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=-Dccx(g m-2s-1): mass density, cF=-DccxF=- DcsxF=-MDacacCx(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 PPn000(/ )( /)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 129IVRSerialNetworkParallelNetworkResistors, r, and Conductors, gOhm’s LawCurrent = Voltage/Resistance10/24/20124Biometeorology ESPM 129111Rr rRrrrrabababGg g12Parallel Resistance/ Serial Conductances ParallelNetworkBiometeorology ESPM 129Rr rab111Gg gGggggabababSerial Resistance/ Parallel Conductance NetworksSerialNetwork10/24/20125Biometeorology ESPM 129Flux-ResistanceFCCraai0FCCrai0Meteorologists:R (s/m)Ecophysiologists:R (mole-1m2s1)F g ms mol mc()(( ))13Biometeorology 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)xyztyz 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 129tuxvywz()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(, , ,)dcdtctucxvcywczHow advection terms arise, relation between total and partial derivatives10/24/20129Biometeorology ESPM 129Incompressible Flowddtuxvywz0( )()uxvywzBiometeorology ESPM 129ctDcxc22Fick’s Second LawTime rate of change in C is related to the second derivative with respect to space10/24/201210Biometeorology ESPM 129FA FFxdx ActAdx()ctFxctDcxc22Conservation Equation, Laminar FlowFick’s Second LawF=-DccxBiometeorology ESPM 129ctucxucxxDcxjjjjjcj''[]ccc' (')(')(')[(')]cctuuccxxDccxjjjjcjConservation Budget, Turbulent Flow10/24/201211Biometeorology ESPM 129ctucxwczzDczc''[]2D SimplificationBiometeorology ESPM 1290 awczFz''Ideal, steady-state, infinite fetch, no advectionConstant Flux Layer, Internal Boundary Layerctucxjj00Integral 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 vegeationaahwc h wc S z dz''() ''() ()z00Constant Flux Layer, Internal Boundary LayerBiometeorology ESPM 129Flow Through SystemCinCout~CFhux~CFhtStatic Chamber SystemStatic vs Dynamic Chamber Systems10/24/201213Biometeorology ESPM 129time (s)0 200 400 600 800 1000 1200CO20100020003000400050006000ctFth()ctFx()()ctFt htCase 1, No Advection, Dynamic Responseucx 0Evaluate Flux at t=0!!() 00cFtthBiometeorology ESPM 129ucxFzucxFhFuhcx Case 2, Steady-State, Advectionct000cFuxh10/24/201214Biometeorology 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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