1Lecture 25, Leaf Boundary Layers and their Resistances and Mass and Momentum Exchange, Part 2 November 1, 2010 Instructor: Dennis Baldocchi Professor of Biometeorology Ecosystem Science Division Department of Environmental Science, Policy and Management 345 Hilgard Hall University of California, Berkeley Berkeley, CA 94720 Topics to be Covered 1. Sherwood Number 2. Schmidt Number 3. Grasshof Number 4. Nusselt Number 5. Prandtl Number b. Characteristic leaf length c. Enhancement Factors d. Leaf Shape e. Pubescence f. clumping and mutual sheltering test with HNO3 g. Characteristics of external fluid flow B. Model Calculations on the effects of leaf boundary layer resistance on varying leaf temperature and evaporation. C. Integrated Canopy Form Drag 1. Skin Friction L25.1 Sherwood Number The Fickian equation for mass transfer is a function of a mass transfer coefficient, gc, and the difference in mole density (Schuepp 1993): Fgccc A dimensionless mass transfer coefficient, the Sherwood number, is the ratio of the turbulence conductance, hc, and the ratio between the molecular diffusivity, Dc, and a2length scale, l (Monteith and Unsworth 1990; Schuepp 1993) This provides a ratio of m s-1 over m s-1. ShglDFlDccsa() Empirical studies show that Sh can be computed as a function of the Sc and Re number: Sh aScbcRe For laminar flow over flat leaves: Sh Sc066033 05.Re.. Forced convection Sh Sc0037033 08.Re.. Let’s compute the laminar and forced conductances based on what we know so far. Lets assume D for vapor is 24.9 mm2 s-1, the kinematic viscosity is 15.5 mm2 s-1 and Sc is 0.622, at 20 C. Shud 05612.( )/ gDShduddudcc056 249155354121212..()..()/// gDShduddudcc00315 2491550086454545 15..()..////3 Figure 1 Conductance for heat as a function of wind speed and length scale The Schmidt Number is the ratio of the kinematic viscosity,  to the mass diffusivity, Dc: ScDDcc So the key difference besides the multiplicative factor is that Re changes from ½ to 4/5 power as one goes from laminar to turbulent flow. For cylinders exposed to Reynolds numbers between 1000 and 50000 one can use: Sh Sc024033 06.Re.. For spheres, the Sherwood number for water vapor is: Sh03406.Re. Under free Convection, the Sherwood number is a function of the Grashof number: u (m s-1)024681012gh (m s-1) 0100200300400 1 mm 1 cm 10 cm flat leaf4 Sh aSc Grbc For free convective transfer there is no reference velocity. In this case a reference velocity is. On this basis, one can derive the Grashof number as the ratio of buoyant forces times an inertial force to the square of a viscous forces GrlgT322  is the coefficient of thermal expansion Free convection occurs when wind is very calm and leaves are illuminated by the sun. In this circumstance, leaf temperature becomes elevated and it establishes a small convective circulation. It has been our experience, from model calculations, that we cannot ignore the onset of free convection. Without the effects of convective transfer model calculations will compute extremely elevated (and potentially lethal) leaf temperatures. The criterion for Gr is that it be much greater than Re2 Figure 1 shows graphically how Sh varies in turbulent and laminar flow.5 Figure 1 Sherwood number with laminar and turbulent flow L25.2 Nussult Number With regard to heat transfer, a version of Fick’s Law is: FCgTHph The heat transfer coefficient, ht, is computed from the Nu, which is a function of Pr and Re (Monteith and Unsworth 1990; Schuepp 1993). NuglDHlCD T Thtphsa() The Nusselt number is the ratio of the heat transfer coefficient, gh and the ratio of the thermal diffusivity, Dt, and a length scale, l. Nusselt number for laminar flow Re0 5000 10000 15000 20000 25000 30000Sh020406080100120140160laminar flowturbulent flow6 Nu abcPrRe For laminar flow Nu066033 05.PrRe.. gDNuddudhh066 08922 215512...(.)/ gudh 33112.()/ (m s-1) For forced convection (Re > 20000) Nu003033 08.PrRe.. For free convection Nu a GrbcPr In this application, the Prandtl Number is the ratio of the kinematic viscosity, , to thermal diffusivity, Dt. Pr Dt The Prandtl and Schmidt number are a function of variables that experience a temperature dependency. Table 1 Prandtl and Schmidt numbers for a variety of temperatures. (D: mm s-1). Adopted from (Monteith and Unsworth 1990; Massman 1998) T Dt Dv Dc Pr Sc h2o Sc co2-5 18.3 12.9 20.5 12.4 0.705 0.629 1.04070 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 1.02915 20.8 14.6 23.4 14.2 0.702 0.624 1.02820 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 1.025 Textbooks, such as Monteith and Unsworth (Monteith and Unsworth 1990)and Campbell and Norman (Campbell and Norman 1998) provide a list of equations for the computation of boundary layer resistances over an variety of objects. The first cohort of formulae have units of m s-1 1. Flat plate, laminar flow grudaHaH166205.(). 2. Cylinders, flow normal to long axis grudaHaH14030604.( ).. 3. Spheres grudaHaH15710604.( ).. A second cohort of formulae have units of mol m-2 s-1 1. forced convection grudaHaH1013505.(). grudavav1014705.(). grudacoaco220510110.().8 2. Free Convection gTTdaHsa005014.( )/ gTTdavsa005514.( )/ gTTdacosa_/.( )2140038 Inter-relation of resistances Most often scientists measure the boundary layer resistance for water vapor and apply gas law relations to obtain resistances for CO2 and other trace gases. Inspecting numerous biophysics textbooks, one may encounter formula showing that the boundary layer resistance for carbon dioxide is related to the resistance to water times the ratio of the molecular diffusivites taken to the 2/3 power rrDDac avvc ()/23 Why does this occur? Using the engineering formula at hand, one can deduce the reason. rrDScDScacavvcReRe////12 1312 13 ScDx rrDDDDDDacavvccvvcReRe() ./////1213121323139 Similar formulae can be derived for heat, water vapor and CO2 exchange. rrDavah v() ./23093 rrDacoah co2223132() ./9And for heat and