Surface Water Quality Modeling Lecture PDF

Surface Water Quality Modeling  Introduction Water quality models try to simulate changes in the pollutants concentration as they move through the environment. A pollutant entering the environment may increase or decrease its concentration due to a large variety of mechanisms....

Surface Water Quality Modeling  Introduction Water quality models try to simulate changes in the pollutants concentration as they move through the environment. A pollutant entering the environment may increase or decrease its concentration due to a large variety of mechanisms. The fate of pollutants is the resultant of interactions between mass transfer and kinetic processes. All these changes are the subject of water quality modelling. Fate and Transport of pollutant in lake  Purposes of water quality modeling The purpose of modelling falls into one of the following categories 1. To understand more fully the transport regime of the pollutant. 2. To quantify the dominant controlling processes. 3. To establish time ranges within which contaminants could have reached specified level in certain areas.  Describing Environmental Models Two broad groups of models are commonly used to address environmental fate and transport of chemicals: compartment models and plume models.  Compartment Models Compartment models, also referred to as Box models , conceptualise the environment as consisting of homogeneous, well-mixed compartments. The principle of mass conservation and mass balance equations are the basis for compartment models. Once a chemical is emitted into a given compartment, it can be transferred to other compartments.  Compartment Models Conceptual representation of a model with three compartments. Chemicals enter a given compartment (straight arrow) and can be transferred into other compartments (curled arrows). As soon as a chemical enters a given compartment, the concentration of the chemical in that compartment is assumed to immediately be homogeneously distributed throughout the whole compartment  Plume models Plume models are used if it is assumed that compartments are not homogeneous and not well mixed. Plume models aim to model the distribution of a chemical within a given compartment.  is a simple representation of the emission of a contaminant and its spreading due to diffusion and dispersion effects.  Plume models Two-dimensional representation of a pollutant plume within a rectangular compartment  Modeling Processes Phase 1 Data Collection (historic, field monitoring) Model Input Preparation Phase 2 Calibration  Validation Verification Phase 3 Analysis of Alternatives  Modeling Processes Phase 1 Two primary information sources feed this phase. The first are management objectives, control options, and constraints, which might include physical constraints as well as legal, regulatory, and economic information. The second source is data related to the physics, chemistry, and biology of the water body and its drainage basin.  Modeling Processes When this phase is complete, the modeler should have a clear idea of the problem objectives and the accompanying water- quality variables required to assess whether the objectives are attainable. The modeler should understand the temporal, spatial, and kinetic resolution that will be compatible with the problem needs.  Modeling Processes Model calibration involves minimization of the deviation between measured field conditions and model output by adjusting parameters of the model. Data required for this step are a set of known input values along with corresponding field observation results.  Modeling Processes Model validation involves the use of extreme values, for required testing  Modeling Processes Model verification is used for the examination of the numerical technique and computer code to ascertain that it truly represents the conceptual model and that there are no inherent numerical problems with obtaining a solution.  Definitions  Model A model is a simplified representation of the real world  Water quality model A mathematical representation of pollutant fate, transport, and degradation within a water body  Mass and concentration In the water quality the amount of pollution in a system is represented by its mass. C = m/v Where: m= mass and V= volume L3  Definitions Rates values normalized to time Mass loading rate (W) To describe waste discharge W= m/t Volumetric flow rate Q=U Ac Where U= velocity of water and Ac = cross sectional area  Definitions Mass flux rate. The term flux is used to designate the rate of movement of an extensive quantity like mass normalized to area. For example the mass flux rate through the conduit can be calculated as J = m/t Ac = W/Ac J= Uc Example: Loading and Flux Partitioning - partitioning between two phases, e.g. air and water, - adsorption/desorption on particles, - uptake into lipid phases. Transport - mixing and dilution, - advection/convection, - diffusion, - dispersion. Transformation - photochemical degradation, - hydrolysis, - microbial biotic degradation  Transport