## Oxygen Diffusion

### Tom Richard

Diffusion is a reflection of the fact that molecules, as they
vibrate with random motion in a gas or liquid, move toward an
equilibrium where all the molecular species in the mixture are
uniformly dispersed, and the concentration of any one species
is the same everywhere.

The diffusion equation (Fick's second law), states that the
rate of molecular diffusion is proportional to the second derivative
of its concentration. It its most general form this can be written:

Source: Bird et al., 1960.

For a one dimensional concentration gradient of oxygen in air,
this simplifies to:

For a one dimensional concentration gradient of oxygen in water,
the simplified equation is:

In a composting system, the concentration gradient is a function
of the rates of oxygen supply and aerobic __biodegradation
and oxygen uptake__ (link coming soon). The O_{2} concentration
gradient is the driving force that moves O_{2} into the pile by diffusion,
and there is a corresponding CO_{2} gradient driving diffusion of
CO_{2} out of the pile. From the practical standpoint of process
management, it is the diffusion of O_{2} that is critical to maintaining
aerobic conditions, so that will be the focus of the present analysis.

A detailed discussion is provided for calculating
the oxygen diffusion coefficient in air, as well as the procedure
for calculating the oxygen diffusion
coefficient in water. Using this analysis, we find that the
O_{2} diffusion coefficient in saturated air (at 15% O_{2} concentration)
is 5700 to 10,800 times greater than in water (at 60°C and
20°C, respectively). When oxygen is forced to diffuse through
water saturated pores, this restriction on oxygen
transport is one of the most important factors
leading to anaerobic conditions.

**Reference**

Bird, R.B., W.E. Stewart, and E.N. Lightfoot.
1960. Transport Phenomena. John Wiley & Sons. NY. 780 pp.
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