topic intro model phil

8/8/2019 Topic Intro Model Phil

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for t > t 0 ,

C ( x, t ) = C 02

exp(U w) x

2 D

erfc

x wt 2 Dt

erfc

x w t t 0( )2 D t t 0( )

+exp (U +w) x2 D erfc x +wt 2 Dt erfc

x +w t t 0( )2 D t t 0( )

for 0 < t t 0 ,

C ( x, t ) = C 02 exp (U w) x2 D

erfc x wt 2 Dt

+exp (U +w) x2 D

erfc x +wt 2 Dt

where w =U 1+ 4 kDU 2

d 2

2

x Skc x

c D+ xcv-=

t c


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1 Introduction

Lesson objectives:To understand conceptually the steps in modeling:

1. Define the problem2. Conceptualization of system--determine what

is known, define system3. Selection or development of mathematical

description (model)--important to understandassumptions, data requirements and model

limitations4. Model calibration--parameter adjustment5. Evaluation of model results--do they make

sense??


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Modeling objectives

Increase understanding--which phenomena areimportant, may provide insight into fundamentalbehavior

Predict effects of mitigation, remediation efforts

Estimate exposure => health & environmental risk

Model Identification


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Modeling Protocol


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Developing conceptual models

1. Establish the PURPOSE of the model.Determine what questions are to be answered.Defines what equations need to be selected and solved.Determines what type of data is necessary.

2. Define the problem : contaminant(s), locationDefine the evaluative environment Source: steady? Plant discharge, leaking tank, old spill? Relevant phases: what media (air, water, soil) are

affected? Properties of media important Model boundaries: scale? Soil grains -> globe

Modeling Protocol


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Example : someone has dumped their houseboat waste into Lake

Powell. Health officials are concerned about immediate and futureeffects on human health. Define the evaluative environment.

soilair

water

sediment

What characteristics of environment could be important?

Air water soil/sedimentP, T, wind velocity density grain density(depends on altitude) mixing, stratification organic matterMixing, stratification T (P) permeabilityAerosol-deposition rates surface tension porosity

flowpHionic strengthredox potentialparticulate matter


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Develop a CONCEPTUAL MODEL of the system.

Must gather and integrate field data.i.e. Need to determine the type of aquifer, its units, andboundaries.

Often includes the conceptual water budget.It is important to visit the field site during this step!


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Select proper GOVERNING EQUATIONS andCOMPUTER CODE

Need equations and code to model your specific site.Should verify equations and code against known analyticalsolutionsor to modeling problems in similar settings.

DESIGN the model

Convert conceptual model to the form necessary for theequations and code.

Includes descritization in time and space.Must set boundary and initial conditions.Must select initial parameters to be used in model.


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CALIBRATE the model

Compare model to experimental data (calibration data set) Calibrate model based on calibration data set--model

coefficients, rate constants initially chosen from lab studiesor literature--tune within a range of observed or reportedvalues, using some mathematically defined criteria (e.g.minimizing sum of squared errors)

Performance criteria--how much error is acceptable? Forboth calibration and verification-can be tight or loose; define

before calibration or verification


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Verify the ModelOnce you have a model you must measure (collect data) to

verify the model(known as model validation).All Models must be verified and validated.All Models must be verified and validated.

Model VerificationModel VerificationModel VerificationModel Verification

Model verification refers to the process of removing computationalor conceptual errors from a model. This process is an attempt tomake the model internally consistent, and is usually performed bythe developers of the model before it is accessible to the general

public. Even after a model is in use, model users may occasional finderrors or inconsistencies. If this information is communicated to thedevelopers, they can improve later versions of the model.


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VALIDATE the model

See if the model can simulate a second set of field data correctly

(validation data set.)Must specify the purpose of model and conditions and range in

which it is valid (all have limitations)

Use the model to PREDICT

Use the calibrated model to determine how it responds to futureevents.Must estimate future conditions.


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Sensitivity and uncertainty analysis

Determine how the model responds to uncertainty inparameter values. May need to do for different stress periods. Can be a lot of uncertainty in values for future conditions.

Determine SENSITIVITY to future parameter uncertainty sensitivity: which input parameters make the mostdifference in the model output uncertainty: which input parameters should we knowmore precisely to make the model more precise?

Both are important

outputinchangeparameterinchange

:asdefinedoftenysensitivit


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Sensitivity analysis

Definitions vary. In most studies, sensitivity analysis is thestudy of model properties through - not necessarily realisticallysized - changes in the input variables and the analysis of itseffect on model outputs. The questions addressed are for

instance:

whether or not some output is affected at all by someinput

continuity, differentiability, monotonic increase ordecrease of the model's response to input variation

Most of the variation of outputs is generally caused by asmall number of inputs.


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UncertaintyIn this context: imperfect knowledge regarding aspects of a model.Uncertainty regarding model variables is usually specified by aprobability distribution or by a sample of measured values (an

empirical probability distribution); sometimes it is specified by a setof possible values. We adhere to the probabilistic concept of uncertainty, and we use variances as measure of uncertainty.


