Notes on Water Quality

7/28/2019 Notes on Water Quality

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Water Quality Modelling March, 2005

IHE 1

1. Introduction

1.1 Introduction

A model is a simplified representation of a real system, in terms of one or more mathematicalequation. The most important word is simplified: a model can never represent the real system in all itsdetails.

Two main types of models exist. Empirical or statistical or black box models are based onexperiments and statistical testing, not on theory behind the behaviour of the system under study.

Deterministic models are based on (fixed) mathematical descriptions of natural processes governingthe system behaviour. Field data are only used to validate the model performance or to calibrate

certain model parameters.

In general there are two types of questions you can answer with the help of a model:

1. why? questions: these focus on gaining insight in a water system, on cause-effect relationships,and on finding knowledge gaps;

2. what if?questions: these focus on the expected changes in the system as a result of measures orchanges in the external conditions (e.g. climate changes).

The use of a model is justified if such questions need a quantitative answer.

In practice, the two types of questions often come together in one project. First the why question is

answered to better understand the system, and afterwards relevant what if questions are addressed.

1.2 Some basic terms

The conceptual model comprises the selection of the equations and the unknowns to be used in themodel. The equations can be time dependent (unsteady, dynamic) or steady state. The number of

spatial dimensions included can be 0, 1, 2 or 3.

A modelling tool is a piece of software that helps you to create models of certain water systems. Theyusually contain a solver for a well-known set of mathematical equations, a user interface to facilitate

the use of the tool, presentation facilities, database facilities, etc. The technology behind such tools is

called hydro-informatics. Sometimes, a modelling tool is called a model.

The input data which make a model represent a certain geographical area (geometry input data) are

also sometimes called a model. Other input data that are needed to make a model are:

The boundary conditions: the values of the unknowns at the boundaries of the spatial domaincovered by the model.

The initial conditions: the values of the unknowns at t=0 (for time dependent models only). External variables or forcing functions: quantities which are considered outside the scope of the

modelling exercise.

Model parameters: certain tuning dials to make a model really represent a certain area. These arecommon to many water quality models which often have a semi-empirical nature.

The present course does not have the scope of a hydro-informatics course. It concentrates on

understanding and using the key equations in water quality modelling. It also provides an introduction

into the use of modelling tools.


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March, 2005 Part 2: practical applications Environmental modelling

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1.3 A modelling project

The notice of Quality Assurance plays an important role in todays professional practice. Many

institutions are in a process of formalising their production process and even try to obtain ISO-9xxx

certificates. In the Netherlands, an initiative was started to develop a Quality Assurance system for

modelling studies. This is called the Good Modelling Practice in water management (GMP)

initiative.The GMP focuses primarily on the definition of models and modelling studies, by improving the

degree of structure in the modelling process. An important aspect is the reproducibility of the process.

The GMP handbook distinguishes 7 basic steps in a modelling project.

1. Starting a logbook

2. Defining the modelling project

3. Building the model

4. Analysing the model

5. Using the model

6. Interpreting the results

7. Reporting and archiving

Step 1 (starting a logbook) is the starting point for obtaining a reproducible modelling study. Step 2(defining the modelling project) is very important, because it involves the definition of the problem to

be solved, as well as the definition of the objectives and the requirements for the modelling exercises.

The underlying principle is that it is not possible to make a good model without knowing exactly

which questions need to be answered and in what context. Relevant aspects can be: the type of

problem, the time scales and spatial scales under consideration, the desired accuracy, and the

constraints for acceptability of the model.

Step 3 (building the model) consists of the selection of the conceptual model and of the actualconstruction of the model, either by programming the associated computer code yourself, or by using

an existing (maybe commercially available) computer programme. The equations can be solved

analytically (simple geometry, simple input data, exact solution) or numerically (all cases,

approximation of the solution). The end of step 3 is the verification of the model: does the model solve

the right equations in a correct way?

With the model available, step 4 (analysing the model) includes activities like mass balance tests,sensitivity analysis, calibration, validation and uncertainty analysis. Not all of these activities are

carried out in every study, but each one of them can be considered to improve a specific aspect of the

models credibility. The word calibration refers to the adjustment of certain model parameters to make

the model behave in agreement with field data. The word validation refers to the checking of the

agreement between the model and the field data.

Paying serious attention to step 4, helps you to carry out step 5 (using the model), which is where theactual purpose of the modelling activity starts. Step 6 (interpreting the results) provides the most vitalinput for the modelling study report, and ends with the following question: did the modelling study

meet its objectives and requirements (defined during step 2)? The final step 7 (reporting andarchiving) includes the formalisation of the end product of the study (the report) and forms the finalstep in the process of assuring the reproducibility of the work.


