# Boundary layer conductance

Boundary layer conductance to H2O.

## source

Nobel, P.S., 1991. Physicochemical and environmental plant physiology. Academic Press.

eqn. 7.8 p. 364; eqn. 8.3 p. 397

## details

### boundary layer thickness

δbl Represents an average thickness of the unstirred air layer adjacent to a leaf. The main factors affecting δblare the ambient wind speed and leaf size, with leaf shape excerting a secondary influence. Partly for convenience, but mainly because it has proved experimentally justifiable, the effect of leaf size on boundary layer thickness is determined by the characteristic dimension l, which is the mean length of a leaf in the direction of the wind. Based on hydrodynamic theory for laminar flow adjacent to a flat surface as modified by actual observations under field conditions (s. Pearman & al., 1972) ther is an approximate, but useful expression for the average thickness of the boundary layer next to a leaf:

$(mm)}^{bl} = 4.0 \cdot \sqrt {\frac{l_{(m)}}{v_{(m /cdot s^{-1})}$ (7.8)

where l(m) is the mean length of the leaf in downwind direction in m, v(m s-1) the ambient wind speed in m/s, and ${\displaystyle \delta _{(mm)}^{bl}}$ the average thickness of the boundary layer in mm (the factor 4.0 has units of mm/s1/2).

### boundary layer conductance

The air boundary layers on both sides of a leaf influence the entry of CO2 and the exit of H2O as was clearly shown by Klaus Raschke in the 1950s. Movement of gas molecules across these layers is by diffusion in response to differnces in concentration. We can represent the conductance and the resistance of a boundary layer of air as follows:

${\displaystyle g_{j}^{bl}={\frac {J_{j}}{\Delta c_{j}^{bl}}}={\frac {D_{j}}{\delta ^{bl}}}={\frac {1}{r_{j}^{bl}}}}$(8.3)

The SI units for the diffusion coefficient Dj are m2s-1 and the thickness of the boundary layer δbl is in (m2s-1/m or m s-1 (often expressed as mm s-1), and ${\displaystyle r_{j}^{bl}}$ is in s m-1. Jj is expressed per unit of leaf area, so ${\displaystyle g_{j}^{bl}}$ and ${\displaystyle r_{j}^{bl}}$ also relate to unit area of a leaf.

Dj is a fundamental measure of conductivity (values in suitable handbooks) describing the diffusion of species j in a given medium. On the other hand, δbl characterizes a particular situation because the thickness of the boundary layer depends on wind speed and leaf size. Thus, ${\displaystyle r_{j}^{bl}}$ as defined by eq. 8.3 describes a particular component of the path, analogous to resistance (R) used in Ohm's law. We recognise that ${\displaystyle {D_{j}}/{\delta ^{bl}}}$ represents the the permeability coefficient for substance j, ${\displaystyle P_{j}=D_{j}K_{j}/\Delta x}$, as it diffuses over an air boundary layer of thickness δbl. When something readily diffuses across a boundary layer, Pj and ${\displaystyle g_{j}^{bl}}$ are large and ${\displaystyle r_{j}^{bl}}$ is small. Using resistances and conductances, we can describe gas fluxes into and out of leaves employing a number of relations originally developed for the analysis of electrical circuits.

## model code

dbl = 4 * sqrt(LeafSize / WindSpeed)
# Diffusion coefficient for water vapor (m2 s-1)
# Temperature sensitivity calibrated (L.R.) from Nobel (1983) Appendix I as p 543
dwv = 2.126e-05 + 1.48e-07 * TemperatureCelsius
return (dwv / dbl)