Day28 - CE 561 Lecture Notes Fall 2009 Days 28: A...

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CE 561 Lecture Notes Fall 2009 p. 1 of 28 Days 28: A digression on the numerical solution of boundary value problems Earlier in the course, we spent substantial time learning how to solve initial value problems. As we have seen, both perfectly mixed batch reactors and completely unmixed plug flow reactors can be described mathematically in terms of an initial value problem. The other prototypical reactor type is the perfectly mixed stirred-tank reactor. It can be modeled by a set of algebraic, rather than differential, equations. We briefly discussed the numerical solution of a set of nonlinear algebraic equations by Newton’s method in the context of implicit methods for initial value problems, and in the context of nonlinear least-squares parameter estimation. We will discuss this further below in the context of steady-state boundary value problems. A reactor with partial mixing generally cannot be described by a set of algebraic equations or by a set of differential equations that take the form of an initial value problem. Instead, a partially mixed reactor will usually lead to some sort of boundary value problem. A prototypical problem of this type is the plug-flow reactor with axial dispersion. It is a 1-dimensional convection- reaction-diffusion problem. The diffusive terms in the equation (which contain second derivatives with respect to position) make this a second order ordinary differential equation. Boundary value problems are sets of differential equations that include derivatives of at least second order and that have boundary conditions specified at more than one location. A boundary value problem that also includes time dependence is an initial-boundary-value problem. A general example of an initial-boundary-value problem is the set of general reactor balance equations for constant physical properties ∇⋅ v = 0 2 1 M k k k k ik i i dC vC D C r dt α = = − ⋅∇ + + 2 1 ˆ () M p ii i dT C v T Hr T dt ρλ =  + ⋅∇ = −∆ + ∇   with boundary conditions specified over the reactor walls, inlet, and outlet, and an initial condition specified over the whole reactor volume. An initial-boundary-value problem like this is a set of partial differential equations in up to three spatial dimensions plus the time dimension. We will not learn how to solve such equations in this course. In many cases, such sets of multi-dimensional partial differential equations are most easily solved by starting from a commercial computational fluid dynamics code that is designed to solve the Navier-Stokes equations and associated conservation equations. We saw some examples of that approach last time. A slightly simpler problem is given by the same set of reactor balance equations at steady state ∇⋅ v = 0 2 1 0 M k k k ik i i r = − ⋅∇ + + =
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CE 561 Lecture Notes Fall 2009 p. 2 of 28 ( ) 2 1 ˆ () M p ii i C v T Hr T ρλ = ⋅∇ = −∆ + With boundary conditions specified over the reactor walls, inlet, and outlet.
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Day28 - CE 561 Lecture Notes Fall 2009 Days 28: A...

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