and transformation processes Diffusion – (molecular) random movement and mixing of molecules – Depend on property of the molecule and the surrounding medium Advection – directed flow of a medium, e.g. water or air flow – e.g. a substance is transported downstream by the flowing of a river Dispersion – flow dynamic process, occurs only in moving media – orders of magnitudes faster than diffusion – e.g. mixing (eddy diffusion)  Diffusion and Dispersion Diffusion: process where a constituent moves from a higher concentration to a lower concentration Dispersion: mixing caused time by physical processes  Volatilisation – diffusive mass transfer between air and water Sedimentation – (effective) sedimentation velocity of particles – diffusive mass transfer into sediment pore water – sediment burial Degradation – hydroloysis – aquatic photolysis – microbial degradation  Bioconcentration Mass Distribution in the System Fish - Water 100 98,4 90 Fish 80 70 60 ____ --- k1 = 0.001 1/h k1 = 0.0001 1/h KFW = 60 50 --- k2 = 0.06 1/h ___ k2 = 0.006 1/h 40 30 20 10 Water 0 1,6 0 50 100 150 200 250 300 350 400 time (h) Kinetic Reactions  Simple decay calcultation In words: “decrease/sec is proportional to amount present” k = decay rate (1/d) dM   kM M = mass (g) dt t = time (d) Used for many processes in water quality modelling, e.g.: Decay of BOD Radioactive decay Mortality of algae etc. 27  Simple decay calculation (2) dM Decay formulation:   kM dt Separating variables: 1  dM  kdt M M t 1 Take the definite integral on both sides:  dM  k  dt M0 M 0 M   M   kt Will lead to: ln     kt  e  M0   M0  Will finally lead to: M (t )  M 0e kt 28  Decay Rate 1 M (t )  M 0e kt Slow rate (low k value) 0.8 High rate (high k value) 0.6 0.4 0.2 0 0 Exercise: Check that it wil take 6.9 days for the Mass to reach 50% of its original value M0 , for k = 0.1 day-1 29 Fate processes of any pollutant in water discharge volatilisation degradation advection deposition of particles (sedimentation)  Types of Models  Prediction water models –These models predict what is happening in the receiving waters, e.g., rivers, lakes, estuaries. (Mathematical Model)  Watershed loading models – These models predict what is happening on “land” that results in an export of pollutant to the river, lake or estuary.  Models vary in complexity… Simple – Long-term average representation of the system. (Typically, an equation) Won’t vary in time or space. Moderately complex – Average representation of the system (monthly, annually). May vary in time or space. Complex – Daily (or less than daily) representation of the system. Varies in time and space Mathematical Models Model Implementation Example: Assimilation Factor  How to build models Input data: – geometry input data (“model”) – boundary conditions: values of unknowns at the boundaries of the spatial domain covered by the model – initial conditions: values of the unknowns at the start of the simulation (t=0) – external variables or “forcing functions”: values of quantities affecting the model results, which are not predicted by the model. Their values are taken from outside (e.g. meteorology data) – model parameters: acceleration of gravity, bottom friction, decay rate, diffusion, dispersion, …… 37  Environmental media can be defined in such a manner as to represent phases or mixtures of phases Rules and laws of chemical equilibrium and kinetics can be applied to environmental systems. Feedback of toxic effects on organisms on the chemicals’ fate is neglected. No mixtures, only single compounds are considered. Interaction between components of mixtures are therefore also not regarded.  MASS BALANCE FOR A WELL-MIXED LAKE A completely mixed system, or continuously stirred tank reactor (CSTR), is among the simplest systems that can be used to model a natural water body. It is appropriate for a receiving water in which the contents are sufficiently well mixed as to be uniformly distributed. MASS BALANCE FOR A WELL-MIXED LAKE A hypothetical completely mixed system The arrows represent the major sources and s i n ks of the pollutant. The dashed arrow for the reaction sink is meant to distinguish it from the other sources and sinks, which are transport mechanisms.  MASS BALANCE FOR A WELL-MIXED LAKE The mass balance for the system can be expressed as:  MASS BALANCE FOR A WELL-MIXED LAKE Thus there is a single source that contributes matter (loading) and three sinks that deplete matter (outflow, reaction, and settling) from the system. The above equation represent a descriptive value, it cannot be used to predict water quality. For this we must express each term as a function of measurable variables and parameters.  MASS BALANCE FOR A WELL-MIXED LAKE Accumulation represents the change of mass (M) in the system over time t: Mass is related to concentration by the equation:  MASS BALANCE FOR A WELL-MIXED LAKE where V = volume of system (L3). Equation 3.3 can be solved for which can be substituted into Eq. 3.2 to give  we assume that the lake's volume is constant.  