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Transport You are on a train to NYC. You are stirring the milk into your coffee. The train and everything in it are moving toward NYC via directed or advective transport As you stir, the milk is moving via turbulent diffusion. This is a random process. Flux Gradient flux law: Flux: mass transfer or Concentration per unit exchange gradient area per velocity (ng/m3) unit time (m/day) (ng/m2-day) aka mass transfer coefficient Fick’s First Law One example of a gradient flux law is Fick’s First Law: Relates the diffusive flux (Fx) of a chemical to its concentration gradient (dC/dx) and its molecular diffusion coefficient (D) Fick’s Second Law The local concentration change with time (dC/dt) due to a diffusive transport process is proportional to the second spatial derivative of the concentration (concentration gradient) Turbulent diffusion In contrast to molecular diffusion, which arises due to thermal molecular motions, turbulent diffusion is based on the irregular patterns of currents in water and air. Turbulent vs. laminar flow is defined by the Reynold’s number: d = spatial dimension of the flow system or objects around which the flow occurs (m) v = typical flow velocity f = dynamic viscosity of the fluid (kg/m-s) rf = density of the fluid (kg/m3) For laminar flow Re < 0.1 Turbulent diffusion the effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick’s first law: the molecular diffusivity (D) is now replaced by the turbulent or eddy diffusion coefficient, E E >>> D anisotropy in natural systems, turbulent diffusion is usually anisotropic, meaning that the magnitude of E depends on the direction. horizontal diffusion is usually much greater than vertical diffusion because: 1. natural systems extend horizontally 2. often the system (ocean, atmosphere) is density stratified Transport through boundaries (Chapter 19) What is a boundary = surface at which properties of a system change extensively or, discontinuously (interface) air-water interface sediment-water interface epilimnion - hypolimnion (thermocline) stratosphere – troposphere (tropopause) What to boundaries do? 1. control the transport of energy and matter 2. control chemical process triggered by the contact of two systems with different chemical composition What is the boundary condition? may be defined by a value (i.e. concentration) or by a flux (i.e. mass flux across the boundary per unit time) What types of boundaries are there? 1. bottleneck 2. wall 3. diffusive classified according to the shape of the diffusivity (D) profile bottleneck boundaries bottleneck bottleneck = mass Diffusivity D(x) crossing must squeeze itself through a zone in turbulent which transport occurs diffusion by molecular diffusion (usually interface) molecular diffusivity distance example: air-water interface Like a toll booth on the turnpike wall boundary wall at a wall boundary, a zone characterized by turbulent diffusion encounters a Diffusivity D(x) zone in which transport is turbulent dominated by a much diffusion slower process, such as molecular diffusion molecular example: diffusion distance sediment-water interface Like an icy stretch of road diffusive boundary diffusive boundary at a diffusive boundary, diffusivity is of similar Diffusivity D(x) C (x) magnitude on either side D (x) diffusivity may be molecular or turbulent example: distance troposphere – stratosphere boundary (tropopause) Air – Water Exchange (Chapter 20) Inputs and outputs of SPCBs (kg y-1) Atm dep Volatilization 18-48 317-846 Advection to Atlantic 130-190 CSOs 67-146 Stormwater 36-140 NY/NJ Harbor STP effluents 32 Estuary Advection from Hudson River 260-470 Dredging 150-290 Storage in sediments 147-307 Totten 2005 Air – Water Exchange the air-water interface can be thought of as a bottleneck boundary (if one phase is stagnant we can think of it as a wall boundary) We already know, from our discussions of mass transfer, that the equation for the air-water exchange flux (Fa/w) should look like this: where va/w is a mass transfer coefficient or air-water exchange velocity (m/s) the second term describes the fugacity gradient and the direction of air-water exchange: Net air-water exchange flux sometimes we divide this into the absorption flux (“gross gas absorption”): and the volatilization flux: total exchange velocity the total exchange velocity can be interpreted as resulting from a two-component (air/water) interface with phase change. if water is the reference state, then: (two resistances in series) va typically is about 1 cm/s vw typically is about 10-3 cm/s Critical Kaw thus if Kaw << 10-3 (dimensionless) or 0.025 L bar/mol then the air-side resistance (va) dominates if Kaw >> 10-3 (dimensionless) or 0.025 L bar/mol then the water- side resistance (vw) dominates both phases important air-phase water-phase controlled controlled va derived