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2. Water quality models for reservoirs and lakes

2.1 Water balance

The simplest way to model a reservoir or lake is by neglecting all space dependent aspects of the

reservoir. The general mass balance equation can be written as follows:

Accumulation = Inflow Outflow

Some factors affecting the inflow are the inflow of rivers, groundwater seepage and rainfall. Some

factors affecting the outflow are spillway release, outflow of rivers and groundwater seepage. The

accumulation can be expressed as the change of the water volume V of the reservoir (m3). We can

express the inflow and outflow as volume rates of flow Qin and Qout respectively (volume per time,

m3/s):

V, c

Qin, cin

Qout, c

Over a given period of time t, the water balance equation reads:

in out

dVt Q t Q t

dt

=

Dividing by t:

in out

dVQ Q

dt

=

2.2 Pollutant balance

The water quality of the lake or reservoir can be studied the simplest by assuming that the water body

is well-mixed: the pollutants are uniformly distributed through the entire water body, it has a

homogeneous concentration c (mass per volume, g/m3). The total mass of a pollutant in the reservoir

equals cV (mass, g), while the pollutant inflow or outflow is given by Qc (mass per time, g/s).

The mass balance equation for a pollutant which does not undergo any decay, removal or

transformation processes, for a given period of time t can be written as follows:

in in out

dcV

t c Q t cQ tdt

=

Dividing by t and expanding the differential term gives:


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in in out

dV dcc V c Q cQ

dt dt

+ =

For the term dV/dt we can substitute Qin Qout, according to the water balance equation derived

earlier:

( )

( )

( )

in out in in out

in in out in out

in in

dcc Q Q V c Q cQ

dt

dcV c Q cQ c Q Q

dt

dcV Q c c

dt

+ =

=

=

Dividing by V provides a first order differential equation for c:

( )in inQdc

c cdt V

=

2.3 Pollutant with decay

The pollutant balance equation can be expanded with a decay term. Many decay processes are

expressed in a linear form:

dc

kcdt

=

With k the decay rate (1 per time, s-1

).

When we include a decay term, the pollutant balance equation is written as follows:

in in out

dcVt c Q t cQ t ckV t

dt

=

The differential equation for c reads:

( )in inQdc

c c kcdt V

=

2.4 Solving the equations

There are two basic ways to solve the mathematical equations derived when we make a conceptual

model.


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2.4.1 Analytical solutions

Sometimes the equations are simple enough to allow a direct mathematical solution in closed form.

The pollutant balance equation derived above can be easily solved analytically for example if we

assume that the water volume and the inflow rate are constant and the concentration in the inflowingstream is zero. The equation then reads:

inQdc c kc Ecdt V

= =

Where E is a constant number. The solution to this equation is:

( )0 exptc c Et==

For more complex cases analytical equations may be obtained from the literature, but the use of

numerical methods provides an alternative.

2.4.2 The Euler method

The Euler method provides an algorithm to solve equations of the kind that we have derived above for

the water and pollutant balance of a reservoir. The pollutant balance equation reads:

( )in inQdc

c cdt V

=

The core of the Euler method is the approximation of the time derivative:

1i ic cdcdt t

+

If we substitute this in the pollutant balance equation we obtain:

( )1i i in ini

c c Qc c

t V+

Now we solve for the concentration at the end of the intervalt:

( )1 ini i ini

Qc c t c cV

+

= +

If we provide the initial concentration c(t=0), the formula above can be used to evaluate the

concentration as a function of time, by proceeding with time steps t.


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3. BOD and DO

3.1 Sources and sinks of DO

The principal DO problems stem from municipal and industrial BOD discharges, the oxidizablenitrogen forms, and nutrients which may stimulate phytoplankton growth. The nature of the aquatic

ecosystem then determines the DO levels through such processes as reaeration, photosynthesis and

sediment oxygen demand.

The DO problem begins with the input of oxygen demanding wastes into a water body. In the water

body itself, the sources of DO are:

1. Reaeration from the atmosphere

2. Photosynthetic oxygen productions

3. DO in incoming tributaries or effluents.

Internal sinks of DO are:

1. Oxidation of carbonaceous waste material

2. Oxidation of nitrogenous waste material

3. Oxygen demand of sediments of the water body

4. Use of oxygen for respiration by aquatic plants and algae.

With the above inputs and sources and sinks, the following general mass balance equation for the DO

concentration (designated by c) in a water volume V, can be written as follows:

reaeration + (photosynthesis - respiration) (2.1)

- oxidation of CBOD, NBOD (from inputs)

- sediment oxygen demand + oxygen inputs

+ oxygen transport (into and out of segment)

The various source-sink components will now be examined in order to further develop the mass

balance equation.

