would be12y32 or 0.375. Adjusting for the primary nitrogensource the mass conversion rates become:

● Ammonium source, 0.3715● Nitrate source, 0.2880

and the carbon biomass production rate can becalculated from the photosynthetic rate by multi-plying by the appropriate factor. Since most meas-urements of algae in rivers use a measure of thepigment in the algae(chlorophyll-a) rather thanthe algal biomass itself, the chlorophyll-a (Chl-a)may be estimated using a fixed carbon to Chl-aratio of 1 g of carbon to 12 mg of Chl-a (Chapra,1997).

5. Modelling dissolved oxygen in lowland rivers

From the preceding discussion, it is clear thatany (mechanistic) model that attempts to simulatethe concentration of DO in a lowland river will,at the very least, have to simulate the processesdescribed in this review and summarised in Fig. 6below.

These processes will occur all along any riverthat is being modelled, and most models will usea discretisation whereby the river is split into aseries of reaches. Within a river reach conservationof mass must be maintained, but changes in theDO concentration may occur due to physical trans-port and transformation processes such as thosedescribed in this review. The description of thoseprocesses in a mathematical model requires ahydraulic model in order to simulate the transportof the solutes along the river as well as includingany biological, chemical and physical conversionprocesses that are to be simulated. The model may

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Fig. 6. A schematic of the major processes influencing the concentration of DO in rivers.

be described by well-known extended transportequations such as(Rauch et al., 1998):

B E™ ™ ™ ™ ™≠c ≠c ≠c ≠c ≠ ≠c™ ™™ C Fsyu yv yw q ´xD G≠t ≠x ≠y ≠z ≠x ≠x

B E B E™ ™≠ ≠c ≠ ≠c ™C F C Fq ´ q ´ qDc (89)y zD G D G≠y ≠y ≠z ≠z

where is an multi-dimensional mass concentra-™ction vector for each of the determinants;t is thetime; x, y and z are spatial coordinates;u, v andw are the corresponding velocity components,´ ,x

´ and ´ are turbulent diffusion coefficients fory z

the directionsx, y and z, respectively; and is a™Dcterm representing the rates of change of determi-nants due to internal transformations in the reach(e.g. nitrification or reaeration). This partial dif-ferential equation(or PDE) can be solved numer-ically, but is usually simplified to some extentbefore solving. For example, if the concentrationsin the reach do not vary greatly over the depth orthe cross-sectional area, the number of dimensionsmay be reduced and this leads to the so-calledadvection–dispersion approach. Alternatively, the

system might be assumed to consist of a numberof interconnected, perfectly mixed tanks or ele-ments, which leads to a set of ordinary differentialequations(or ODEs).

A model of this type(simulating the processesin Fig. 6) will invariably require a large amountof data, especially if it is to simulate time-varying(i.e. dynamic) changes in the system. The first setof data required describes the physical character-istics of the system such as the layout of thenetwork, the division of the river system intoreaches, and the widths, depths and lengths ofthose reaches. The reach descriptions will alsoinclude the type and dimensions of any flowstructures such as weirs whose effects are intendedto be simulated. At the top of the system, a modelneeds the upstream boundary conditions of flow,DO, BOD, ammonium, nitrate and chlorophyll-a(or some other indicator of the plant density) andin addition to this, the pH and temperature will berequired. The temperature determines the rate atwhich the various reactions will take place, and, ifthis is to be a dynamic model, these conditionswill need to be time series for each determinantfor the period of interest. In more complex hydrau-lic models, boundary conditions must also besupplied at the bottom of the system. At each

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reach a similar set of data will be required for anyinfluences such as tributaries or effluent dischargesthat enter the river at that point, together with theflow rates of any abstractions.

5.1. Data availability

Often models are designed with little regard tothe data requirements for that model. This isunfortunate, because such models will be imprac-tical if it turns out that certain parameters are notgenerally measured or worse are impossible tomeasure. Thus, it is common for field studies toaccompany modelling exercises so that therequired data might be obtained, but if a modelcan be designed that is satisfactory using onlycommonly measured determinants then this willclearly be preferable. Much of the required datamay be available having been collected by organ-isations such as the Environment Agency orresearch institutes, or as a part of an ongoing studysuch as LOIS.

Within each reach, these determinants may betransformed according to the processes described,and the rate parameter for each reaction in eachreach will be required. It has been shown inSection 4.3 that these rates are all obtainable eitherfrom theoretical considerations or by experiment,if the necessary data are available. They may alsobe obtained by calibration of each rate in turn ina specific order, so that the act of variation of oneparameter has as small an affect as possible on thedeterminants that have already shown a good ‘fit’to observed data. However, this ‘blind calibration’method should ideally require a large number ofobserved data points in order to ensure a ‘goodfit’. Furthermore, this method should only be usedas a last resort because one cannot be certain thatthere will be a unique set of parameters thatfacilitates a good fit of the simulated and observeddata. It is therefore suggested that as many para-meters as possible should first be estimated usingtheoretical or experimental techniques beforeresorting to ‘blind’ calibration, although it is per-haps reasonable that minor adjustments may bemade to estimates of rate coefficients once theyhave been determined if this improves modelperformance

6. Conclusions

This review has described the processes thatinfluence DO, in lowland river systems and thedata requirements for using models that simulatethese processes have been outlined. It is has beenshown how mathematical models can be used todescribe these processes so that one can simulateboth the condition and the variability in thesesystems. Although mechanistic models have beendescribed, it is clear that there is some degree ofuncertainty in these equations—relating to thereaction rate parameters in particular. The mostcommonly used techniques for estimating the ratesof these processes have been described, but it isclear that there will be inherent uncertainty inapplying any model based on the process formu-lations described here. Therefore, the effects ofuncertainty must be analysed in any water qualitymodel so that the results obtained may be inter-preted correctly before any management decisionsare made based on those results.

Acknowledgments

This review results from a NERCyEnvironmentAgency funded Ph.D.(GT2y97y3yEA), and theauthor is grateful to NERC and the EnvironmentAgency for the financial support granted, also toPaul Whitehead(The University of Reading) andColin Neal (CEH, Wallingford) for supervisingthe project.

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