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Application of linear algebric equation in chemical engineering

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Here.two simple applications of linear algebraic equations is explained with example.

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Application of linear algebric equation in chemical engineering

  1. 1. APPLICATION OF LINEAR ALGEBRIC EQUATION IN CHEMICAL ENGINEERING Prepared By: Lakhlani Nishith
  2. 2. Linear Algebraic Equation  The definition of a linear equation is an algebraic equation in which each term has an exponent of one.  The graphing of the equation results in a straight line.  One or more variables in the equation.
  3. 3. Application In Chemical Engineering  One of the most important organizing principles in chemical engineering is the Conservation of Mass.  The principle of mass conservation, states that for any closed system the mass of the system must remain constant over time.
  4. 4. Cont..  independent of any chemical and physical changes taking place within the system.  For stable condition(i.e., steady-state) it can be represented as: Input = output
  5. 5. Example-1  Suppose we are performing a mass balance for a conservative substance (i.e., one that doesn’t increase or decrease due to chemical transformation) in a reactor, we would have to quantify the rate mass flows into the reactor through the two inflow pipes and out of the reactor through outflow pipe.
  6. 6. Cont..  For pipe 1,product flow rate Q1=2 m3/min , Q2= 1.5 m3/min and concentrate C1=25 mg/m3 C2= 10 mg/m3 ; therefore, the rate at which mass flows into reactor through pipe 1 and pipe 2 accordingly.
  7. 7. Cont..  Because of Steady state of Reactor  Input = Output , according to that Q1C1 + Q2C2 = Q3C3 50 + 15 = 3.5 C3 C3 =18.6 mg/m3
  8. 8. Example-2 Problem  By examination we can see that the following equation is not balanced. CH4 + O2 CO2+H2O
  9. 9. Cont.. Solution  First assign variables to each of the unknown coefficients in the equation which gives us: wCH4 + xO2 yCO2 + zH2O
  10. 10. Cont.. CARBON w = y There is 1 carbon atom in the w term and 1 in the y term. HYDROGEN 4w = 2z There are 4 Hydrogen atoms in the w term and 2 in the z term. OXYGEN 2x = 2y + z There are 2 oxygen atoms in the x term, 2 in the y term, and 1 in the z term.
  11. 11. Cont..  Rewrite the linear equations in standard form to get a homogeneous system of equations with 4 variables: w-y = 0 4w-2z = 0 2x-2y-z = 0  Create a matrix for the above systems of equations augmented with zero’s (left) and perform the Gauss-Jordan elimination method to reduce the matrix(right).
  12. 12. Cont..  This gives us the following values for our variables. w = 1/2z x = 1z y = 1/2z  above equations we calculate the values of our 4 variables to be: w x y z 1 2 1 3
  13. 13. Cont..  Replace these values as the coefficients to our original equation. CH4 + 2O2 CO2 + 2H2O

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