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Sources of uncertainty

Uncertainty exists at the level of inputs and output of the model.Uncertainty of model formulation also exists. We will assumethat the model is deterministic, and that uncertainties aresolely introduced via the inputs.

Input uncertainty is caused by natural variation (e.g. weather, soilor water variation) as well as by imperfection of data. Although thecauses of uncertainties may differ, their effect is the same, namelyuncertainty about model outputs. Up to the modeler whether or notto incorporate natural variation in the model; the choice dependsalso on the spatial or temporal scale at which the model is used.

The input uncertainty of different parameters may containcorrelations caused by biological or physical mechanisms, e.g.correlation between photosynthesis rate during the day or night, or

between weather at two consecutive days.


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PRESENT RESULTS of model and model design

Probably one of the most important steps.You may have a great model, but you need to show to others.Difficult to do for laypersons.

Remember that modeling is an iterative process.After the above steps are completed, it is important to re-examine the model,Find where it is successful and where it is less than satisfactory,

then improve it.


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This is science and philosophy, like thescientific method.

It has changed the way science is done and isthe basis of a modern technologically basedapproach.


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Some Thoughts on Modeling the Environment &Models used in Environmental Evaluation

Essential skills for successful model development.

1. Identifying the problem variables accurately.

2. Constructing appropriate relations between these variables.

3. Taking measurements judging the size of quantities that aresignificant and the limitations for these relationships. Set limits.


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4. Collecting data, decide how to use them and test yourmodel before your finalize it. Modify the model with dataand with additional theoretical relationships that you findthrough this process.

5. Know and document the limits of the model by estimating

the parameters within the model that cannot be measured orcalculated from data.

6. Limit the use of the model to its functional and appropriatecapabilities excluding the inappropriate use of the model.


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It has been suggested that canisters of nuclear waste can safely be driven into theocean floor and stored without contaminating large regions of ocean.Model Proposed by Researcher

Assumptions made by the model

1. The canisters will penetrate the ocean floor to a minimum depth of 50 Meters.2. No disturbance of the ocean water will be caused by the presence of thecanisters.3. Canisters will last for over 1,000 years.4. After 1,000 years, leaching will occur.5. A diffusion model was constructed to distribute the radioactive leachedisotopes. Taking into consideration exclusively brownian movement anddiffusion aspects of the dispersion in the water.

Example:Ocean Disposal of Nuclear Waste


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Q uestions:

How was the model constructed?What are the important variables and parameters?What are the relationships?

What is the mathematical relationship? (an example of atypical diffusion algorithm)


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The process of solute movement are commonly described by the advection-

dispersion equation.

c/ t = D m(2c/ x2) - v c/ x

Where: c/ t is the change in concentration with the change in time, andc is solute concentration

t is time

D is the dispersion coefficient ( D = Do+ ev )where: e is the coefficient of solute dispersivity in soil, and Do is themolecular diffusion coefficient

v is pore water velocity (given by water flux (q) divided by the volumetricwater constant (theta)

Ref. Modeling Chemical Transport in Soils: Natural and AppliedContaminants, Hossein Chadiri and Calvin Rose, Lewis Publishers, pg. 147,1992.


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Then, the following steps were taken and questions asked:

Model Verification, Evaluation, Validation, and Limitation:

Were the assumptions valid?Is this an appropriate model?If it was not a good model, why and how would you improve it?


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Model Type

Deterministic Stochastic

Steady State non-Steady State Steady State non-Steady State

Conservative non-Conservative Conservative non-Conservative


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Some model types

Deterministic : a partial differential is solved, numerically oranalytically, for a given set of input values, atmosphere,stream, lake or aquifer parameters, and boundary conditions.

The resulting output variable has a specific value at a givenplace in the system. There is a fixed relationship between inputand output.

Steady state : system does not vary with time. Inputs are constantand system eventually reaches some equilibrium condition.a. conservative parameter: material doesn't react or decay.b. nonconservative: waste undergoes reactions, transformations,decay.

Dynamic : system changes with time due to changing conditionsor inputs.a. conservativeb. nonconservative


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Stochastic : a model for which there is a statistical

uncertainty in the value of the output variables due touncertainties in the system parameters, input parameters,or errors in measurement. Models allow for randomfluctuations. Uses mean and standard deviations of parameters.

