problem_set2 - 10.34 Fall 2006 Homework#2 Due Date...

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10.34 – Fall 2006 Homework #2 Due Date: Wednesday, Sept. 20 th , 2006 – 9 AM Problem 1: Linear Regression Complete problem 1.B.2 in Beers’ textbook (page 82). Submit a graph with the rate data and fitted expression on the same plot, along with the values of the parameters determined by the linear regression. Problem 2: Fitting heat capacity data sets using various functional forms Part A: Complete problem 1.A.3 in Beers’ textbook. Part B: Apply your function calc_poly_coeff.m to interpolate between the following data for the heat capacity (C V [=] cal/mole-K) of CO 2 using the following polynomial form: ( ) 23 01 2 3 V CT a a Ta T a T =+ + + Temp (K) 300 600 900 1200 C V 6.91 9.32 10.68 11.48 i.e. write a function Cv = Cv_CO2_poly(T,T_data,Cv_data) that takes in a vector of T values and returns estimates of the corresponding C V (T) vector, given the data vectors for temperature and C V . Note that in this special case, N data = N param , so the parameters are determined by solving a linear system (not regression). These sorts of interpolations are needed for estimating the thermodynamic properties of gases. (In fact, often we need to extrapolate to predict the behavior of gases at very high T, where it is difficult or impossible to measure the properties directly.) Report the parameter values determined for the polynomial. Also report the condition number of the linear system matrix ( : X in X a f = , where a are the parameters). Part C: As T gets very high, it is known that molecular heat capacities are asymptote to values predicted by classical mechanics. For CO 2 , the asymptotic value is C V = 6.5R. Also, for CO 2 , the heat capacity is expected to be a monotonically increasing function of T. Does your program Cv_CO2_poly.m give reasonable estimates? Does it give accurate extrapolations at high temperature? Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (, Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Often a better extrapolation can be obtained using the Pade form: () 2 01 2 2 32 1 V V aa T C a T CT aT aT ++ = Write a program calc_Pade_coeff.m that determines a 0 , a 1 , a 2 , and a 3 and a program Cv_CO2_Pade(T,T_data,Cv_data) that estimates the C V (T) of CO 2 using the Pade form. (Note that the Pade form can be posed as linear in the parameters, so a nonlinear solver is not needed) Report the parameter values determined for the Pade form. Compare the two interpolations graphically, and make
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problem_set2 - 10.34 Fall 2006 Homework#2 Due Date...

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