Chemical engineering lesson 2 math review systems and processes

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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review 2.0 context and direction Imagine a system that varies in time; we might plot its output vs. time. A plot might imply an equation, and the equation is usually an ODE (ordinary differential equation). Therefore, we will review the math of the first-order ODE while emphasizing how it can represent a dynamic system. We examine how the system is affected by its initial condition and by disturbances, where the disturbances may be non-smooth, multiple, or delayed. 2.1 first-order, linear, variable-coefficient ODE The dependent variable y(t) depends on its first derivative and forcing function x(t). When the independent variable t is t0, y is y0. a (t) dy + y( t ) = Kx ( t ) dt y( t 0 ) = y 0 (2.1-1) In writing (2.1-1) we have arranged a coefficient of +1 for y. Therefore a(t) must have dimensions of independent variable t, and K has dimensions of y/x. We solve (2.1-1) by defining the integrating factor p(t) p(t ) = exp ∫ dt a(t ) (2.1-2) Notice that p(t) is dimensionless, as is the quotient under the integral. The solution t p( t 0 ) y( t 0 ) K p( t ) x ( t ) y( t ) = + dt p( t ) p( t ) t∫0 a ( t ) (2.1-3) comprises contributions from the initial condition y(t0) and the forcing function Kx(t). These are known as the homogeneous (as if the right-hand side were zero) and particular (depends on the right-hand side) solutions. In the language of dynamic systems, we can think of y( ...
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Need help with Healthcare Marketing and Strategy

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