Solving_Linear_Equations_with_CVX.m

%% Solving Linear Equations with CVX
%
% CVX is a Matlab toolbox for the modeling and solution of convex
% optimization problems. That may sound intimidating, but it turns out
% convex optimization applies to many material and energy balances used in
% process flowsheeting. In particular, the CVX toolbox provides an  elegant 
% way to represent and solve the linear material and energy balances 
% encountered in this course.

%% Downloading and Installing CVX
% 
% *Step 1* If you haven't already done so, download and install the current
% version of Matlab. Follow the instructions at 
% <https://oit.nd.edu/software-downloads/>
%
% *Step 2* Download CVX from <http://cvxr.com/cvx/download>. Download 
% |cvx-rd.zip| which is the redistributable version for Mac and PC
% systems. Double-click to unpack the downloaded package into a folder.
% Once complete you should have a folder name |cvx|. Move that folder into 
% your Matlab working directory.
%
% *Step 3* Start Matlab, then in Matlab navigate to the |cvx| directory you
% created in the previous step. Run the command |cvx_setup| in the Matlab 
% command window. Depending on your setup, you'll see a report describing 
% the installation and testing of CVX.

%% Example: Solve a simple system of linear equations
% This first example demonstrates solution of a simple set of linear
% equations. 
%
% <latex>
% \begin{eqnarray*}
% 3x + 4y = 12 \\
% 2x - 3y = 10
% \end{eqnarray*}
% </latex>
%
% CVX consists of specialized statements that are contained between
% |cvx_begin| and |cvx_end|.  The |variables| statement identifies the
% mathematical unknowns you wish to compute with CVX. Then the equations 
% are written with a double 'equals' sign |==| to distinguish them from 
% standard Matlab assignments. Following the |cvx_end| statement, CVX
% attempts to find a solution for the unknowns that satisfy the given
% equations.

cvx_begin
    variables x y 
    3*x + 4*y == 26;
    2*x - 3*y == -11;
cvx_end

display(['x = ',num2str(x)]);
display(['y = ',num2str(y)]);

%% Example: Murphy Example 2.8 -- Ammonia Synthesis
% The following model is a solution to Example 2.8 from the course
% textbook. Variable |X| denotes the extent of reaction.

cvx_begin quiet

    variables H1 N1 H2 N2 A2  X
    
    % Specifications
    N1 == 150;
    H1 == 4*N1;
    N2 == 0.3*N1;
    
    % Material Balances
    0 == N1 - N2 - X;
    0 == H1 - H2 - 3*X;
    0 ==    - A2 + 2*X;
    
cvx_end

display(['Extent of Reaction = ', num2str(X)]);