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Berkeley ESPM C129 - Conservation Budget overheads

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10/24/20121Biometeorology ESPM 129Lecture 23, Fluxes and the Conservation Budget• Fick’s First Law• Resistors and Conductors• Continuity Equation – Concept– Derivation – local and total derivatives– constant density, incompressible flow• Conservation of mass for multicomponent system– diffusive flux densities on molar and mass bases – Fick's Second Law • Conservation of Mass, turbulent flow– bulk flux density on molar and mass bases– Reynolds decomposition– derivation 10/24/2012Biometeorology ESPM 129Diffusion is defined as:process resulting from random motion of molecules by which there is a net flow of matter from a regionof high concentration to a region of low concentration.10/24/20122Biometeorology ESPM 129Fick’s Law of Diffusion• a chemical species diffuses in the direction of decreasing mole fraction. the flux density is proportional to a diffusion coefficient and a gradient Fkccxx~1212 F k c c x x ~ 1 21 2  c 1,x 1c 2,x 2 Biometeorology ESPM 129F=-Dccx(g m-2s-1): mass density, cF=-DccxF=- DcsxF=-MDacacCx(mol m-2s-1): mole density, c(g m-2s-1): mass fraction, s(mol m-2s-1): mole fraction, CcComputing Flux Density, F10/24/20123Biometeorology ESPM 129DDTT PPn000(/ )( /)TDh2oDco2Do2oC mm2s-1mm2s-1mm2s-10 21.2 13.9 17.710 22.6 14.8 18.820 24.0 15.7 20.030 25.4 16.7 21.240 26.9 17.7 22.5Molecular Diffusivity, DBiometeorology ESPM 129IVRSerialNetworkParallelNetworkResistors, r, and Conductors, gOhm’s LawCurrent = Voltage/Resistance10/24/20124Biometeorology ESPM 129111Rr rRrrrrabababGg g12Parallel Resistance/ Serial Conductances ParallelNetworkBiometeorology ESPM 129Rr rab111Gg gGggggabababSerial Resistance/ Parallel Conductance NetworksSerialNetwork10/24/20125Biometeorology ESPM 129Flux-ResistanceFCCraai0FCCrai0Meteorologists:R (s/m)Ecophysiologists:R (mole-1m2s1)F g ms mol mc()(( ))13Biometeorology ESPM 129rmol m s rm sVPTPTooo()() 121 1gms gmolm sVPTPTooo()( ) 121Vo= 0.0224 m3mol-1at STPEcophysiological, Alternative, View of Resistance10/24/20126Biometeorology ESPM 129Bath tub analogy, change of height of water in a volumeBiometeorology ESPM 129Same Principle with Economy$$$$ in$$ outMoney in Bank10/24/20127Biometeorology ESPM 129uc(x)uc(x+dx)wc(z)wc(z+dz)xyztyz u uxxx[| | ]yx w wzzz[| | ]How air density, , of a volume changes with timeBalance of mass fluxes in and out of horizontal and vertical wallsBiometeorology ESPM 129tuxvywz()Continuity Equation, how air density, , changes with timeu, longitudinal velocityv, lateral velocityw, vertical velocity10/24/20128Biometeorology ESPM 129ddt tuxvywzuxvywz()Expansion of termsu, longitudinal velocityv, lateral velocityw, vertical velocityBiometeorology ESPM 129dc t x y zdtctdxdtcxdydtcydzdtczctucxvcywcz(, , ,)dcdtctucxvcywczHow advection terms arise, relation between total and partial derivatives10/24/20129Biometeorology ESPM 129Incompressible Flowddtuxvywz0( )()uxvywzBiometeorology ESPM 129ctDcxc22Fick’s Second LawTime rate of change in C is related to the second derivative with respect to space10/24/201210Biometeorology ESPM 129FA FFxdx ActAdx()ctFxctDcxc22Conservation Equation, Laminar FlowFick’s Second LawF=-DccxBiometeorology ESPM 129ctucxucxxDcxjjjjjcj''[]ccc' (')(')(')[(')]cctuuccxxDccxjjjjcjConservation Budget, Turbulent Flow10/24/201211Biometeorology ESPM 129ctucxwczzDczc''[]2D SimplificationBiometeorology ESPM 1290  awczFz''Ideal, steady-state, infinite fetch, no advectionConstant Flux Layer, Internal Boundary Layerctucxjj00Integral of dF/dZ equals a CONSTANT10/24/201212Biometeorology ESPM 129Integrate from Ground up and Define Flux as Sum of flux at the ground and the sum of the Diffusive source-sink from the vegeationaahwc h wc S z dz''() ''() ()z00Constant Flux Layer, Internal Boundary LayerBiometeorology ESPM 129Flow Through SystemCinCout~CFhux~CFhtStatic Chamber SystemStatic vs Dynamic Chamber Systems10/24/201213Biometeorology ESPM 129time (s)0 200 400 600 800 1000 1200CO20100020003000400050006000ctFth()ctFx()()ctFt htCase 1, No Advection, Dynamic Responseucx 0Evaluate Flux at t=0!!() 00cFtthBiometeorology ESPM 129ucxFzucxFhFuhcx Case 2, Steady-State, Advectionct000cFuxh10/24/201214Biometeorology ESPM 129Homework• Use a unit-correct form of the conservation equation to evaluate the change of CO2concentration with time (up to 1000 s) in a closed chamber that has horizontal cross section of 0.1 (x) and 0.1 m (y). – Perform the calculations for cases where the chamber is 0.1, 0.3, and 0.5 m tall. – Start with a CO2 concentration of 350 mol mol-1. – The initial flux density is 2 mol m-2s-1, the exchange conductance, g, is 4.30 10-4mol m-2s-1and the reference deep soil CO2concentration is 5000 mol mol-1. – In performing these calculations consider feedback between the flux density () and build up of CO2in the head space. Assume the concentration in the chamber is well mixed.Fgctcref(() )Biometeorology ESPM 129• Use the advection form of the conservation equation to evaluate the flux density of CO2 into an open chamber. • The chamber is 0.5 (x) by 0.5 (y) by 0.1 (z). The incoming and outgoing CO2 concentrations are 350 and 355 mol mol-1, respectively. Perform calculations for cases where the flow velocity is 1, 3 and 6 m s-1(the units of flux density should be mol m-2s-1• What is the flux of CO2into an open chamber, where the volumetric flow rate is 1, 3 and 6 liters per minute? Use the same


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