csh_lecture7_advection

csh_lecture7_advection - CWR 6537 Subsurface Contaminant...

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Unformatted text preview: CWR 6537 Subsurface Contaminant Hydrology Lecture 7 1 SOLUTE TRANSPORT: ADVECTION OR MASS FLOW A. Lecture Goals: To understand mass flow transport and, ultimately, mechanical dispersion, a known manifestation of spatial variations in advective transport. B. Advective Transport: Let us consider solute transport by advective flow alone; ignore diffusive transport for the moment. The total amount of solute transported per unit time across a unit cross section, designated by J c , is given by the product of volumetric flux (L 3 T -1 ) and solution concentration (ML-3 ). Thus, UnitArea 1 tion VolumeSolu MassSolute UnitTime tion VolumeSolu = J a ( 1 ) C q = C A Q = J L L a ( 2 ) where, q is a vector equal to Darcy's flux (LT -1 ). Note that J a has units of mass per time per unit area, (ML-2 T-1 ). Considering transport in one spatial direction, the continuity equation for simultaneous transport of water and solute is x ) C q (- = x J- = t ) C ( m L x m a L ( 3 ) where C L represents the total solute mass per unit soil volume. For steady one-dimensional water flow alone, and q x are constants. Thus, Eq. (3) can be restated as x C q- = t C m L x L ( 4 ) Dividing both sides by , and defining v x = (q x / ), CWR 6537 Subsurface Contaminant Hydrology Lecture 7 2 x C v- = t C m L x L ( 5 ) Note that v x is the linear flow velocity within the pores, while q x is water flux per unit area (L 3 L-2 T-1 ); v x is referred to as the pore-water velocity and has the units of (LT -1 ). In order to solve Eq.[5], we use a MOVING COORDINATE system (rather than a FIXED COORDINATE system). Let x 1 and t 1 be the variables in the new coordinate system, where t v- x = x x m 1 ( 6 ) t = t 1 ( 7 ) Notice the following relationships: v- = t x x 1 ( 8 ) 1 = t t 1 ( 9 ) 1 = x x m 1 ( 1 ) t t t C + t x x C = t C 1 1 L 1 1 L L ( 1 1 ) Now, we need to transform C L (x m ,t) to C L (x 1 ,t 1 ). The total change of C L with respect to time can be written using the chain rule: Substituting (8) and (9) into (11) gives t C + x C v- = t C 1 L 1 L x L ( 1 2 ) Then from using the chain rule with Eq. (10), CWR 6537 Subsurface Contaminant Hydrology Lecture 7 3 x C = x x x C = x C 1 L m 1 1 L m L ( 1 3 ) recalling Eq.(5), x C v- = t C m L x L ( 1 4 ) and combining it with equation (13) gives x C v- = t C 1 L x L ( 1 5 ) A comparison of equations (12) and (15) reveals the governing differential equation for the moving coordinate system: = t d ) t , x ( C d 1 1 1 L ( 1 6 ) or, C L (x 1 ,t 1 ) = constant in the moving coordinate system. Now, transforming back to the FIXED coordinate system, t v < x ; C = t) , x ( C x m o m L ( 1 7 ) t v > x ; = t) , x ( C x m m L ( 1 8 ) Equations (17) and (18) reflect a C Heaviside function (also called a unit step function)....
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This note was uploaded on 09/24/2011 for the course CWR 6537 taught by Professor Jawitz during the Spring '08 term at University of Florida.

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csh_lecture7_advection - CWR 6537 Subsurface Contaminant...

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