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Urban Water Global water aspects
Department of Hydro Sciences, Institute for Urban Water Management Urban Water Global water aspects Introduction to urban water management Basics for systems description Water transport Matter transport Introduction to water supply Water extraction Water purification Water distribution Introduction to wastewater disposal Urban drainage Wastewater treatment Sludge treatment Peter Krebs Dresden, 2010
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach 4.4 Advection-dispersion approach
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach 4.4 Advection-dispersion approach
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Characteristics of compounds
Passive solubles Travel ~ with water Often used to indicate velocity and residence-time distribution Solids Transport decoupled from flow Suspended solids and gravel Sedimentation and Erosion Reactive matter Can be solubles or solids Residence time and conditions in reactor important Reaction must be known for balancing
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Milk and sugar in a cup of coffee
Quiescent conditions Molecular diffusion Stirring Turbulent diffusion
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Tracer in a full pipe L, t Transport with flow
Longitudinal extension of tracer cloud Decrease of peak concentration
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Residence-time distribution in a clarifier
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach 4.4 Advection-dispersion approach
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Advection Transport with water flow no relative movement Flux
Example: Transport of compound with constant concentration C in a tube with cross section A:
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Molecular diffusion Transport in the direction of decreasing concentration 1st Fick law 1D approach; it also applies in a 2D or 3D system Dmd,M is a specific value for a certain compound M Dmd,M is a function of temperature
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Turbulent diffusion Process similar to molecular diffusion, but some orders of magnitude more efficient C x Diffusive flux Dtd is dependant on flow and state of turbulence, not on the compound itself Concentration gradients decrease !!
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Dispersion Dispersion is not transport relative to water, but inhomogeneous advection In 1D formulation, dispersion “collapses on diffusion”
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Sedimentation Suspended particles have a transport component in gravity direction In reactors this effect is used for particle separation In transport systems, a sink or source term - depending on the operation conditions - is needed Sedimentation flux v vS Examples: - 1D clarifier model - Sewer sediments
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach CSTR Plug-flow reactor CSTR in series 4.4 Advection-dispersion approach
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Continuously stirred tank reactor (CSTR)
Constant volume Immediate mixing Complete mixing no concentration gradients CReactor = COutlet Q Cin C V r Mass balance
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CSTR: steady state Mass balance 0-order reaction 1st-order reaction
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CSTR: residence-time distribution (RTD)
Tracer pulse is introduced to the inlet tracer concentration is measured in the outlet Mass balance No input, no reaction
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach CSTR Plug-flow reactor CSTR in series 4.4 Advection-dispersion approach
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Plug-flow reactor Constant volume Constant cross section
No mixing (ev. lateral) Concentration gradients along flow axes Mass balance x dx A
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Plug-flow reactor: steady state
Outlet concentrations: with x = L L/v = Mass balance 0-order 1st order
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Plug-flow reactor: RTD
Tracer pulse is introduced to the inlet tracer pulse appears in the outlet unchanged !!
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach CSTR Plug-flow reactor CSTR in series 4.4 Advection-dispersion approach
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CSTR cascade Q C2 V2 r Q Ci Vi r Ci-1 Cn Vn Cn-1 Q Q r Cin C1 V1 C1
Reactor i 1st order reaction
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CSTR cascade: 1st order reaction (i)
n = number of reactors Total volume 2 Reactors n Reactors
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CSTR cascade: 1st order reaction (ii)
n Reactors
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CSTR cascade: RTD (i) Initial condition in 1st reactor c0,1 as reference concentration 1st reactor 2nd reactor i-th reactor
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CSTR cascade: RTD (ii) Solving the coupled equations with Laplace transformation yields Mean value Variance Peak value at time
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CSTR cascade: RTD (iii)
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4.1 Introduction to transport phenomena 4.2 Transport processes
Peter Krebs Department of Hydro Sciences, Institute for Urban Water Management Urban Water 4 Matter transport 4.1 Introduction to transport phenomena 4.2 Transport processes 4.3 Reactor approach 4.4 Advection-dispersion approach
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Advection-dispersion approach (i)
Analytical solution for a tracer pulse u = mean velocity Ddisp = dispersion coefficient m = total amount of tracer introduced A = cross-section area t = time from dosage
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Advection-dispersion approach (ii)
Standard deviation Dispersion coefficient Shear velocity cf = Fischer coefficient = (-) b = width of water surface h = water depth Sf = friction slope
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A-D approach: effect of dispersion/diffusion
Advection Dispersion, estimated by diffusion approach Standard deviation
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A-D approach: tracer curves
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1 2 3 4 5 A-D approach: dispersion in a river tracer model
50 100 150 200 250 300 350 400 450 Time (minutes) tracer model 1 2 3 4 5 Boeije (1999)
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A-D approach: reactor approximation (i)
Normalisation by length L of reactor Peclet number Pe large Advection dominant plug flow behaviour Pe small Diffusion dominant CSTR behaviour small < Pe < large CSTR cascade or A-D approach
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A-D approach: reactor approximation (ii)
Relation of turbulence/dispersion and standard deviation Simplification for Pe > 100 (applies to conditions in sewers and rivers) CSTR approximation, „hydrologic model“ Turbulence can be estimated from RTD (i.e. )
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