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Numerical Differentiation and Quadrature (Integration) 1Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich

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Numerical Differentiation Problem: Though an analytical derivative can be found for all differentiable functions, it is often impractical to calculate it Solution: Approximate the derivative numerically Method of finite differences: Remember that: Therefore:for small h 2Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Numerical Quadrature (Integration) f(x) [a,b] Problem: Generally, it is not possible to find the antiderivative (Stammfunktion) of a function f(x) in order to solve a definite integral in the interval [a,b] Solution: Approximate the area under the curve numerically Trapezoidal rule: Divide the interval into sub-intervals and approximate the integral by the sum of the areas of the resulting trapezoids 3Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Trapezoidal rule 4Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature ax1x1 x2x2 x n-1 b

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How does Matlab do it? quad quad : Low accuracy, non-smooth integrands, uses adaptive recursive Simpson rule quadl quadl : High accuracy, smooth integrands, uses adaptive Gauss/Lobatto rule (degree of integration routine related to number of points) quadgk quadgk : High accuracy, oscillatory integrands, can handle infinite intervals and singularities at the end points, uses Gauss/Konrod rule (re-uses lower degree results for higher degree approximations) Degree p of an integration rule = Polynomials up to order p can be integrated accurately (assuming there is no numerical error) 5Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Matlab Syntax Hints result = quadl(int_fun, a, b,...); All the integrators use the syntax result = quadl(int_fun, a, b,...); int_fun is a function handle to a function that takes one input x and returns the function value at x; it must be vectorized f_new K); Use parametrizing functions to pass more arguments to int_fun if needed f_new K); f_new is now a function that takes only x as input and returns the value that int_fun would return when K is used as second input Note that K must be defined in this command result = K), a, b); This can be done directly in the integrator call: result = K), a, b); integral(...) Matlab 2012a and newer: integral(...) 6Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Assignment 1 Consider the function f(x)x = 1h logspace(-15, -1, 200) h 1.Use the method of finite differences to approximate the derivative of f(x) at x = 1. Vary h between and using logspace(-15, -1, 200), and calculate the error of the finite differences approximation compared to the analytical solution for each h. hloglog 2.Plot the error vs. h using loglog. What do you observe? What could be the cause for this behavior? 7Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Exercise Mass Transfer into Semi-Infinite Slab Consider a liquid diffusing into a solid material The liquid concentration at the interface is constant The material block is considered to be infinitely long, the concentration at infinity is therefore constant and equal to the starting concentration inside the block 8Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature c(z, t) 0 z c 0 = const.c ∞ = const.

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Exercise (continued) Using a local mass balance, we can formulate an ODE where j is the diffusive flux in [kg m -2 s -1 ] With Δz 0, we arrive at a PDE in two variables 9Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature z z+Δz j in j out

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Exercise (continued) By combining this local mass balance with Fick’s law, a PDE in one variable is found: The analytical solution of this equation (found by combination of variables) reads: 10Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Assignment 2 1.Write a Matlab program to calculate and plot the concentration profile in the slab Use the following values: c ∞ = 0; c 0 = 1 zeta = linspace(1e-6, 3) czetaczeta Create a vector zeta = linspace(1e-6, 3), calculate the value of c for each zeta, then plot c vs. zeta integral quadl Use integral or quadl for the integration 2.Create a function which calculates an integral with the trapezoidal rule function F = trapInt(f, n, a, b) Use the form: function F = trapInt(f, n, a, b) Where f is a function handle to the function that is to be integrated, n is the number of points and a and b denote the interval quadlintegral Use your function to solve the diffusion problem and compare it to the result you obtained with quadl or integral. 11Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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Assignment 2 (continued) 3.Improve your function by adding a convergence check: In addition to computing the integral with n points, simultaneously calculate it with 2n points while while the results differ by more than 10 -6, double n and iterate the calculation Terminate the calculation and issue a warning if n exceeds Daniel Baur / Numerical Methods for Chemical Engineers / Numerical Quadrature

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