3.1.1 Biochemical oxygen demand (BOD)

BOD (mgO2/l) is defined as the oxygen consumption for the breakdown of organic material in the

water. One can differentiate between the carboneceous BOD (CBOD) and nitrogeneous BOD

(NBOD).

BOD (CBOD) can be related with the theoretical oxygen demand for converting the organic waste to

CO2 and H2O; e.g. for glucose:

C6H12O6 + 6 O2 6 CO2 + 6 H2O(M.W.= 180)

Theoretically, 6x32/180 = 1.07 mg O2/mg glucose are necessary. In practice, the CBOD is only a

fraction (40-80%) of this theoretical value, due to e.g. the partial non-biodegradability of the material.Also, in the laboratory procedure the BOD degradation may be measured only for the first 5 days; in

this way a BOD520

, depicting the BOD after 5 days at 20 C, may be defined.

Vdc

dt=


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Nitrogeneous materials (e.g. proteins, urea, amino acids) are first hydrolyzed to ammonia, NH4+. Via

the nitrification reaction this is then oxidized to NO3-:

NH4+

+ 2 O2 NO3-+ 2 H

++ H2O

Theoretically, 4.57 g O2 are used per gram NH4+-N. In practice, some of the ammonia will be used for

cell production, and the above oxygen utilization will have a value of 4.2-4.5.

Kinetics of BOD degradation

Although CBOD and NBOD may have different degradation kinetics, they are, as a first

approximation, often combined. Assuming also first order decay, then:

dL/dt = - k1 L , in which:

L = remaining BOD at time t (mg/L)

k1 = BOD decay rate constant (day-1

)

Integrating; at t = 0, L = L0:

L = L0 exp (-k1 t) (2.2)

K1 = 0.1 - 0.4 day-1

(dependent on e.g. river characteristics, wastewater purification, time after

discharge). The temperature effect on k1 is often expressed as:

k1(T) = k1(20) 1.04(T-20)

3.1.2 Atmospheric reaeration

Transfer of O2 into the water is brought about by the "driving force" cs - c, in which c and cs are the O2

concentration (mg/L) at time t, and the O2 saturation concentration in the water, respectively:


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dc/dt = k2 (cs c)

k2 (day-1

) is the reaeration coefficient; integrating:

c = cs - (cs c0 ) exp (-k2 t) (2.3)

The reaeration coefficient k2 is a function of:

Turbulence caused by currents and waves Waterfalls, dams and other hydraulic structures

Surface films, e.g. by detergents Temperature, often expressed as: k2(T) = k2 (20) 1.02(T-20)

In many rivers over the world, the k2 behaviour has been studied in detail. The following semi-

empirical formula is commonly used to describe k2 as a function of water depth H and water velocity

U:

(2.4)

in which b, m and n are constants.

It can tentatively be shown that the power of H will be >1: reduced water depth gives rise to higher

water turbulence; also, a proportionally smaller water volume has to be supplied with the oxygen. The

power of U will be between 0.5 and 1. In other words, reaeration is much more important for shallow,

fast flowing rivers.

Over the years the results of three main researches have been used (Thomann & Mller, 1987, cf. Fig

2.2):

O'Connor (1958) for relatively deep, moderate velocity streams:b = 3.863 , m = 0.5 , n = 1.5 (for U in m/s and H in m)

Churchill (1962), for deep, fast flowing rivers:b = 5.026 , m = 0.969 , n = 1.673 (for U in m/s and H in m)

kbU

H

m

n2=


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Owens (1964), for shallow streams:b = 5.322 , m = 0.67 , n = 1.85 (for U in m/s and H in m).

Fig. 2.2. Reaeration coefficient k2 (day-1) at 200C as a function of water depth (feet) and velocity (feet/s) (1 foot = 0.3 m).

Data were taken from three main researchers, using Eq. 2.4.

3.1.3 Photosynthesis and respiration

The presence of aquatic plants, including algae, causes a diurnal variation of the O2 levels in the water.

