# Exercise 4

Aerobic biodegradation of petroleum hydrocarbons present in the vadose zone is manifest by a predominately upward flux of carbon dioxide. The magnitude of CO_{2} flux is an indicator of the rate of depletion of the liquid-phase hydrocarbon. Among the various methods that have been used to estimate the CO_{2} flux in the field is the so-called gradient method, by which the upward flux is computed from a measured concentration distribution and the effective diffusion coefficient. More than two gas components are involved in the depletion of a liquid-phase hydrocarbon by aerobic biodegradation. Nevertheless, it is instructive to revise the gradient method using the flux equations for a binary gas in which air and carbon dioxide are assumed to be the only two constituents.

Refer to Figure Exercise 4-1 and derive an equation by which the steady CO_{2} flux can be calculated from a measured concentration distribution and a known effective diffusion coefficient. Assume the molecular regime prevails, pressure diffusion is negligible, and air can be considered to be a single species (species *A*).

Compute the mole flux of carbon dioxide, phase (bulk gas) flux, non-equimolar flux, and viscous flux using the following data from Tracy (2015).

*p* = 8.3 × 10^{4} Pa

*T* = 21.6 °C

*D* = 4.7 × 10^{-6} m^{2}/s

*z*_{2} – *z*_{1} = 1.29 m

*x*_{B}(*z*_{1}) = 0.0583

*x*_{B}(*z*_{2}) = 0.0013