Consider the absorption of water through a plant root like that in Figure P8.27. The plant is a desert plant that has just experienced its first rain storm in over four months. The surface of the ground is saturated, but because of the dry conditions, the water does not seep into the ground very well. In fact, the concentration of water in the ground varies linearly with depth. Assuming the geometry shown in the figure, and physical properties given in the problem:

a. Derive an expression for the water concentration profile in the root. You may assume a constant concentration at the root base, an impermeable root tip, and steady-state operation.

b. Derive an expression for the effectiveness and efficiency of this root. For the physical properties listed, what is the effectiveness and efficiency?
\[\begin{aligned}& L=5 \mathrm{~cm} \quad d=0.005 \mathrm{~m} \quad c_{a o}=50 \mathrm{moles} / \mathrm{m}^{3} \\& c_{\infty L}=10 \mathrm{moles} / \mathrm{m}^{3} \quad c_{\infty o}=1000 \mathrm{~mole} / \mathrm{m}^{3} \\& D_{a b}=1 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s} \quad k_{c}=1 \times 10^{-6} \mathrm{~m} / \mathrm{s}\end{aligned}\]

Transcribed Image Text:

FIGURE P8.27 Cao Coco k L Dab CooL Ground Root