Upload
carmella-ray
View
213
Download
0
Tags:
Embed Size (px)
Citation preview
MSE 430 © 2006, J.C.LaCombe 1
Numerical Methods in Diffusion
Numerical Methods in Diffusion
Portions of this lecture were adapted fromElements of Heat and Mass Transfer, 3rd ed., F.P. Incropera, and D.P. De WittJohn Wiley & Sons, NY, 1990
J.C. LaCombeUniversity of Nevada, Reno
Reno, NV, [email protected]
These lecture notes complement an online learning module and diffusion simulation software that can be found at:
http://unr.edu/homepage/lacomj/Diffusion/index.htm
MSE 430 © 2006, J.C.LaCombe 2
Numerical MethodsIntroduction
In many real-world problems, the details of the system may not correspond to a known solution to the diffusion equation. In these cases, it is usually possible to produce a solution where the equation is solved through iterative techniques. This is generally a lot of work to do by hand. However, with the aid of a computer, this becomes possible. The techniques used to reach such solutions are known as numerical methods.
Situations that often require numerical methods include
• Multi-dimensional problems (not simply 1-D)
• Complex Problems
•Complex Geometry/Shape
•Complex boundary conditions
•Complex initial conditions
MSE 430 © 2006, J.C.LaCombe 3
Elements and NodesDiscretization of the
Problem
The numerical methods we will use in this course solve complex diffusion problems by breaking up the system into manageable parts and solving the diffusion equations for each part simultaneously with all the other parts.
To do this, we discretize the problem mathematically by dividing up space into little elements or nodes and treating time as moving forward in small steps.
Element
Node
Component divided into elements
MSE 430 © 2006, J.C.LaCombe 4
Discretization of the ProblemDiscretization of the
Problem
Each element is identified using subscripts, and has a finite dimensions.
2
x
2
yNodem,n
x
y
MSE 430 © 2006, J.C.LaCombe 5
Fick’s Laws in 2-DThe Finite-Difference
Approach
Recall: Fick’s 1st law simply tells us how solute will flow if there is a concentration gradient.
Recall: Fick’s 2nd law is simply a combination of Conservation of Mass with Fick’s 1st law.
2
2
2
2
2
1
y
C
x
C
t
C
D
CDt
C
Fick’s 2nd Law in 2-D Cartesian Coordinates.
Before we solve this, we need to re-write the equation into a discretized form. The approach presented here is known as a finite-difference solution approach.
(1)
MSE 430 © 2006, J.C.LaCombe 6
Discretizing the Spatial DerivativesThe Diffusion Equation
Consider first, the spatial 2nd derivatives on the RHS of Equation (1).
The 2nd derivatives can be thought of more simply as the slope of the 1st derivatives. This is written (approximated) in the x-direction as
2
2
2
21
y
C
x
C
t
C
D
(1)
xx
C nmxC
nmxC
nm
,,
,
2
221
21
(2)
Note that the 1st derivatives here are simply the concentration gradients. We can use the central difference approximation to determine these…
Slope of the 1st derivative
Gradient at RHS of CV
Gradient at LHS of CV
MSE 430 © 2006, J.C.LaCombe 7
Discretizing the Spatial DerivativesThe Diffusion Equation
m,n m+1,nm-1,n
m-1,n+1 m+1,n+1m,n+1
Eq. (2) further simplifies if we can determine the concentration gradient at the midpoint between nodes.
The gradient can be estimated using the concentration values at the neighboring nodes and the distance between the nodes.
Thus,
x
CC
x
C nmnm
nm
,,1
,21
m m+1m-1
21m 2
1m
x x
x
CC
x
C nmnm
nm
,1,
,21
(3)
(4)
Evaluate gradient here
C(x)
xx
C nmxC
nmxC
nm
,,
,
2
221
21
(2)
MSE 430 © 2006, J.C.LaCombe 8
Discretizing the Spatial DerivativesThe Diffusion Equation
Equations (3) and (4) can now be substituted into (2) to produce the discretized form of the 2nd spatial derivative in the x direction.
