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1-, 2-, and 3-D Analytical Solutions to CDE

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1-, 2-, and 3-D Analytical Solutions to CDE. Equation Solved:. Constant mean velocity in x direction!. Resident and Flux Concentrations. Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85. ‘Instantaneous’ Source. - PowerPoint PPT Presentation
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1-, 2-, and 3-D Analytical Solutions to CDE
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Page 1: 1-, 2-, and 3-D Analytical Solutions to CDE

1-, 2-, and 3-D Analytical Solutions to CDE

Page 2: 1-, 2-, and 3-D Analytical Solutions to CDE

Equation Solved:

• Constant mean velocity in x direction!

t

CR

z

CD

y

CD

x

CD

x

Cv zzyyxx

2

2

2

2

2

2

Page 3: 1-, 2-, and 3-D Analytical Solutions to CDE

Inlet BC Exit BC Solution Reference

A-1 0),0( CtC

0),(

tx

C

2/12/10 2

exp2

1

22

1

DRt

vtRxerfc

D

vx

DRt

vtRxerfc

C

C

Lapidus and Amundson, 1952. The mathematics of adsorption in beds. IV. The effect of longitudinal diffusion in ion exchange chromatographic columns. J. Phys Chem. 56:984-988.

A-2 0

0

vCvCx

CD

x

0),(

tx

C

2/1

2

22/12

2/10

2exp1

2

1

4exp

22

1

DRt

vtRxerfc

D

vx

DR

tv

D

vx

DRt

vtRx

DR

tv

DRt

vtRxerfc

C

C

Lindstrom, Hague, Freed, and Boersma, 1967. Theory on the movement of some herbicides in soils: linear diffusion and convection of chemicals in soils. Environ. Sci. Technol. 1:561:565

A-3 0),0( CtC

0),(

tLx

C

02

)cot(

22

42expsin2

11

22

2

22

0

D

vL

D

vL

D

vL

RL

Dt

DR

tv

D

vx

L

x

C

C

mm

m

m

mmm

Cleary and Adrian, 1973. Analytical solution of the convective dispersive equation for cation adsorption in soils. Soil Sci. Am. Proc. 37:197-199.

A-4 0

0

vCvCx

CD

x

0),(

tLx

C

04

)cot(

22

42expsin

2cos

2

1

2

12

22

2

2

22

0

D

vL

vL

D

D

vL

D

vL

D

vL

RL

Dt

DR

tv

D

vx

L

x

D

vL

L

x

D

vL

C

C

mmm

m

mm

mmmmm

Brenner, 1962. The diffusion model of longitudinal mixing in beds of finite length. Numerical values. Chem. Eng. Sci. 17:229-243.

Page 4: 1-, 2-, and 3-D Analytical Solutions to CDE

Resident and Flux Concentrations

x

CDCuCu r

rf

Page 5: 1-, 2-, and 3-D Analytical Solutions to CDE

•Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85.

Page 6: 1-, 2-, and 3-D Analytical Solutions to CDE

‘Instantaneous’ Source• Solute mass only

– M1, M2, M3

• Injection at origin of coordinate system (a point!) at t = 0

• Dirac Delta function– Derivative of Heaviside:

)(x

dx

xHd

)()()( afdxaxxf

Page 7: 1-, 2-, and 3-D Analytical Solutions to CDE

‘Continuous’ Source• Solute mass flux

– M1, M2, M3 = dM1,2,3/dt

• Injection at origin of coordinate system (a point!)

Page 8: 1-, 2-, and 3-D Analytical Solutions to CDE

Instantaneous and Continuous Sources

• 1-D

tD

vtx

tD

MC

xxxx

i 4exp

2

21

tD

vtxerfc

D

vx

tD

vtxerfc

D

vx

v

Dxv

M

C

xxxx

xxxxxxc

22exp

22exp

2

2exp1

Page 9: 1-, 2-, and 3-D Analytical Solutions to CDE

2-D Instantaneous Source

tD

y

tD

vtx

DDt

MC

yyxxyyxx

i 44exp

4

222

Page 10: 1-, 2-, and 3-D Analytical Solutions to CDE

2-D Instantaneous Source (MATLAB)• %Hunt 1978 2-D dispersion solution Eqn.14.• clear• close('all')

• [x y] = meshgrid(-1:0.05:3,-1:0.05:1);

• M2=1• Dyy=.0001• Dxx=.001• theta=.5• V=0.04

• for t=1:25:51

• data = M2*exp(-(x-V*t).^2/(4*Dxx*t)-y.^2/(4*Dyy*t))/(4*pi*t*theta*sqrt(Dyy*Dxx));