from evaporation of water Kaw (water) = 2.3 10-5, so air side resistance dominates wind speed is important va increases linearly with wind speed up to ~8 m/s va (water) = 0.2u10 + 0.3 where u10 is the wind speed (m/s) at 10 meters vw derived from tracers with high KH O2, CO2, He, Rn, SF6 Influence of wind speed, but also wave field Liss and Merlivat 1986 See Table 20.2 for equations Air-water exchange models for lakes, oceans Whitman Two-Film Model (1923) considers two bottleneck boundaries, stagnant films on the air and water side of the interface where transport occurs by molecular diffusion Surface Renewal Model interface is described as a diffusive boundary. parcels of air or water undergo a/w exchange to eqbm, then are replaced (air is replaced more often than water b/c less viscous) Boundary Layer Model (Deacon, 1977) considers changes in turbulence and molecular diffusivity (due to changes in T) separately Whitman two-film model Whitman Two-Film Model each stagnant boundary layer has a characteristic thickness : If we assume that the layer thickness is the same for all chemicals then we can easily convert the transfer velocity for water or CO2 to a velocity for our chemical: Diffusivity In air: In water: Air-water exchange in flowing waters Physics of boundary now influenced by both wind and water movement Turbulence in rivers is primarily introduced by shear at the bottom Water side: vw is affected by flow Air side: va is not affected by flow Two models: Small Eddy Model (Lamont and Scott, 1970) Large Eddy Model (O’Connor and Dobbins, 1958) small vs. large eddy Small Eddy Model The turbulent eddies produced by water flowing over the rough bottom are small compared to the depth of the river (bottom is smooth and/or river is deep) Sciw = Schmidt number = vi/Diw = viscosity/diffusivity vw = kinematic viscosity of water u* = shear velocity h = water depth Large Eddy Model The turbulent eddies produced by water flowing over the rough bottom are large compared to the depth of the river (river is shallow and/or bottom is rough) Constant ~ 1 u = mean flow velocity of river h = mean river depth Summary We have moved from the smooth flow (small eddy) regime to the rough flow (large eddy) regime. At even rougher flow, bubbles (foam) are formed which further enhance air-water exchange. Note that when we apply either the large or small eddy model, we necessarily assume that air-water exchange is enhanced (greater than the stagnant flow models). Thus the vw we calculated from either the large or small eddy model must be greater than the vw we get from the stagnant two-film model! Study Site Need: • Large fetch upwind of site • Easy access • Power Description of the Micrometeorological Technique • Uses two systems to determine turbulent fluxes in the near surface atmosphere: – Aerodynamic Gradient (AG) Method • determine profile of wind speed, temperature and water vapor, which along with concurrent measurements of PCB air concentration at two heights are use to determine vertical fluxes of PCB emanating from the water column. – Eddy Correlation system • directly measure fluxes of momentum, sensible heat and latent heat, which can be used for correction of PCB concentration profile for non-adiabatic conditions. Calculation of Fluxes and va/w • Vertical PCB fluxes (FPCB) were calculated using the Thornthwaite-Holtzman equation : k = von Karman’s constant u* = friction velocity Need C1 = upper concentration measurabl C2 = lower concentration e conc z1 = upper height gradient! z2 = lower height • Every term can be measured except C , which is the atmospheric stability factor. C can be determined from M, H, and W which are the Micrometeorological Measurement PCB Concentration gradients 6 5 4 Concentration ng m-3 3 Upper Sample Lower Sample 2 1 0 PCB samples Results: PCB fluxes Low MW congeners Heavier congeners 25 have higher fluxes due volatilize more slowly b/c to higher Cw and faster they are sorbed to solids 20 and have slow va/w va/w Fluxes pg/m2 s 15 10 5 0 101+89+90 24+27 52+43 47+48 70+76 63 15 18 25 28 22 53 42 64 55 91 7+9 5+8 110 154 147 146 132 138+163 159 157 178 185 177 172+192 1 3 PCB Congeners MTC m/d 0 2 4 6 8 12 14 10 1 7+9 11 17 25 20+21+33 53 49 42 PCB MTC 40 70+76 Congeners Results: va/w 93+95 101+89+90 151 153 138+163 157 174+181 180 How to use va/w It’s hard to use net flux, because it is dependant on both Ca and Cw, and is not, therefore, pseudo first order with respect to either of them. By dividing the flux in to the absorption and volatilization fluxes, you can model each as a pseudo first order process. Pseudo first order rate constants To obtain a pseudo first order rate constant, you need to get va/w into units of 1/time: Mass lost = volatilization flux times surface area To convert to concentration change, divide by volume: Define a pseudo first order rate constant kaw = va/w/d (d = depth)