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Class Note for BSysE 595 at WSU 06

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Washington State University taught by a professor in Fall. Since its upload, it has received 23 views.

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Date Created: 02/06/15
Mass Transport Equations 1 Mass Conservation Statement Mass in ow rate Mass out ow rate Net mass in ow rate Change in mass storage with time MT 1 l 2 General Form For a REV let J be the mass ux vector molesLZT the LHS of Eq 1 divJ denoting net mass in ow rate per unit volume If changes in mass in the REV oc changes in 6 Cn 6t denoting time rate of change of mass per unit volume 9 concentration the RHS of Eq 1 Combining LHS and RHS gives aCn 6t divJ 2 3 The Diffusion Equation Fick s law for diffusion in sediments J D ngradC 3 D bulk diffusion coefficient accounting for tortuosity Substituting 3 into 2 leads to divD ngradC 6quot 4 If n not changing with space amp time and D constant spatially 4 becomes the diffusion equation 1 divgradC 5 at or 2 2 2 6C6C6CZV2CZ 1g 6 6x2 6y2 6z2 D8t If the net concentration does not change With time 6 is simplified to Laplace equation V2C0 4 The Advection Diffusion Equation Combining the advective and diffusive uxes 6C Jx Dn van 6x 6C Jy Dy na y 1nd 6C Jz Dz Ila z szn For isotropic media J DngradCan 7 J v vectors With x y 2 components Substituting 7 into 2 yields VDnVC Vvcn aftquot 8 Expanding the advection term and assuming constant D over space and n over time amp space gives DV2C vVC CVv 2 9 For steadystate ow Vv 0 and DV2C vVC 6 10 A noadvection condition reverts 10 to 6 16C 16t V2C A steadyCprofile condition leads to DV2C vVC0 11 Eq 10 11 are called the advection diffusion equations 5 The Advection Dispersion Equation Combining diffusion and mechanical dispersion processes and assuming the hydrodynamic dispersion coefficient D D D constant over space yields aC 6t DVzC vVC CVv 12 Eq 12 is the advection dispersion equation If D small 12 reverts to the advection diffusion equation further if no advection it reduces to the diffusion equation

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