three minerals-2acid by newton raphson

7/31/2019 Three Minerals-2acid by Newton Raphson

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THREE MINERALSTWO ACID MODEL-1D

For the 3 minerals2 acid model with the following stoichiometry,

Solve the system of equations numerically and plot the result at the end of the simulation

Where


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Solution

The discretization was carried out for fully implicit solution.

Note that,

[ ]

[ ]

Where represents location and represents time

We have

[ ]Similarly for the second mineral

[ ]

[ ]


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[

]

For the third mineral

[ ]

Discretizing the concentrations using central differencing

( ) (

)[ ]

Where

So,


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[ ]

Then we have

[ ]

Doing the same for the second acid,

( ) (

)

Then this gives

To solve all these equations simultaneously,


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[ ]

[ ]

[ ]

[ ]

Then we implement Newton Raphsons iteration

The derivative of residual function of concentration of acid 1 at with respect to the variables at

[ ]


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The derivative of residual function of mineral 1 volume fraction at with respect to thevariables at

[ ]

[ ]


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[ ]

[ ]


* +[ ]

[

]

[ ]


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Then for the differentiation of the residuals at with respect to variables at neighboring grids

The derivative of residual function of concentration of acid 1 at with respect to the variables at {


{


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The derivative of residual function of mineral 1 volume fraction at with respect to thevariables at neighboring grids



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Adapting the boundary cells for boundary conditions

The injection dimensionless concentration is 1 for acid 1 and zero for acid 2.

For the first cell,

( ) [ ]


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[ ]

Then we have

[ ]

The residual is thus

[ ]

And

[ ]

Repeating the same for the second acid dimensionless concentration

For


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( )

Then this gives

So


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All derivatives for the residuals of the mineral volume fractions remain the same

The derivatives respect to the neighbors give

For the last cell, since the effluent concentration is unknown, we use backward differencing for

the space differential coefficient


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[ ]

[ ]

Then we have

[ ]

The residual is thus

[ ]

And

[ ]


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Repeating the same for the second acid dimensionless concentration

Then this gives

Thus


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All derivatives for the residuals of the mineral volume fractions remain the same

The derivatives respect to the neighbors give


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So the 5 variables will be taken to the new time level all together. In the MATLAB code, theconcentration of acid 1 will be referred to as that of acid 2 will be the mineral 1volume fraction as that of mineral 2 will be and for mineral 3 we will have it atWe will be solving a system of the form

[ ]

[ ]

[ ]

[ ]


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We conduct matrix multiplication of f and its transpose and use this as the error that must be

driven below the tolerance, the maximum value of the three is used

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Using the input data below, the numerical simulation was done using the MATLAB file below

and the results follows

three minerals-2acid by newton raphson

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