Steady state , conservative and nonconservative

Dynamic , conservative and nonconservative


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Model Type

Deterministic Stochastic

Steady State non-Steady State Steady State non-Steady State

Conservative non-Conservative Conservative non-Conservative


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Contaminant Transport Mass Balance Advective transport Diffusion and

dispersion Volatilization Adsorption

Biodegradationprocesses Chemical Reaction


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1.2.4 Define fate and transport phenomena

physical basic mass balance, reactor theoryadvection : transport of contaminants due to bulk flowof water or air diffusion : spread of contaminants due to randommolecular motion dispersion : spread of contaminants due to mechanicalmixing

Can have transport between phases (aqueous, vapor,solid), transport within a phase, complete mixing withina phase (Topics 4 & 5)

chemical and biological rxns (Topic 2)


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Subdivisions of Transport Process

Point Sourcesintentional

well defined sourceboundary

Nonpoint sourcesFugitive emissions

unintentional emissionsIll-defined source boundary

Loading Advective

physical movementof contaminants

Dispersive

spreading to down gradientmixing due to turbulence etc.

Diffusivemovement from higherto lower concentration

Transport

Reactions

HomogeneousSingle phase

HeterogeneousMultiple phases

Mass Balance


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Sources of Contaminate Loading

Industrial spills,discharges and leaks

Surface impoundments Storage tanks and

pipes Landfills Burial areas and

dumps

Injection wells


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Areas of Contamination Surface soils

Subsurface soils Shallow ground water Deep ground water Vapors above water

table Drinking water wells Receiving

streams/lakes

Air


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Mechanisms of Contamination

ConfiningUnit

Water table

SalineWater

Lateralintrusion ofsaline water

Ocean

Municipalwater well

Abandonedoil well

DeepAquifer

pond

Infiltration of

pesticides andfertilizers fromfarmlands

Brine leakage fromruptured well casing

septic tankleakageFreshwater

Accidentalfuel spill

Municipallandfill

Leakage from

hazardouswaste siteContaminated

shallowwell

Leakingpetroleum

tank

ConfiningUnit


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Leachate togroundwater

Landfill

Volatilization

To upper atmosphere

Wind dispersed contaminants

Precipitation

Inflow

Sewer

PhotochemicalsSunlight

Volatilization

PhotolysisHydrolysisAcid-base equilibria

Sediment

Adsorption/ desorption

Sedimentation

Bioaccumulation

Biodegradation

Leachate togroundwater

Sun


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1.2.5 Define the state of the system

Steady or unsteady stateSteady state : no change of mass &/or concentration at

a point with timedmdt

=0, dC dt

=0

m or C

t (at a point )

Opposite is unsteady state,

or Dynamic


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Equilibrium or nonequilibrium (partitioning between 2 phases)

Chemical equilibrium : net rate of mass transfer betweenphases is zero (or rate of forward and backward reaction is 0)

Ratio of concentrations in phases = a specific constant calledthe equilibrium partition coefficient = K ij

AirCA

WaterCw

C AC w

=K AW at equilibriumC

t

CA

Cw

Concs. change,ratio is constant


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Example :Given the system is at equilibriumand steady state;

each phase is well mixed.

C AC w

=20

AirV=100 m 3

CA=8 mol O 2 /m 3

WaterV=100 m 3

Cw=0.4 mol O 2 /m 3

a. AirCA=8 mol

O2 /m 3

WaterCw=0.4 mol

O2 /m 3

CA,in = 8 molO2 /m 3

Cw,in = 0.4 molO2 /m 3

CA,out =8 molO2 /m 3

Cw,out =0.4 molO2 /m 3

All concs. constantEquilibrium? Steady State?

b. Air

Water

CA,in = 8 mol

O2 /m 3Cw,in = 0.1 mol

O2 /m 3

(both constant)

CA,out =6 molO2 /m 3

Cw,out =0.2 molO2 /m 3

(constant)

Equilibrium? no--ratio 20Steady State?


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c.

Both C ins vary

Both C outsvary butratio isalways 20

Example (cont.)

d.

Equilibrium? (notethat influent streams do NOT

need to be at equilibrium forthe system to be at equilibrium)Steady State? no

Equilibrium? noSteady State? no

Both C ins vary

Both C outsvary andratio 20


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1.3 Developing mathematical models

Mass balance approach (engineering fundamental)--mass cannotbe created or destroyed (except nuclear rxns), but it can betransferred or transformed

Define control volume (evaluative environment) Define inputs and outputs across and within boundaries =>mass balance equation (mathematical model)

Outputs

Boundary

Inputs Accumulation

Decay

Accumulation or rate of change of storage = input - output +reaction (+ generation - consumption)


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Example (will cover in more depth in Topic 4): closed, steady

state system--no mass flow into or out of system

AirVA, C A

WaterVw, C w

SedimentVs, C s

Total mass in this evaluative environment = sumof mass in each phase

m total = C AVA + C wVw + C sVs

Given C w, V is-- relate C A and C s to C w by

equilibrium partition coefficients--Henrys law K aw = C A /C w--sediment/water partition coeff sw = C s /C w

m total = K awCwVA + C wVw + swCwVs= (K awVA + V w + swV s) C w


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Well look at 2 good models with sound physical basis

completely mixed compartment (box) modelNot spatially dependent--concentration is uniform in

each phase

advection/dispersion modelsChanges in concentration with both time and space

topic intro model phil

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