Application of Linear Equation in Chemical Engineering Problem

Application of linear algebraic equation for chemical engineering problem The chemical engineering system models often outcome of set of linear algebraic equations. These problems may range in complexity from a set of two simultaneous linear algebraic equations to a set involving 1000 or even 10,000 equations. The solution of a set two or three linear algebraic equations can be obtained easily by the algebraic elimination of variables or by the application of cramer’s rule. However for systems involving five or more equations the algebraic elimination method becomes too complex. In this section, we give several examples of systems drawn from chemical engineering applications which yield sets of simultaneous linear algebraic equations. In the following sections of this unit we discuss few methods for the numerical solution of such problems and demonstrate the application of these methods on the personal computer. Material and energy balances are the primary tools of chemical engineers. Such balances applied to multistage or multicomponent processes result in sets of equations. The general form of conserved quantity (Φ) in the chemical engineering system is given below

Rate of input of Φ - Rate of output of Φ + Rate of generation of Φ = Rate of accumulation of Φ

Rate of input of Φ or rate of output of Φ = mass flow rate or mole flow rate or volumetric flow rate Rate of generation of Φ = generation rate or depletion rate per unit volume (r) x volume of the system Depletion rate = - generation rate

Where r is rate of reaction

Rate of accumulation of Φ = the time rate of change of that particular quantity within the volume of the system At steady state, the accumulation term become zero Steady state The term steady state means that at a particular location in space the dependent variable does not change as a function of time. If the dependent variable is Φ, then

(∂Φ/∂t)x,y,z = 0
A classical example of the use of these techniques is in the analysis of