2,,1,1
,
2
2 2
x
CCC
x
C nmnmnm
nm
(5a)
The 1st and 2nd derivatives in the y (and z) directions can also be evaluated in a similar manner…
2,1,1,
,
2
2 2
y
CCC
y
C nmnmnm
nm
(5b)
These make up the RHS of Fick’s 2nd Law (Eq. 1)
MSE 430 © 2006, J.C.LaCombe 9
Discretizing the Time DerivativeThe Diffusion Equation
Now that we have discretized Fick’s 2nd law in space (1), we must discretize it in time as well. To do this, we will introduce a new variable, p, that is an integer that represents the time step. The duration of each step is t. Thus, the total time, t, is written…
t
CC
t
C pnm
pnm
,1
,
tpt (6)
The finite-difference approximation to the time derivative (the LHS of Fick’s 2nd law) is then expressed as…
(7)
The superscript, p, denotes the time dependence. The time derivative is expressed in terms as the difference in concentrations between the new time (step p+1) and the previous time (step p).
This is the LHS of Fick’s 2nd Law (Eq. 1)
MSE 430 © 2006, J.C.LaCombe 10
The 2-D Diffusion EquationFick’s 2nd Law in Discretized Form
We present here a solution approach known as the explicit method. In this finite-difference scheme, the concentration at any node m,n at time t+t is calculated from knowledge of the concentration at the same and neighboring nodes for the preceding time t.
We now can combine (5a,b) and (7) to produce the discretized form of the diffusion equation, (1).
(8)
Note:
Other approaches, such as the implicit method, are more efficient with a computer, but require more complex algorithms. Nonetheless, the fundamental principles are the same as we are applying here. We will not be covering these other methods in this course.
2,1,1,
2,,1,1,
1, 221
y
CCC
x
CCC
t
CC
D
pnm
pnm
pnm
pnm
pnm
pnm
pnm
pnm
Fick’s 2nd Law in discretized form...
MSE 430 © 2006, J.C.LaCombe 11
The Diffusion EquationThe Fourier Number
Equation (8) is the general form of our solution. We can simplify the notation a bit if we use square elements, so that x = y. Additionally, we can form the following group of parameters, which is commonly known as the dimensionless Fourier Number, Fo.
(9)
pnm
pnm
pnm
pnm
pnm
pnm CFoCCCCFoC ,1,1,,1,11
, 41
2x
tDFo
Now, we can re-arrange (8) to solve for the concentration in node m,n at the new time step, p+1. This equation applies to any element/node on the interior of a component. The expression simplifies to…
(10)2-D Interior Node
The NEW composition in an element is calculated using the PREVIOUS compositions in the element and its neighbors.
MSE 430 © 2006, J.C.LaCombe 12
The 1-D Diffusion EquationExplicit Method & Solution Stability
For the case of 1-D transport, Equation (8) would instead develop into the form of
pm
pm
pm
pm CFoCCFoC 2111
1
The accuracy of finite-difference solutions may be improved by decreasing the values of x and t (I.e., finer discretization). On the other hand, making these values larger will allow the calculation to proceed more quickly.
One additional limitation of the explicit method is that it is not always a stable solution. If the values of x and t are not small enough, it can cause the solution to oscillate (even when this is physically impossible).
(11)1-D Interior Node
MSE 430 © 2006, J.C.LaCombe 13
Critical Values of FoStability Criteria
To prevent such erroneous results when the solution is “unstable”, the values of x and t must meet certain criteria (details omitted). For interior nodes, these are,
21Fo
41Fo
1-D Stability Criteria
2-D Stability Criteria
2x
tDFo
Recalling,
So, once you pick a value of either x or t , the other value must be chosen so that the stability criteria is met. Simply re-arrange the equation for Fo to calculate the acceptable value.
MSE 430 © 2006, J.C.LaCombe 14
Other Element ConfigurationsZero-Flux Elements
and Surfaces
The equations presented so far (10, 11) are for interior nodes. I.e., each element’s surroundings are geometrically the same in all directions.
We can develop similar equations for different element types, but we need to be clever, or it gets messy.
A surface node (with no flux flowing through the surface), can be modeled using the same equation as an interior node. All we need to do is include an imaginary node just outside the surface and set its composition to the same as the node just inside the surface. This has the effect of producing a zero net gradient through the surface.