Under light conditions, i.e. during the daytime, primary production takes place, leading to algal cell

production ():

CO2 + H2O + O2

Respiration, with use of oxygen in the living material, takes place during day and night:

+ O2 CO2 + H2O

As a result, the O2 concentration will show a day/night cycle , with DO maxima during midday and

minima just before sunrise. In rivers (in contrast to lakes), the diurnal DO variation is generally not

very large, i.e. in the range of 1-2 mg/L.


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3.1.4 Sediment Oxygen Demand (SOD)

Especially sediments with high organic matter contents may exert a substantial oxygen demand. Some

typical SOD values (gO2/m2/day) are given in Table 2.1. (Thomann & Mller, 1987). It will be

apparent that SOD will especially be important for shallow waters; in deeper rivers, the SOD term can

often be neglected in comparison with the processes in the large water volume above the sediment.

The SOD is strongly dependent on temperature: ST = ST(20) 1.07T-20

, for T between 10 and 30 C; forT< 5 C, the SOD approaches zero.

Table 2.1: Some Sediment Oxygen Demand values (gO2/m2/day) at 20 C)

Bottom type and location Range Average

Sphaerotilus (10 g dry wt/m2) 7Municipal sewage sludge outfall vicinity 2-10 4

Municipal sewage sludge aged, downstream outfall 1-2 1.5

Estuarine mud 1-2 1.5

Sandy bottom 0.2-1.0 0.5

Mineral soils 0.05-0.1 0.07

3.2 Reservoir model for DO

Combining the reservoir mass balance equation derived before and the sink and source terms for BOD

and DO we obtain the following equations for BOD (L) and DO (c) in a reservoir:

( ) 1in

in

QdLL L k L

dt V

=

( ) ( )1 2in in sQdc c c k L k c cdt V = +

These equations can be solved either analytically, or by the Euler method.

3.3 Steady state river model for BOD and DO

The reservoir equations for BOD and DO, can be used to set up a steady state river model as well. The

first step n this procedure. is the evaluation of the reservoir equations for steady state conditions. For

BOD:

( ) 1

1

1

0in in

in inin

inin

in

QdL L L k Ldt V

Q Qk L L

V V

QL

VLQ

kV

= =

+ =

=

+

and for DO:


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( ) ( )1 2

2 1 2

1 2

2

0in in s

in inin s

inin s

in

Qdcc c k L k c c

dt V

Q Qk c c k L k c

V V

Q c kL k cVc

Qk

V

= + =

+ = +

+=

+

The next step is to represent the river as a sequence of reservoirs with length x:

BOD

x=0x

Qup (m3/s)

Lup,cup (g/m3)

Qeff(m3/s)

Leff,ceff(g/m3) x=x x=2x x=3x x=4x x=5x

c1 c2 c3 c4 c5 c6Lin,cin

We assume that the discharge Q in the river is constant and homogeneous. Therefore, the inflow is

equal to the outflow for all river segments: Q in = Qout = Q.In a river, the discharge can be expressed

as the cross section A (m2) multiplied with the velocity U (m/s, average over the cross-section):

Q A U=

The volume of each of the river segments is equal to the cross-section A multiplied with the segment

length x:

V A x=

Therefore, we can write U/x rather than Q/V in the mass balance equations for the river segments.

We apply the mass balance equations to the individual segments, in a downstream direction. For BOD:

1 2

1 2 3

1 1 1

, , , .inU U UL L L

x x xL L L etcU U U

k k kx x x

= = =+ + +

and for DO:

1 1 2 1 1 2 2 2 1 3 2

1 2 3

2 2 2

, , , .in s s s

U U Uc kL k c c k L k c c kL k c

x x xc c c etcU U U

k k kx x x

+ + + = = =

+ + +


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Note: these equations can be solved analytically as well, by the famous Streeter Phelps equations:

/t x U=

( )0 1expL L kt=

[ ]1 0

1 2 0 2

2 1exp( ) exp( ) ( )exp( )s s

kL

c c kt k t c c k tk k=


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4. Modelling other pollutants

This section provides an overview of relevant processes and simple modelling strategies for different

pollutants. For more information, we refer to text books on the subject of water quality modelling,

such as Thomann & Mller, 1987 (R.V. Thomann & J.A. Mller, 1987. Principles of water qualitymodelling and control. Harper & Row, New York, 644 pp.) and Chapra, 1997 (Surface Water-Quality

Modeling. Chapra, Steven C., McGraw Hill, 1997, ISBN: 0-07-011364-5).

4.1 Coliform bacteria

Coliform bacteria are a pollutant used as an indicator for the presence of waste from human origin

with possible pathogenic effects on human health.

