NTU, adsorption

8/13/2019 NTU, adsorption

1/16

Introduction to Computational Methodsin Chemical Engineering

by

Santosh Ansumali

and

Kwak Sang Kyu


2/16

2CH2007: Computational Methods in Chemical Engineering

Ordinary Differential Equations

For most of the Chemical engineering problems in real lifeanalytical solution is not possible!

With Initial condition

We need to solve them numerically!

We can rewrite our equation as:


3/16


Eulers Method

This method has an error of

Where

In general

Where


4/16


Implicit Methods

Forward Euler Method:

Evaluation of (n+1)th stage depends only information on nth stage !

Backward Eulers Method:

In general, we would need an iterative scheme to advance further !

In general, implicit method allows for much larger time steps!


5/16


Backward Euler Method: Linear

Case

1+!= ydt

dy

0)0( =yWith Initial Condition

Which can be made explicit as


6/16


Backward Euler Method: Non-

Linear Case

With Initial Condition


7/16


Multi-Step Method

We Know from central difference formula:

Leap-Frog Method:

Compare it with the trapezoidal rule result:

This is an example of 2-Step method!


8/16


Explicit Multi-Step Method

Explicit Adams-Bashforth Methods:

1 Step :

2 Step :

3 Step :

Accuracy is for a r-step method !


9/16


10/16


Single-Step versus Multi-Step

Method Single step methods are self starting while multi-

step methods require some other method to start

with.

The time step can be changed at any stage in

single step methods. Single step methods can work much better in

presence of discontinuity.

Higher order single step method may be more

expensive due to evaluation of function toomany time per time step.


11/16


Accuracy

Eulers Method:

TaylorsSeries:


12/16


Stability:Forward Euler

dy

dt= "100y +1

Forward Euler Scheme:

Take

Method is unstable !!!


13/16


Stability: Backward Euler

Both methods have similar error why forward Euler is unstablebut Backward Euler is stable?


14/16


Stability Analysis: Growth of

Error

Eulers Scheme:

Taylor Series for Exact Solution:

Subtracting the two equations give equation for error as:

By taking absolute values on both sides:


15/16


Stability Analysis

Which means:

Or,

Thus, we can say that error is growing if:

So Method is stable or error is bounded with time if:

Or,


16/16


Stability Analysis

Thus, forward Eulers method is stable only if:

Thus, we cannot take step size larger than this limit !

For Backward Euler:

Taking absolute magnitude, we see that:

So Method is stable for every value of h!

NTU, adsorption

Documents

Transcript of NTU, adsorption