I.e., if the surface node is then the no-flux condition is modeled by adding an imaginary node at m+1 and setting to achieve a state of no-flux at node m (no gradient means no net flux).
m,nm-1,n
m-1,n+1 m,n+1
mm-1
21m
m,n-1m-1,n-1
Ex
tern
al
Su
rfa
ce
(ze
ro-f
lux
pla
ne
)
m+1Imaginary
node
pm
pm CC 11
pmC
MSE 430 © 2006, J.C.LaCombe 15
Other Element ConfigurationsZero-Flux Elements
and Surfaces
So, we can model a surface node by modifying the equation for an interior node. Recalling Equation (10) for 2-D,
pm
pm
pm CFoCFoC 212 1
1
pm
pm CC 11
We then incorporate the imaginary node…
pnm
pnm
pnm
pnm
pnm
pnm CFoCCCCFoC ,1,1,,1,11
, 41 2-D Interior
Node
And are left with the equation for a surface node…
pnm
pnm
pnm
pnm
pnm CFoCCCFoC ,1,1,,11
, 412 2-D Surface
Node
In 1-D, this would work out to…
1-D Surface Node
(12)
(10)
(13)
MSE 430 © 2006, J.C.LaCombe 16
Developing Expressions for OtherElement Types
Other Solution Approaches
The method used on the previous slides to discretize the problem is not the only way to produce equations such as Equations (10)-(13).
Another method can be used to provide even greater flexibility with boundary conditions. It is simply based on conservation of mass (Recall that Fick’s 2nd law is also essentially this as well).
Let us consider the element surrounding each node to be subject to conservation of mass. This would be written as…
Solid State Diffusive Flux
Solute “Generated”
Stored Mass
+ =
In practice, this can be something like solute entering an element at external surface
MSE 430 © 2006, J.C.LaCombe 17
Other Ways to Discretize the Problem
Other Solution Approaches
Writing this for a generic interior node, we account for all possible influences. As before, minor changes can be made for an external node. Note that here, flux into the node is considered “positive”.
x
CCDJ
x
CCDJ
pnm
pnm
m
pnm
pnm
m
,1,1
,1,1
1mJ
1nJ
1nJ
1mJ
genM
storedC
xACMAJAJAJAJ storedgennnmm
1111
y
CCDJ
y
CCDJ
pnn
pnm
n
pnn
pnm
n
,1,1
,1,1
Where,
t
CCC
pnm
pnm
stored
,1
,
(14)
unit timeper "Created" MassgenM
MSE 430 © 2006, J.C.LaCombe 18
Other Ways to Discretize the Problem
Diffusion with “Mass Generation”
When massaged, Equation (14) evolves into the same form as the earlier equations (10)-(13), except now, we have added in the solute generation term.
pnmgen
pnm
pnm
pnm
pnm
pnm CFoMCCCCFoC ,1,1,,1,11
, 41 (15)
And in 1-D, this is
pnmgen
pnm
pnm
pm CFoMCCFoC ,,1,1
1 21 (16)
Thus, there are a variety of approaches to produce the discretized diffusion equations for a variety of different element types.
MSE 430 © 2006, J.C.LaCombe 19
1-D Thick Diffusion CoupleExample
An earlier topic presented the analytical solution to the case of a binary diffusion couple. Let’s analyze this using a finite-difference model. Assume D = 110-9 cm2/s, and the initial compositions are Cl = 0.75, and CR = 0.25.
x110-3 cm
1 2 3 4 5 675.0lC 25.0RC
st
cmt
scm
500
101
101
2
129
23
First, the stability criteria for this 1-D arrangement is that Fo ½.
2
12
x
tDFo
Thus the maximum time step for a stable solution of this problem is 500 seconds.