They can be modelled by a linear decay process:

ColidColi k Colidt =

where the kColi increases with:

Intensity of UV radiation Increase of Water temperature Increase of Salinity

The minimum value is in the range of 0.1 d-1

. UV radiation is a very important factor. In clear water,

for instance sea water, and at high radiation intensity, mortality rates up to and over 50 d-1

have been

observed.

It is important to realise that the UV-radiation at the water surface can not penetrate fully into the

water column. In practice, the intensity of the radiation varies with the depth, as follows:

( )ezIzI = exp/)( 0

where:

I intensity of radiation (W/m2)

I0 intensity of radiation at water surface (W/m2)

z distance from water surface (m)

e extinction coefficient for UV light (m-1)

The value of e depends on the turbidity of the water. Typical values range from 0.2 (very clear) to 5

(very turbid).

A mathematical expression for the coliform decay rate can be obtained from Mancini, J. L., 1978.

Numerical estimates of coliform mortality rates under various conditions. Jour. Water Poll. Control

Fed. : 2477-2484.


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4.2 Suspended sediments

Small particles (typically < 50-100 m) can remain suspended in the water, and change the visual

appearance of the water (reduce transparency). The particles can be inorganic silt or clay particles or

organic particles stemming from aquatic life forms such as algae.

The concentration of suspended particles (SS) can be modelled by a linear decay-like process:

1s

cr

dSS vSS

dt H

=

where:

SS concentration of suspended particles (g/m3)

vs settling velocity (m/d)

H water depth (m) shear stress (N/m

2), exercised by currents, function of velocity

cr critical shear stress (N/m2)

This formula represents a decay caused by settling of particles to the bottom. Note that the

equivalent decay rate vs/H is inverse proportional to the water depth. Values of the settling velocity

are in the range of 0.1 to 10 m/d.

The shear stress can be approximated by:

2

1VC

where V is the velocity in m/s, and C1 is in the range 1-10 (e.g. 4).

The factor (1-/cr) expresses that strong currents cause a lot of turbulent mixing which keep the

particles in suspension, while weak or absent currents allow particles to settle.

The critical shear stress depends on the grain size of the sediments and the degree of consolidation. It

is in the range 0.2-5 N/m2.


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4.3 Nutrients and eutrophication

Nutrients, nitrogen and phosphorus, show a complex behaviour characterised by:

formation of algae and other aquatic life, on the basis of inorganic nutrients (ammonium, nitrates,phosphates); mortality of aquatic life; mineralisation of organic matter to inorganic forms.

See the figure below.

Life cycle of algae

Inorganic matter

(C, N, P, Si)

algae

sedimentation

mineralisation O2

mortality

pr.production

Organic matter(C, N, P, Si)

solar radiation

respiration

Formulating a model including these processes reaches too far in this course.

By approximation, nitrogen and phosphorus can be considered as decaying pollutants, where the

decay consists of:

denitrification at the water sediment interface (nitrogen);

settling of particles with a significant phosphorus content.

Both processes can be modelled like a settling process:

N PdN v dP vN Pdt H dt H

= =

The parameters vN and vP are strongly dependent on the water system in question, and need to be

derived from field data.


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4.4 Heavy metals

Heavy metals, like cadmium, zinc, lead, mercury, copper, etc., are present in most water systems. Due

to industrial activities, their concentrations can be much higher than usual, with a possible toxic effect

on humans and aquatic life.

Heavy metals are partly dissolved and partly present in particles. The distribution is often expressed by

a partition coefficient:

solid

diss

C

CP =

Cdiss concentration of metals, dissolved (g/m3)

Csolid concentration of metals, solid phase (g/g of suspended particles)

P partition coefficient (m3/g)

The heavy metal fraction in the particles can settle to the bottom, together with the particles. The

mathematical expression for the heavy metal (HM) reads:

1s

cr

dHM dSS vP P SS

dt dt H

= =

4.5 Organic pol lutants (pesticides, chemicals)

Mankind has created many chemicals and pesticides which may cause problems in the water quality.

These pollutants are usually decaying very slowly. Depending on the properties of the pollutant,

different processes can be important:

Biodegradation or photolysis, which can both be modelled as a simple linear decay process (SeeSection 2).

Settling of a fraction of the pollutant which is present in particles. This can be modelled in a waysimilar to heavy metals.

Volatilisation (exchange between the air and the atmosphere), which can be modelled similar toreaeration of dissolved oxygen.

As a first approximation, organic pollutants can be considered conservative (non-decaying).

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Notes on Water Quality

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