Updated: 23 October 2017 Print version Lecture #9 (Diffusion) Chapra L8 David A. Reckhow CEE 577 #9 1
Forge Pond project Continued discussion Update questions EPA 305(b) listing for Forge Pond David A. Reckhow CEE 577 #9 2
Liquid Water Transport Advection: unidirectional flow Diffusion: movement of mass that is not unidirectional flow; usually movement in an unorganized fashion Dispersion Eddy Diffusion Molecular Diffusion David A. Reckhow CEE 577 #9 3
Incorporates molecular movement Mass Diffusion and interfacial area Bulk Diffusion (m 2 /yr) T=0 V 1 , c 1 V 2 , c 2 dc ( ) ′ = − 1 V D c c 1 2 1 dt T=1 Concentration Gradient T=2 T=large David A. Reckhow CEE 577 #9 4
Fick’s First Law Mass flux is proportional to the concentration gradient and a diffusion coefficient dc = − J x D dx David A. Reckhow CEE 577 #9 5
Bulk Diffusion Coefficient The mixing dc = − 1 V JA length V 1 , c 1 V 2 , c 2 1 c dt − dc c c dc ≅ = − 2 1 J x D dx dx dc DA = − And combining all three: 1 c V ( c c ) 1 2 1 dt D’ EA ′ E = c Similar for Eddy Diffusion David A. Reckhow CEE 577 #9 6
Dispersion Differences in velocities of parallel flow paths David A. Reckhow CEE 577 #9 7
Embayment Model Q 2 Main W 1 Bay Lake (2) (1) W 2 Q 1 dc ′ = − − + + − 1 ( ) V W Q c k V c Q c E c c 1 1 1 1 1 1 1 2 2 2 1 dt dc ′ = − − + − 2 V W Q c k V c E ( c c ) 2 2 2 2 2 2 2 1 2 dt David A. Reckhow CEE 577 #9 8
Embayment Model with a Conservative Substance Conservative substances (s) are those that do not undergo degradation, thus k=0 The mass balance on the bay (2), then becomes: ds ′ = − + − 2 V W Q s E ( s s ) 2 2 2 2 1 2 dt And solving for the bulk diffusion coefficient: − W Q s ′ = 2 2 2 E − s s 2 1 David A. Reckhow CEE 577 #9 9
Map of Huron/Saginaw System Lake Huron Saginaw Bay David A. Reckhow CEE 577 #9 10
Parameters for Saginaw Bay Units Parameter Symbol Value 10 9 m 3 Volume V 2 8 m Depth H 2 5.81 10 6 m 2 Surface Area A 2 1,376 10 9 m 3 /yr Outflow Q 2 7 mg/L Chloride Conc. s 2 15.2 10 12 g/yr Chloride Loading W s2 0.353 10 12 mg/yr Phosphorus Loading W p2 1.42 David A. Reckhow CEE 577 #9 11
Parameters for Lake Huron Units Parameter Symbol Value 10 9 m 3 Volume V 1 3507 m Depth H 1 60.3 10 6 m 2 Surface Area A 1 58,194 10 9 m 3 /yr Outflow Q 1 161 mg/L Chloride Conc. S 1 5.4 10 12 g/yr Chloride Loading W s1 ~0 10 12 mg/yr Phosphorus Loading W p1 4.05 David A. Reckhow CEE 577 #9 12
Chloride Tracer Model − [ ] W Q s ′ ( ) = 2 2 2 − E 12 9 0 . 353 x 10 7 x 10 15 . 2 − ′ = s s E 2 1 − 15 . 2 5 . 4 = 9 3 25 . 2 10 / x m yr EA ( ) ′ E = c ′ 9 3 E 2 . 52 x 10 10 x 10 m = = E 6 2 A 0 . 17 x 10 m c = 9 2 1 . 48 10 / x m yr = 5 2 4 . 7 x 10 cm / s David A. Reckhow CEE 577 #9 13
To next lecture David A. Reckhow CEE 577 #9 14
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