EVEN 6318 Environmental Systems Modeling

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EVEN 6318 Environmental Systems Modeling

• Venki Uddameri
• Dept. of Environmental Civil Engineering, MSC
  213
• Texas AM University-Kingsville
• Ph 361-593-2742 Fax[phone]
• E-Mail vuddameri_at_tamuk.edu

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Administrative

• Course Web-site
• www.even.tamuk.edu/vuddameri/even6318
• Not up yet probably by this weekend

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Recap

• Systems philosophy
• Contrast with reductionism
• Characteristics of a system
• Boundary
• Transport in and out
• Reactions inside the system
• Reactor configurations
• BR, CFSTR, PFR, PFD
• Is it mixed or not (thats the question?)

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Goals

• Fundamental Mass-balance equation
• Flux and Loadings
• Box versus control volume formulations
• Heterogeneity
• Advection Process
• What is it
• Role of energy balances in characterizing
  advection
• Momentum balances
• How to quantify it in various systems of interest
• Introduce diffusion/dispersion processes
• Time permitting

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Mass Balance

• Mass can neither be created or destroyed
• Except under extreme conditions (usually not of
  concern to environmental engineers)
• Mass can be transferred from one place to another
• Location A to location B
• Mass can be transferred from one phase to another
• Water to ice

Mass Balance is the cornerstone of Environmental
Models
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Mass Balances Transport Equation

• We are often interested in dynamic behavior of
  material movement
• Rate of Accumulation (within a system)
• Inflows into the system Outflows into the
  system Source/sinks
• Inflows and outflows occur at system boundaries
• Source/sinks occur within the system

Sink
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Mass Balance Equation

• A separate mass balance equation must be written
  for each compartment and each phase within the
  compartment
• If a system has 2 compartments and
• compartment 1 has 1 phase and,
• compartment 2 has 3 phases
• for a pollutant that can exist in all phases we
  can need
• 4 mass balance expressions

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State Variables and Expressions

• Mass Flux Mass crossing a boundary of known
  cross-sectional area in an unit period of time.
• Mass/(areatime) has dimensions of M/T
• Volumetric Flux Volume of fluid crossing a
  boundary of known cross-sectional area in an unit
  period of time
• Volume/(areatime) has dimensions of L/T
• Mass Loading Mass moving into (or out of) a
  system per unit time
• Mass/time
• Mass loading Mass flux (cross-sectional area)
• Mass Flux Mass loading integrated over a
  cross-sectional area
• Volumetric Loading Volume moving in (or out of)
  a system per unit time
• Volume/time
• Has units of discharge
• Volumetric loading volumetric flux
  (cross-sectional area)

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Flux and Loading
10 kg/hr
Area 5 Ac.
Area 3 Ac
5 kg/hr
Mass flux in 2 kg/(ac-hr)
Mass flux out 5/3 kg/(ac-hr)
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State Variable

• Master Variable or the output from the model
• Concentration is a typical state variable for
  tracking pollutants
• Flowrate is a typical state variable for tracking
  flow of fluids
• Other state variables are also possible
• Fugacity
• Escaping tendency of a gas
• Proportional to concentration for dilute
  solutions
• Concentration Normalized mass of the substance
• Mass is often normalized to volume of fluid in
  which it is present
• Concentration Mass of benzene / Volume of water
• Mass can be normalized to mass of substance in
  which it is present
• Concentration Mass of benzene / Mass of Soil

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Mass Balances Role of Heterogeneity

• To carry out mass balances we need to draw system
  boundaries
• The system represents the domain of interest
• We also need to identify if the domain is
  homogeneous or heterogeneous
• Homogeneous Well mixed can be treated as a
  single entity
• Concentration is the same through out the system
• Heterogeneous Incompletely mixed
• Concentration varies within the system

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Control Volume

• Control volume is the region within the system
  that exhibits uniform properties at a given scale
  of measurement
• Things are changing within the control volume at
  scales smaller than the measurement scale
• However, on average things look the same within
  the control volume

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• Control volume represents the volume within which
  the properties can be considered uniform

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System and Control Volumes

• For homogeneous systems the system volume and the
  control volume can be the same

