Modeling Tools for Environmental Engineers and Scientists

By introducing a proportionality constant, D i , called the molecular diffusioncoefficient (L 2 T –1 ), and a negative sign to indicate that the flux is positive inthe x-direction,4.4.1.2 Unsteady State ConditionsJ x,i = –D i A x d Ci (4.18)dxThe diffusive transport rate under time-dependent, unsteady state can bequantified using Fick’s Second Law:∂ ∂ x –D i ∂ C∂ i x ∂ Ci∂ (4.19)tThe above equations can be applied to diffusive transport through gases orliquids. The diffusion coefficient (or diffusivity), D i , is an intrinsic molecularproperty for a chemical-solvent system. Tabulated numerical values for D canbe found in handbooks; they can also be estimated from chemical and thermodynamicproperties following empirical correlations such as the Wilkie-Chang equation for diffusion of small molecules through water and theChapman equation for diffusion in gases.4.4.1.3 Multiphase DiffusionIn certain environmental systems, molecules may diffuse through a matrixof multiple phases. A typical example is the diffusion of chemical vaporsthrough the vadose zone matrix that may consist of air, water vapor, purechemical liquid, and soil. The effective diffusion coefficient under these conditionswill be dependent upon the pore characteristics and can be accountedfor by the tortuosity factor, τ, to modify the pure phase diffusivity as follows:D pore, j = D i, j τ (4.20)where D pore, j is the diffusivity in the pores filled with phase j, D i,j is themolecular diffusivity in phase j, and θ is the porosity of the matrix.Worked Example 4.4The molecular diffusivity of nitrates in water is 19 × 10 –6 cm 2 /s. In a river,nitrate concentration in the water column is 20 mg/L, and in the sedimentpore waters, at a depth of 10 cm, it is 0.05 mg/L. Estimate the diffusive fluxof nitrate into the sediments, assuming sediment bed porosity of 65% and atortuosity factor of 3.© 2002 by CRC Press LLC