Leaf boundary layer model

The boundary conductance controls the transfer of heat and mass (both, H\(_2\)O and CO\(_2\)) from the leaf surface to the surrounding air. The Nusslet number, \(Nu\) [-], is the ratio of convective to conductive heat transfer, while the Sherwood number, \(Sh\) [-], is the ratio of convective to conductive mass transfer that are given by

\[\begin{eqnarray} \label{eqn_definitation_of_numbers} \text{Nu} &=& \frac{g_{bh}d_\ell}{\rho_m D_h} \\[0.2em] \text{Sh}_w &=& \frac{g_{bw}d_\ell}{\rho_m D_w} \\[0.2em] \text{Sh}_c &=& \frac{g_{bc}d_\ell}{\rho_m D_c} \end{eqnarray}\]

where \(g_{bh}\) [mol m\(^{-2}_{leaf}\) s\(^{-1}\)] is boundary conductance for heat , \(g_{bv}\) [mol m\(^{-2}_{leaf}\) s\(^{-1}\)] is boundary conductance for H\(_2\)O, \(g_{bc}\) [mol m\(^{-2}_{leaf}\) s\(^{-1}\)] is boundary conductance for CO\(_2\), \(D_h\) [m\(^2\) s\(^{-1}\)] is the molecular diffusivity for heat, \(D_w\) [m\(^2\) s\(^{-1}\)] is the molecular diffusivity for H\(_2\)O, \(D_c\) [m\(^2\) s\(^{-1}\)] is the molecular diffusivity for CO\(_2\), \(\rho_m\) [mol m\(^{-3}\)] is molar density, \(d_\ell\) [m] is the representative leaf dimension, and \(\text{Sh}_w\) and \(\text{Sh}_c\) are Sherwood number for water vapor and CO\(_2\), respectively.

Empirical studies have developed relationship for \(\text{Nu}\), \(\text{Sh}_w\), and \(\text{Sh}_c\) for laminar flow:

\[\begin{eqnarray} \text{Nu}^{Laminar} &=& b_1 0.66 \text{Pr} ^{0.33}Re^{0.5} \\[0.4em] \text{Sh}_w^{Laminar} &=& b_1 0.66 \text{Sc}_w^{0.33}Re^{0.5} \\[0.4em] \text{Sh}_c^{Laminar} &=& b_1 0.66 \text{Sc}_c^{0.33}Re^{0.5} \end{eqnarray}\]

and turbulent flow:

\[\begin{eqnarray} \text{Nu}^{Turbulent} &=& b_1 0.036\text{Pr} ^{0.33}Re^{0.8} \\[0.4em] \text{Sh}_w^{Turbulent} &=& b_1 0.036\text{Sc}_w^{0.33}Re^{0.8} \\[0.4em] \text{Sh}_c^{Turbulent} &=& b_1 0.036\text{Sc}_c^{0.33}Re^{0.8} \end{eqnarray}\]

where \(\text{Re}\) [-] is the Reynolds number that is a ratio of inertial forces to viscous forces, \(\text{Pr}\) [-] is the Prandtl number that is a ratio of diffusivity of momentum to diffusivity of heat in fluid, \(\text{Sc}_w\) [-] and \(\text{Sc}_c\) [-] are the Schmidt numbers that are ratio of diffusivity of momentum to diffusivity of mass for H\(_2\)O and CO\(_2\)) in fluid, respectively, and \(b_1 = 1.5\) is a typical value that converts the empirical relationship developed for a flat rectangular plate to for leaves.

The Prandtl, Reynolds, and Schmidt numbers are

\[\begin{eqnarray} \text{Re} &=& \frac{u d_\ell}{\nu} \\[0.4em] \text{Pr} &=& \frac{\nu}{D_h}\\[0.4em] \text{Sc}_w &=& \frac{\nu}{D_w} \\[0.4em] \text{Sc}_c &=& \frac{\nu}{D_c} \end{eqnarray}\]

where \(\nu\) [m\(^2\) s\(^{-1}\)] is the kinematic viscosity. The forced flow due to laminar and turbulent flow is given as

\[\begin{eqnarray} \text{Nu} ^{forced} &=& \max(\text{Nu}^{Laminar}, \text{Nu} ^{Turbulen}) \\[0.4em] \text{Sh}_w^{forced} &=& \max(Shv^{Laminar}, \text{Sh}_w^{Turbulent}) \\[0.4em] \text{Sh}_c^{forced} &=& \max(Shc^{Laminar}, \text{Sh}_c^{Turbulent}) \end{eqnarray}\]

In free convection, The Nusselt and Scherwood number are described in terms of Grashof number, \(\text{Gr}\) [-], as

\[\begin{eqnarray} \text{Nu}^{Free} &=& 0.54\text{Pr} ^{0.25}\text{Gr}^{0.25} \\[0.4em] \text{Sh}_w^{Free} &=& 0.54\text{Sc}_c^{0.25}\text{Gr}^{0.25} \\[0.4em] \text{Sh}_c^{Free} &=& 0.54\text{Sc}_c^{0.25}\text{Gr}^{0.25} \\[0.4em] \end{eqnarray}\]

The Grashof number is given as

\[\begin{equation} \text{Gr} = \frac{gd_\ell^3 (T_\ell - T_a)}{\nu^2 T_a} \end{equation}\]

where \(g\) [m s\(^{-2}\)] is the gravitational acceleration, \(T_\ell\) [K] is the leaf temperature, and \(T_a\) [K] is the air temperature. Lastly, the combined Nusselt and Scherwood number for forced and free flow are given as

\[\begin{eqnarray} \label{eqn_combined_numbers} \text{Nu} &=& \text{Nu} ^{Forced} + \text{Nu} ^{free} \\[0.4em] \text{Sh}_w &=& \text{Sh}_w^{Forced} + \text{Sh}_w^{free} \\[0.4em] \text{Sh}_c &=& \text{Sh}_c^{Forced} + \text{Sh}_c^{free} \end{eqnarray}\]

The leaf boundary conductances for heat, H\(_2\)O, and CO\(_2\) are given by combining equations \eqref{eqn_definitation_of_numbers} and \eqref{eqn_combined_numbers}

\[\begin{eqnarray} g_{bh} &=& \frac{D_h \times \text{Nu}}{d_\ell} \rho_m\\[0.4em] g_{bw} &=& \frac{D_w \times Sh_v}{d_\ell} \rho_m\\[0.4em] g_{bc} &=& \frac{D_c \times Sh_c}{d_\ell} \rho_m \end{eqnarray}\]

The kinematic viscosity and molecular diffusivities are adjusted to account for air temperature and air pressure, \(P_a\) [Pa], as

\[\begin{eqnarray} \nu &=& f \times \nu_0 \\ D_h &=& f \times Dh_0 \\ D_w &=& f \times Dv_0 \\ D_c &=& f \times Dc_0 \\ f &=& \frac{10325}{P_{a}}\times \left(\frac{T_a}{272.15}\right)^{1.81} \end{eqnarray}\]

where \(\nu_0\) is kinematic viscosity at 0\(^0\) C, and \(Dh_0\), \(Dv_0\), \(Dc_0\) are molecular diffusivity for heat, H\(_2\)O, and CO\(_2\) at 0\(^0\) C.