Numerical Methods in Diffusion

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

mailto:[email protected]


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


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


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


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)


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



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)



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)


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)


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...


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.


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


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.


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


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)


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


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


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.


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.



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)



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…



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!


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

Numerical Methods in Diffusion

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