Leaf Boundary Layer Resistances and Mass and MtEh tIIMomentum Exchange , part II:• Dimensionless Numbers– Sherwood Number, Sh–Schmidt Number, Sc– Grasshof Number, Gr– Nusselt Number, Nu– Prandtl Number, PrESPM 129 Biometeorology 1Re ulReynolds number Re Inertial to visous forcesSchmidtScKinematicScDDccSchmidtScKinematic viscosity to molecular diffusivityPrandtl Pr Kinematic viscosity toPrDtglviscosity to thermal diffusivitySherwood Sh Dimensionless mass transfer conductance (dtShglDcc(conductance divided by the ratio of the molecular diffusivity and a length scale l)GrlgT322length scale, l)Grasshof Gr Buoyant force times an inertial force to the square of the viscous forceNuglDhtviscous forceNusselt Nu Dimensionless heat transfer conductacneESPM 129 Biometeorology 2Conductance is f(dimensionless Number, Diffusivity andLength scale)e gt sca e)gDNudhhgDShdcc(m/s)dd•gh: heat conductancegh: heat conductance• gc: scalar conductance• d: length scale•Dh: thermal diffusivityDl diff i it•Dc: scalar diffusivity•Nu: Nusselt Number•Sh: Sherwood NumberESPM 129 Biometeorology 3NlNbNNusselt Number, NuNusselt number, is the ratio of the turbulence conductanceghand a length scaleltoconductance, gh, and a length scale, l, tothe thermal diffusivity, Dt, NbcPRlHlNuabcPrReNuglDHlCD T Thtphsa()gDNudhhConductance for Heat is a f(Nu)ESPM 129 Biometeorology 4gdhPrandtl Number, PrRatio between kinematic viscosity, and thermal diffusivity, Dty,tPrPrtatDDESPM 129 Biometeorology 5Defining whether the flow is turbulent or laminarReynolds NumberDefining whether the flow is turbulent or laminar. Reynolds NumberRe duReis the ratio between inertial and viscous forcesReis the ratio between inertial and viscous forces d, physical dimensionu, fluid velocityRe < 2000 laminar, kinematic viscosityESPM 129 Biometeorology 6Osborne ReynoldsRe < 2000, laminarFl t Pl tFlow Chart for boundary layer conductance for heat, ghdNuDghhFlat PlatecbaNuRePraNuRePrLaminarFree Convection5.033.0RePr66.0NucbGraNu PrForced ConvectioncNu8.033.0RePr03.0223TglGrESPM 129 Biometeorology 7Laminar FlowNu066033 05.Pr Re..Forced ConvectionNu003033 08.Pr Re..Free ConvectionbGr >> Re2Nu a GrbcPrESPM 129 Biometeorology 8Grashof Number, GrBuoyant force times an inertial forceto the square of the viscous force GrlgT3222l, length scaleg,acceleration due to gravity, air densityT, air temperatureESPM 129 Biometeorology 9T, air temperature, viscositySherwood Number, Sh,Sh d b i th ti f th t b l d tSherwood number, is the ratio of the turbulence conductance, gc, and a length scale, l, to the molecular diffusivity, Dc, ShglDFlDccsa()DShgDShdccConductance =f(Sh)ESPM 129 Biometeorology 10Fl t Pl tFlow Chart for boundary layer conductance for mass, gchcDShgdFlat PlateRebcSh aScReSh aScLaminarFree Convection0.33 0.50.66 ReSh ScbcSh aSc GrForced Convection0.33 0.80.03 RecSh Sc223TglGrESPM 129 Biometeorology 11Sherwood Number, Sh,ShaScbcReShaScReLaminar Flow, Flat PlateForced Convection, Flat PlateSh Sc 066033 05.Re..Sh Sc0037033 08.Re..a a o , at ateo ced Co ect o , at ateRe, Reynolds NumberSc, Schmidt NumberESPM 129 Biometeorology 12Sc, Schmidt NumberSchmidt Number, ScRatio of kinematic viscosity to molecular diffusivityat o o e at c scos ty to o ecu a d us tySScDDccESPM 129 Biometeorology 13Sherwood Number, Free ConvectionbSh aSc GrbcGr >> Re2ESPM 129 Biometeorology 14140160120140laminar flowSherwood number withSh80100number with laminar and turbulent flow4060turbulent flow20Re0 5000 10000 15000 20000 25000 300000ESPM 129 Biometeorology 15Prandtl and Schmidt numbers for a variety ofPrandtl and Schmidt numbers for a variety of temperatures. (D: mm s-1) Tk n DvDcPr Sc h2o Sc co2-5 18.3 12.9 20.5 12.4 0.705 0.629 1.0400 18.9 13.3 21.2 12.9 0.704 0.627 1.0315 19.5 13.7 22 13.3 0.703 0.623 1.03010 20.2 14.2 22.7 13.8 0.703 0.626 1.0291520 814 623 414 20 7020 6241 0281520.814.623.414.20.7020.6241.02820 22.2 15.5 24.9 15.1 0.698 0.622 1.02625 22.5 15.75 25.3 15.3 0.700 0.623 1.02930 22.8 16 25.7 15.6 0.702 0.623 1.02635 23.5 16.4 26.4 16 0.698 0.621 1.025ESPM 129 Biometeorology 16Forced ConvectiongrudaHaH166205.().Flat PlateaHu106.grudaHaH140304.( ).CylindergrudaH15710604.( )..SphererdaHESPM 129 Biometeorology 17Re ulReynolds number Re Inertial to visous forcesSchmidtScKinematicScDDccSchmidtScKinematic viscosity to molecular diffusivityPrandtl Pr Kinematic viscosity toPrDtglviscosity to thermal diffusivitySherwood Sh Dimensionless mass transfer conductance (dtShglDcc(conductance divided by the ratio of the molecular diffusivity and a length scale l)GrlgT322length scale, l)Grasshof Gr Buoyant force times an inertial force to the square of the viscous forceNuglDhtviscous forceNusselt Nu Dimensionless heat transfer conductacneESPM 129 Biometeorology 18Conductance for heat as a function of wind speed and length scale400flat leaf1) 300 1 mm 1 cm 10 cm gh (m s-1200100 ( -1)0246810120ESPM 129 Biometeorology 19u (m s-1)Flat Plate Theory vs ObservationsESPM 129 Biometeorology 20Stokes et al 2006 AgForMetComputations of Sherwood number for real and theoretical leavesESPM 129 Biometeorology 21ESPM 129 Biometeorology 22Application of leaf resistance theory to compute nitric acid vapor fluxESPM 129 Biometeorology 23Impact of nonlinearity and nonGaussian turbulence onImpact of non-linearity and non-Gaussian turbulence on the evaluation of leaf resistance models ESPM 129 Biometeorology 24Leaf angles andand momentum transfertransfer ESPM 129 Biometeorology 25Why is CO2 resistance related to water vapor resistance in terms rrDacavv()/23ypof the ratio of Diffusivities Raised to the 2/3 power?Dacavc()rrDScDScacavvcReRe////12 1312 13rDDDacvcvRe()///121323139rDDDacavvccvvcRe().///121323139ESPM 129 Biometeorology 26SummarySummary • A laminar sublayer always exists close to the surface of leaves, even when experiencing turbulent floweven when experiencing turbulent flow• The Reynolds’ number quantifies whether a leaf is experiencing turbulent or laminar flow and increases with characteristic leaf size.• The conductance for mass transfer is proportional to the molecular ppdiffusivity and the Sherwood number and is