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Unformatted text preview: 1 Numerical Solution of Partial Differential Equations Parabolic Equations: The Heat Equation 2 TimeDependent PDEs z There are a number of important timedependent partial differential equations (PDEs) that arise from transport problems in chemical engineering: • Heat equation (heat conduction) • Diffusion equation (mass diffusion) • NavierStokes equations (momentum diffusion – viscous fluid flow) z These are all examples of parabolic differential equations – named by the geometry of the “characteristic curves” z Another important class of timedependent PDEs are wave equations , which are normally hyperbolic in character 1D wave equation 3 A TimeDependent Heat Conduction Example z Consider an insulated metal bar suspended in air and in contact with two thermal reservoirs at each end (T 1 < T 2 ): z Fourier’s law + heat conservation leads to the following initial value/boundary value (IVBV) problem: T 1 T 2 x=0 x=L x Dirichlet BCs Initial temperature profile “thermal diffusivity” “Heat Equation” 4 Solution of TimeDependent PDEs z We normally solve timedependent PDEs such as the parabolic heat equation, or the hyperbolic wave equation by applying finite differences in time – just like IV ODE problems z Define a uniform grid of time points (time step Δ t): z Let z A forward difference approximation to the time derivative then gives an equation very similar to the forward Euler formula for integrating an ODE: z Or, We can use this explicit formula to step forward in time from the initial condition (n=0) Finite differences or spectral formulas can be used to compute the spatial derivatives Convention: superscripts denote time indices 5 Forward Time—Centered Space (FTCS) Algorithm z A common approach is to apply a secondorder accurate central difference approximation to the spatial derivative z This is the FTCS algorithm: z Or, time index space index z This can be viewed as a linear mapping of the N+1 component spatial vector {T n } into the N+1 component spatial vector {T n+1 } z The first and last rows of [M] are used to impose the BCs as before! [M] is tridiagonal 6 Accuracy and Stability of the FTCS Algorithm z The FTCS algorithm suffers from two major flaws – low accuracy and poor stability characteristics z Accuracy: • The error in approximating the time derivative is O( Δ t) and the error in the spatial derivative is O(...
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This note was uploaded on 12/29/2011 for the course CHE 132b taught by Professor Ceweb during the Fall '09 term at UCSB.
 Fall '09
 Ceweb

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