Consider a highly idealized model of the root of a leguminous plant shown in Figure P8.25. A root of diameter, \(d\), and length, \(L\), is composed of two sections. The section near the base is catalytic and consumes substrate along its length to produce a more useful fuel. The catalytic section absorbs no nutrients through its surface. This part corresponds to the root nodules where bacteria convert nitrogen into nitrate for absorption by the plant. The partition coefficient between the root and the soil is assumed to be one for convenience. The substrate undergoes a first-order conversion:

\[-r_{s}=k^{\prime \prime} c_{s}\]

The outer region of the root collects the substrate for the catalytic part. The base of the root is free of substrate while the tip of the root is impermeable.

a. Determine the substrate concentration profile in both root sections. You may assume the diffusivity of substrate is the same in both sections.

b. How does the rate of reaction in the nodule or catalytic part control the mass transfer in the passive section, i.e., how does the root's effectiveness depend upon \(k^{\prime \prime}\) ?

Transcribed Image Text:

FIGURE P8.25 Dab Dab -Ta L/2 L kc 0=0% deg/dx = 0