Under suitable circumstances, the rate of oxygen metabolism by bacterial cells is very nearly zero order with respect to oxygen concentration. We examine such a case here and focus our attention on a spherical aggregate of cells, which has a radius $R.$ We wish to determine the total rate of oxygen uptake by the aggregate as a function of aggregate size, oxygen mass concentration at the aggregate surface $({}_0),$ the metabolic activity of the cells, and the diffusional behavior of the oxygen. For simplicity we consider the aggregate to be homogeneous. We then approximate the metabolic rate by an effective volumetric reaction rate $r_{O_2}=-k_0$ and the diffusional behavior by Fick's law, with an effective pseudobinary diffusivity $D_{O_{2}m}.$ Because the solubility of oxygen is very low in this system, both convective oxygen transport and transient effects may be neglected.

$($a$)$ Show that the pseudo$-$steady$-$state oxygen concentration profile in the spherical aggregate of cells is described by the differential equation:

$\frac{1}{{}^2}\frac{d}{d}({}^2\frac{d}{d})=N$

where $=\frac{{}_{O_2}}{{}_0},=\frac{r}{R},$ and $N=k_0\frac{R^2}{{}_0{D}_{O_{2}m}}.$

$($b$)$ There may be an oxygen$-$free core in the aggregate, if $N$ is sufficiently large, such that $=0$ for Write sufficient boundary conditions to integrate the differential equation above for this situation. To do this, it must be recognized that both $$ and $d\frac{}{d}$ are zero at $={}_0.$ What is the physical significance of this last statement?

$($c$)$ Perform the integration of the differential equation and show how ${}_0$ may be determined.

$($d$)$ Plot the total oxygen uptake rate and ${}_0$ as functions of $N,$ and discuss the possibility that no oxygen$-$free core exists.