• contour(x, y, data)• axis equal• hold on• clear data

• end

tD

y

tD

vtx

DDt

MC

yyxxyyxx

i 44exp

4

222

Page 11: 1-, 2-, and 3-D Analytical Solutions to CDE

2-D Instantaneous Source Solution

Dyy

Dxx

Back dispersion Extreme concentration

t = 1t = 25

t = 51

Page 12: 1-, 2-, and 3-D Analytical Solutions to CDE

3-D Instantaneous Source

tD

z

tD

y

tD

vtx

DDDt

MC

zzyyxxzzyyxx

i 444exp

8

222

33

3

Page 13: 1-, 2-, and 3-D Analytical Solutions to CDE

3-D Instantaneous Source (MATLAB)• %Hunt 1978 3-D dispersion solution Eqn.10.

• clear• close('all')• [x y z] = meshgrid(-1:0.05:3,-1:0.05:1,-1:0.05:1);

• M3=1• Dxx=.001• Dyy=.001• Dzz=.001• sigma=.5• V=0.04

• for t=1:25:51

• data = M3*exp(-(x-V*t).^2/(4*Dxx*t)-y.^2/(4*Dyy*t)-z.^2/(4*Dzz*t))/(8*sigma*sqrt(pi^3*t^3*Dxx*Dyy*Dzz));

• p = patch(isosurface(x,y,z,data,10/t^(3/2)));• isonormals(x,y,z,data,p);• box on

• clear data• set(p,'FaceColor','red','EdgeColor','none');• alpha(0.2)• view(150,30); daspect([1 1 1]);axis([-1,3,-1,1,-1,1])• camlight; lighting phong;• hold on

• end

tD

z

tD

y

tD

vtx

DDDt

MC

zzyyxxzzyyxx

i 444exp

8

222

33

3

Page 14: 1-, 2-, and 3-D Analytical Solutions to CDE

3-D Instantaneous Source SolutionDzz

Dxx

Back dispersion

Extreme concentration

t = 1t = 25

t = 51

Dyy

Page 15: 1-, 2-, and 3-D Analytical Solutions to CDE
Page 16: 1-, 2-, and 3-D Analytical Solutions to CDE
Page 17: 1-, 2-, and 3-D Analytical Solutions to CDE

3-D Continuous Source

tD

vtRerfc

D

Rv

tD

vtRerfc

D

Rv

DDR

Dxv

M

C

xxxx

xxxx

zzyy

xxc

22exp

22exp

8

2exp3

zz

xx

yy

xx

D

Dz

D

DyxR 222

Page 18: 1-, 2-, and 3-D Analytical Solutions to CDE

StAnMod (3DADE)

• Same equation (mean x velocity only)

• Better boundary and initial conditions

• Leij, F.J., T.H. Skaggs, and M.Th. Van Genuchten, 1991. Analytical solutions for solute transport in three-dimensional semi-infinite porous media, Water Resources Research 20 (10) 2719-2733.

Page 19: 1-, 2-, and 3-D Analytical Solutions to CDE

Coordinate systems

• x increasing downward

x

z

y

x

z

y

r

Page 20: 1-, 2-, and 3-D Analytical Solutions to CDE

Boundary Conditions

• Semi-infinite source

x

z

y

-∞

-∞

Page 21: 1-, 2-, and 3-D Analytical Solutions to CDE

Boundary Conditions

• Finite rectangular source

x

z

y

-b

-a

b

a

Page 22: 1-, 2-, and 3-D Analytical Solutions to CDE

Boundary Conditions

• Finite Circular Source

x

z

y

r = a

Page 23: 1-, 2-, and 3-D Analytical Solutions to CDE

Initial Conditions

• Finite Cylindrical Source

x

z

y

r = a

x1

x2

Page 24: 1-, 2-, and 3-D Analytical Solutions to CDE

Initial Conditions

• Finite Parallelepipedal Source

x

z

y

x1

x2

b

a

Page 25: 1-, 2-, and 3-D Analytical Solutions to CDE

Comparing with Hunt

• M3 = r2 (x1 – x2) Co (=1, small, high C)

• Co = 1/[r2 (x1 – x2)] = 106 for r = x= 0.01 x

z

y

r =

a

x 1 x 2

Page 26: 1-, 2-, and 3-D Analytical Solutions to CDE

Wells?• Finite Parallelepipedal Source

x

z

y

x1

x2

b

a


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