Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug...
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Transcript of Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug...
![Page 1: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/1.jpg)
Finite Difference Solutionsto the ADE
![Page 2: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/2.jpg)
t
c
x
cv
x
cD
2
2Simplest form of the ADE
t
c
x
cv
Even Simpler form
Plug FlowPlug Source
Flow Equationt
hS
x
hT
2
2
![Page 3: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/3.jpg)
Effect ofNumerical Errors
(overshoot)
(MT3DMS manual)
![Page 4: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/4.jpg)
t
c
x
cv
(See Zheng & Bennett, p. 174-181)
v
j-1 j j+1
x
x
t
cc
x
ccv
nj
nj
nj
nj
11 )(Explicit approximation
with upstream weighting
![Page 5: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/5.jpg)
t
c
x
cv
t
cc
x
ccv
nj
nj
nj
nj
11 )(Explicit;
Upstream weighting
(See Zheng & Bennett, p. 174-181)
v
j-1 j j+1
x
x
![Page 6: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/6.jpg)
Example from Zheng &Bennett
v = 100 cm/h
l = 100 cm
C1= 100 mg/l
C2= 10 mg/l
With no dispersion,breakthrough occursat t = v/l = 1 hour
![Page 7: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/7.jpg)
nj
nj
nj
nj ccc
l
tvc
)( 1
1
t
cc
x
ccv
nj
nj
nj
nj
11 )(
v = 100 cm/hrl = 100 cmC1= 100 mg/lC2= 10 mg/lt = 0.1 hr
Explicit approximation with upstream weighting
![Page 8: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/8.jpg)
t
cc
x
ccv
nj
nj
nj
nj
111
11 )
2(
Implicit;central differences
t
cc
x
ccv
nj
nj
nj
nj
1111 )(
t
cc
x
ccv
nj
nj
nj
nj
111
1
)(
Implicit;upstream weighting
Implicit Approximations
![Page 9: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/9.jpg)
![Page 10: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/10.jpg)
= Finite Element Method
![Page 11: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/11.jpg)
t
c
x
cv
x
cD
2
2
Governing Equationfor Ogata and Banks solution
![Page 12: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/12.jpg)
j-1 j j+1
x
x
j-1/2 j+1/2
![Page 13: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/13.jpg)
t
c
x
cv
x
cD
2
2Governing Equationfor Ogata and Banks solution
t
cc
x
ccv
x
cccD
nj
nj
nj
nj
nj
nj
nj
11
2
11)()
)(
2(
Finite difference formula:explicit with upstream weighting, assuming v >0
)()2()(
11121 n
jnj
nj
nj
nj
nj
nj cc
x
tvccc
x
tDcc
Solve for cj n+1
![Page 14: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/14.jpg)
Stability Constraints for the 1D Explicit Solution(Z&B, equations 7.15, 7.16, 7.36, 7.40)
Courant Numberx
tvCr
Cr < 1
1)(
22
x
tv
x
tDStability Criterion
Peclet Numberx
D
xvPe
Controls
numerical dispersion& oscillation, see Fig.7.5
![Page 15: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/15.jpg)
Co
Boundary Conditions
a “free massoutflow” boundary(Z&B, p. 285)
Specifiedconcentrationboundary
Cb= Co Cb= Cjj j+1j-1 j j+1j-1
![Page 16: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/16.jpg)
Spreadsheet solution(on course homepage)
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We want to write a general formof the finite difference equation allowing foreither upstream weighting (v either + or –) or central differences.
![Page 18: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/18.jpg)
j-1 j j+1
x
x
j-1/2 j+1/2
![Page 19: Finite Difference Solutions to the ADE. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d355503460f94a0d599/html5/thumbnails/19.jpg)
Upstream weighting:
In general:
jjj ccc 12/1 1(
See equations7.11 and 7.17 inZheng & Bennett