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System and Control Volumes

• For heterogeneous systems the control volumes are
  different from the system volume
• Control volume is a subset of the system volume
• The system is envisioned to comprise of
  interconnected elements (control volumes)
• For extremely heterogeneous cases, the control
  volume is assumed to be infinitely small
• The total accumulation in the system is obtained
  by carrying out mass balance on the control
  volume and integrating it over

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Mass Balances

• All the terms in the mass balances are loadings
• Have dimensions of mass/time
• Volume/time used for constant density fluid flows
• Mass loadings into the system need to be
  specified
• Direct specification of mass loadings is
  sometimes possible
• Emission Inventories provided by state and
  federal agencies

Most often Mass Loadings into the system are due
to different Processes
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Mass Loadings

• Two important physical processes affect mass
  loadings of pollutants and fluids into
  environmental systems
• Advection
• Dispersion and Diffusion

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Advection

• Pollutants usually do not enter or exit the
  system on their own
• Pollutants are dissolved in other fluids
• Chloride in water
• Pollutants that exist as a separate phase, but
  are transported by the movement of another phase
• Suspended solids are transported due to water
  movement
• Aerosols in air
• Pollutants are sorbed onto solids which move in
  and out of the system
• Phosphorus sorbed onto particulate matter
  suspended in water

Movement of Pollutants due to the movement of
other fluids is advection
Moving fluid is the solvent and the pollutant is
the solute
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Advection

• Advection represents dove-tailing effects of
  pollutant transport
• A tennis ball moves from Point A to Point B
  because it floats on the water
• Which of the following represent an advection
  process
• A blob of oil floating on the water a river
• Oil leaking from an underground storage tank site
  moving the soil
• Decane moving in the soil due to an oil leak from
  an UST
• A suspended particle settling on the bottom of a
  lake
• Suspended particle being eroded from the soil
  during a rainstorm

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Advective Fluxes and Loadings

• The advective loadings and fluxes can be
  calculated as follows
• Loadingadvection Flowrate x Concentration of
  the pollutant
• Flowrate is in volume of solvent/time
• Concentration is in mass of pollutant/volume of
  solvent
• Fluxadvection Flowrate x Concentration of
  pollutant / Area
• Flowrate / Area Velocity (continuity principle)
• Flux velocity x concentration of the pollutant

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Advective Loadings Multiphase systems

• Velocities of fluids can be defined in different
  ways in Multiphase systems

Area of C/s total AT LB Area occupied by
fluid A1 Fraction of Fluid Area n A1 / AT
Flux based Velocity VF Vol/(LBt) Actual
Fluid Velocity VA Vol/(A1t) Actual Fluid
Velocity VA VF/(n)
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MBE Advection Box Formulation

• The system is completely mixed
• Inflow and outflow of the solvent (fluid) across
  the boundaries
• Outflow concentration is equal to the
  concentration in the system
• Inflow concentration needs to be specified
• Can be a function of time

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MBE Box Formulation

• Two substances
• Fluid (solvent) flowing in and out
• Solute pollutant moving with the fluid
• The system comprises of one phase
• Only one phase is considered
• Need two mass balance expressions
• One for flowing fluid
• One for Solute
• Fundamental transport equation
• Accumulation In Out /- Reactions

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MBE - Fluid

• Mass balance for the fluid
• For the fluid, using volumetric fluxes, MBE can
  be mathematically expressed as
• Assuming no sources or sinks within the
  system

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MBE for the Pollutant

• Assume no reactions in the system

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Advective Transport Box Formulation

• Additional expressions need to solve the
  simultaneous differential equations
• Initial state of the system w.r.t the fluid
• Initial volume occupied by the fluid
• Initial state of the system w.r.t. the pollutant
• Initial concentration of the pollutant in the
  system
• The solution to Advective transport using a Box
  Formulation is a Initial Value Problem

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Advective Transport Steady- State flow

• A common simplification is to assume that there
  is no net accumulation of the fluid in the system
• The pollutant can still accumulate
• The Mass balance for the fluid can now be written
  as
• The Pollutant Mass Balances Reduces to

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Advection Control Volume Approach

• Control volume approach is useful when the system
  is heterogeneous
• The flowrates change inside the system
• The pollutant concentration is not uniform within
  the system