MASS BALANCE FOR A WELL-MIXED LAKE  Finally ∆t can be made very small and Eq. 3.6 reduces to  MASS BALANCE FOR A WELL-MIXED LAKE Thus mass accumulates as concentration increases with time (+ dc/dt) and diminishes as concentration decreases with time (- dc/dt). For the steady state case, mass remains constant (dc/dt = 0).  MASS BALANCE FOR A WELL-MIXED LAKE Loading Mass enters a lake from a variety of sources and in a number of different ways. where W(t)= rate of mass loading (M T- 1) and (t) signifies that loading is a function of time.  MASS BALANCE FOR A WELL-MIXED LAKE where Q = volumetric flow rate of all water sources entering the system (L3 T- 1)  Cin(t) = average inflow concentration of these sources (M L -3).  MASS BALANCE FOR A WELL-MIXED LAKE Average inflow concentration can be related to loading by equating Equations. 3.8 and 3.9 and solving for  MASS BALANCE FOR A WELL-MIXED LAKE Outflow: The rate of mass transport can be quantified as the product of the volumetric flow rate Q and the outflow concentration Cout (M L -3). Because of our well-mixed assumption, the outflow concentration by definition equals the in-lake concentration Cout = C, and the outflow sink can be represented by:  MASS BALANCE FOR A WELL-MIXED LAKE Reaction: first-order representation  Where  k = a first-order reaction coefficient (T-1 ).  Thus a linear proportionality is assumed between the rate at which the pollutant is purged and the mass of pollutant that is present.  MASS BALANCE FOR A WELL-MIXED LAKE Equation 3.12 can be expressed in terms of concentration by substituting Eq. 3.4 into Eq. 3. 12 to yield  MASS BALANCE FOR A WELL-MIXED LAKE  Settling: losses can be formulated as a flux of mass across the surface area of the sediment- water interface.  MASS BALANCE FOR A WELL-MIXED LAKE  Thus by multiplying the flux times area, a term for settling in the mass balance can be developed as:  Where:  v = apparent settling velocity (L T-1)  As = surface area of the sediments(L 2).  MASS BALANCE FOR A WELL-MIXED LAKE Because volume is equal to the product of mean depth H and lake surface area, Eq. 3. 14 can also be formulated in a fashion similar to the first- order reaction, as:  Where:  ks = a first-order settling rate constant = v/H. Notice that the ratio v/H has the same units (T -1) as the reaction rate k.  MASS BALANCE FOR A WELL-MIXED LAKE  Total balance The terms can now be combined into the following mass balance for a well-mixed lake:  MASS BALANCE FOR A WELL-MIXED LAKE Concentration (C) and time (t) are the dependent and the independent variables, respectively, because the model is designed to predict concentration as a function of time. The loading term W(t) is referred to as the model's forcing function because it represents the way in which the external world influences or "forces" the system.  MASS BALANCE FOR A WELL-MIXED LAKE The quantities V, Q, k, v, and A., are referred to as parameters or coefficients. Specification of these parameters will allow us to apply our model to particular lakes and pollutants.  STEADY-STATE SOLUTIONS If the system is subject to a constant loading (W) for a sufficient time, it will attain a dynamic equilibrium condition called a steady- state.  In mathematical terms this means that accumulation is zero (that is, dc/dt = 0). For this case Eq. 3. 16 can be solved for  STEADY-STATE SOLUTIONS or using the format where the assimilation factor is defined as  STEADY-STATE SOLUTIONS Steady-state solution has successfully yielded a formula that defines the assimilation factor in terms of measurable variables that reflect the system's physics, chemistry, and biology. STEADY-STATE SOLUTIONS STEADY-STATE SOLUTIONS STEADY-STATE SOLUTIONS STEADY-STATE SOLUTIONS  STEADY-STATE SOLUTIONS These results along with the loading can be displayed as  TEMPORAL ASPECTS OF POLLUTANT REDUCTION Suppose that a system is at steady-state. At a specific time a waste removal project is implemented. two interrelated questions arise: How long will it take for improved water quality to occur?  What will the "shape" of the recovery look like?  TEM PORAL ASPECTS OF POLLUTANT REDUCTION (a) A waste load reduction along with (b) four possible recovery scenarios for concentration  TEM PORAL ASPECTS OF POLLUTANT REDUCTION To determine the correct trajectory, let's start with the mass-balance model(Eq. 3. l 6) Before solving this equation, we can divide it by volume to yield  TEM PORAL ASPECTS OF POLLUTANT REDUCTION Collecting terms gives Where in which is called an eigenvalue (that is, a characteristic value).  TEM PORAL ASPECTS OF POLLUTANT REDUCTION If all the parameters (Q, V, k, v, H) are constant, Eq. 3.33 is a nonhomogeneous, linear, first-order, ordinary differential equation.  where  cg = general solution for the case W(t) = 0  cp = particular solution for specific forms of.  The General Solution If C = C0 at t = 0, Eq. 3.33 with W(t)= 0 can be solved by the separation of variables Thus we have arrived at an equation that describes how the lake's concentration changes as a function of time following the termination of waste loading.