MSE 430 © 2006, J.C.LaCombe 20
1-D Thick Diffusion CoupleExample
Equation (11) is the suitable solution form:
To handle the “infinite” ends, we treat them as having no-flux conditions (I.e., the concentration gradient is zero at the ends) using Equation (13). This will be ok, provided that the concentration field never reaches the end during our simulation. The equations are written for each of the 6 elements…
pm
pm
pm
pnm CFoCCFoC 21111
,
ppp
pppp
pppp
pppp
pppp
ppp
CFoCFoC
CFoCCFoC
CFoCCFoC
CFoCCFoC
CFoCCFoC
CFoCFoC
651
6
5461
5
4351
4
3241
3
2131
2
121
1
212
21
21
21
21
212
These 6 equations must be solved at each time step, p.
There will be one equation for each node. Models with lots of elements involve solving lots of equations.
(1-D Interior Node)
MSE 430 © 2006, J.C.LaCombe 21
1-D Thick Diffusion CoupleExample
This is expressed more concisely in matrix form.
16
15
14
13
12
11
6
5
4
3
2
1
2120000
21000
02100
00210
00021
0000221
p
p
p
p
p
p
p
p
p
p
p
p
C
C
C
C
C
C
C
C
C
C
C
C
FoFo
FoFoFo
FoFoFo
FoFoFo
FoFoFo
FoFo
The new concentrations at each node are calculated by solving this matrix at each time step. At each step, you use the resulting concentrations from the previous step, Cp+1, as the new values of Cp. Likewise, to get it all started, you just use the initial concentrations at each node.
You can use whatever methods or software you want to solve the matrix. Even a spreadsheet will work…
MSE 430 © 2006, J.C.LaCombe 22
1-D Thick Diffusion CoupleExample
MS Excel Worksheet…Fo= 0.05 dt 50
Node Old C New Cp=1 1 0.9 0.1 0.75 0.750
2 0.05 0.9 0.05 0.75 0.7503 0.05 0.9 0.05 0.75 0.7254 0.05 0.9 0.05 0.25 0.2755 0.05 0.9 0.05 0.25 0.2506 0.1 0.9 0.25 0.250
p=2 1 0.9 0.1 0.75 0.7502 0.05 0.9 0.05 0.75 0.7493 0.05 0.9 0.05 0.725 0.7044 0.05 0.9 0.05 0.275 0.2965 0.05 0.9 0.05 0.25 0.2516 0.1 0.9 0.25 0.250
p=3 1 0.9 0.1 0.75 0.7502 0.05 0.9 0.05 0.7488 0.7473 0.05 0.9 0.05 0.7038 0.6864 0.05 0.9 0.05 0.2963 0.3145 0.05 0.9 0.05 0.2513 0.2536 0.1 0.9 0.25 0.250
p=4 1 0.9 0.1 0.7499 0.7502 0.05 0.9 0.05 0.7466 0.7443 0.05 0.9 0.05 0.6856 0.6704 0.05 0.9 0.05 0.3144 0.3305 0.05 0.9 0.05 0.2534 0.2566 0.1 0.9 0.2501 0.250
Fo = 0.05
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1 2 3 4 5 6Node
C
p=1
p=2
p=3
p=4
Initial
Exact 200s
RRl C
Dt
xCCtxC
2
erfc 2
),(
Solution, Analytical theRecalling
Double-click above (ppt only) to open the actual spreadsheet!
MSE 430 © 2006, J.C.LaCombe 23
2-D Explicit Finite Difference Equations (x=y)
2-D Equation Summary
m,n
m+1,nm-1,n
m,n-1
m,n+1
m,n
m+1,nm-1,n
m,n-1
m,n+1
m,n
m+1,nm-1,n
m,n-1
m,n+1
m,n
m+1,nm-1,n
m,n-1
m,n+1
Interior Node
Interior Corner
Plane Surface
Exterior Corner
p
nm
pnm
pnm
pnm
pnm
pnm
CFo
CCCCFoC
,
1,1,,1,11
,
41
41
CriterionStability
Fo
41
CriterionStability
Fo
41
CriterionStability
Fo
41
CriterionStability
Fo
p
nm
pnm
pnm
pnm
pnm
CFo
CCCFoC
,
1,1,,11
,
41
2
p
nm
pnm
pnm
pnm
pnm
pnm
CFo
CCCCFoC
,
1,1,,1,1321
,
41
22
pnm
pnm
pnm
pnm
CFo
CCFoC
,
1,,11
,
41
2