11.6 More Detailed Energy Budget

So far, the energy budget equation has contained only the leaf width as the characteristic dimension for convective heat transfer and gas diffusion. Such a simple expression is only strictly correct for leaves which are nearly square; however, relatively few leaves are of this form. Most leaves have a long dimension which is considerably greater than the narrow dimension. Some years ago, Gates (1968) reported investigations of the heat and water vapor transfer from simulated leaves of blotting paper of a variety of lengths and widths. He found if \(D\) is the characteristic dimension of the leaf in the direction of air flow and W is the dimension transverse to the wind, that the energy budget is given by \[\begin{equation} Q_a = \varepsilon \sigma [T_l + 273]^4 + k_1 \bigg( \frac{V}{D} \bigg)^{1/2} [T_l - T_a] + L(T_l \frac{_sd_l(T_l)-r.h. {_sd_a(T_a)}}{r_l + k_2 [D^{0.30}W^{0.20}]/V^{0.50}}) \tag{11.7} \end{equation}\]

where \(k_1 = 9.14 J m^{-2} s^{-1/2} °C^{-1}\) and \(k_2=183 s^{-1/2} m^{-1}\)

Gates elected to include W in the evaporation term only since the amount of data was not sufficient to reveal the functional relationship in both the convective and evaporative terms simultaneously. Equation (11.7) is more realistic than Equation (11.4), which, nevertheless, is a good approximation. If the wind is blowing across the leaf in the direction of the leaf width, then, the characteristic dimension \(D\) is the average leaf width and \(W\) is the average length. If, on the other hand, the wind is blowing along the length of the leaf, then, \(D\) is the average length and \(W\) is the average leaf width.