Accumulation in the Control Volume In Out /-
Sources
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Advection Control Volume

• Consider two materials
• Fluid flowing through the system
• Pollutant moving because of the fluid
• A control volume is selected
• Smaller than the system size
• Represents a volume in which the properties are
  approximately constant
• Let us consider a infinitely small control volume
• Extreme case of continuous variability

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Advection Control Volume Approach

• Mass Balance for the Fluid
• Assume no sources/sinks

0
Dt
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Advection Control Volume

• Mass Balance for the Pollutant
• Assuming no reactions

0
Dt

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Advection Control Volume Approach

• Need to Know the initial state of the system
• Fluid volume at initial time
• Pollutant concentration at initial time
• Also, need to know the what happens at the
  boundary of the system
• Fluid flowrates at inlet and outlet boundaries at
  all times
• Pollutant concentrations at inlet and outlet
  boundaries at all times
• The control volume formulation is a
  Initial-Boundary Value Problem
• IBVP formulation
• The mathematics of CV approach is more rigorous
  than Box formulation
• Simplifications are often made made

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Simplified Example

• Advection in system where fluid flow is constant
  (steady-state) along one direction
• The system is well-mixed in other directions

Control Volume A.Dl Fluid flowrate Q
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Simplified Example

• Accumulation In Out /- Reactions
• Mass balance for the fluid
• No accumulation of the fluid in the control
  volume or the system
• Q constant
• Mass balance for the pollutant

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Simplified Example
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Advective Fluxes

• Need two Parameters to measure advective fluxes
  and loadings
• Velocity or flowrate
• Concentration
• Concentration is the state variable of interest
  in many environmental problems
• We do the modeling to estimate this parameter
• We will need velocity
• How to get velocity when we dont want a separate
  mass balance?

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Obtaining Velocities for pollutant MBE

• Some methods to obtain velocities include
• Direct measurements in the system of interest
• Best approach but costly and cumbersome
• Not suited for forecasts with significant
  disturbances
• Use energy or momentum balances
• Energy balance is probably easier when energy is
  not wasted
• Not much losses in form of heat and sound
• Momentum balances are better when energy is
  wasted
• Use empirical correlations
• Site-specific data
• Interpolate with caution

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Energy Balances

• Energy/Momentum balance is necessary for each
  flowing phase
• Bulk fluids like air and water, suspended solids
  that enter and exit a system
• Need to calculate advective fluxes
• Energy Balance for Fluids is based on Bernoullis
  principle
• Internal energy of the system is equal to sum of
  potential, kinetic and intrinsic energies
• Intrinsic energies are due to pressure, density
  or temperature
• Incompressible fluids
• Pressure, Kinetic and Potential energies
• Compressible fluids
• Requires knowledge of thermodynamics as well
• Kinetic, Potential, pressure and density
  (temperature)
• Need an Equation of state (EOS)

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Energy and Momentum Balances

• Bernoullis Equation is used for steady-state
  energy balances (especially incompressible
  fluids)
• Energy does not accumulate within the system

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Next Class

• Two Important Physical Processes
• Advection
• Dispersion

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Commonly used approaches in different Systems

• Atmospheric systems
• Direct measurements
• Forecasts based on momentum balance
• MM5 model Meteorological forecasts
• Runoff to Lakes
• Flow gages
• Energy balances (account for frictional losses)
• Laminar flow conditions
• Water budgets
• Empirical equations for runoff
• SCS curve number

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Flowrates in Environmental Systems

• Rivers and Streams
• Direct gaging
• Energy balances
• Laminar flow conditions
• Moderate frictional losses
• Momentum balances
• Turbulent flow conditions
• Surrogate measures
• Mannings equation

R A/P
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Flowrates in Environmental Systems

• Soils and Aquifers
• Application of Darcys Law
• Flux proportional to the energy gradient
• Soils and the Vadose Zone
• Energy gradient depends upon
• Capillary pressures
• Surface pressures
• Gravity
• Hydraulic conductivity and capillary pressure is
  a function of saturation
• Aquifers
• Energy gradient depends upon
• Gravity
• Compressibility of the fluid and the media