10.34 – Fall 2006 Homework #2 Due Date: Wednesday, Sept. 20th, 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 (CV [=] cal/mole-K) of CO2 using the following polynomial form: ()2301 2 3VCT a aTaT aT=+ + + Temp (K) 300 600 900 1200 CV6.91 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 CV(T) vector, given the data vectors for temperature and CV. Note that in this special case, Ndata = Nparam, 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 (:Xin 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 CO2, the asymptotic value is CV = 6.5R. Also, for CO2, 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 (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Often a better extrapolation can be obtained using the Pade form: ()201 22321VVaaTCaTCTaT aT∞++=++ Write a program calc_Pade_coeff.m that determines a0, a1, a2, and a3 and a program Cv_CO2_Pade(T,T_data,Cv_data) that estimates the CV(T) of CO2 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 some comments regarding the accuracy of their interpolations and extrapolations. Part D: Suppose that the problem is posed in a slightly different way, such that the variable is converted from T Æ T/1000. The polynomial would be: ()2301 2 3:1000VTCaaaa whereττττ τ=+ + + ≡ Using the previously written calc_poly_coeff.m, calculate the parameter values for this new scaled variable. Are these the expected values given the previous solution with the variable T? Calculate the condition number of the new linear system matrix. Typically a very large condition number implies the problem is near-singular and/or poorly scaled, which do you think is the problem in this case, and why? Part E: Suppose the CV data for CO2 were actually the following: Temp (K) 300 600 900 2200 CV6.91 9.32 10.68 11.48 Compare the interpolation and extrapolation abilities for this set of CV data by providing a plot comparing the polynomial and Pade forms. Comment on the results in this case. Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Problem 3: Solving Linear Systems for “Real” System Problems Consider the reactor-separator esterification system discussed in class: 12 3645ReactorSeparator Contrary to the idealized assumptions made in class, the separators are not perfect. The product stream 6 is actually a mixture of ester, acid, and acid. The steam stream 5 is actually a mixture of H2O and alcohol. There is some ester in the recycle stream. The input stream is a mix of H2O, alcohol and acid. The molecular weight of the alcohol is 32 g/mole. The molecular weight of the acid is 176 g/mol. The reaction is acid + alcohol = H2O + ester. The symbols used for this problem are explained below. The symbols: Wi : water mass flow rate in stream i. Aci: Acid mass flow rate in stream i. Ali: Alcohol mass flow rate in stream i. Ei: Ester mass flow rate in stream i. Mi: Total mass flow rate in stream i. Fi: Total molar flow rate in stream i. The measurements available to you are: 1) The mass flow rates of Alcohol and Acid in the input stream. (Ac1, Al1) 2) The total molar flow rate of stream 3, stream 5 and stream 6. (F3, F5, F6) 3) Performing an IR spectroscopy gives the ratio of Acid and Alcohol in stream 4 and stream 3. (r3, r4) All the measurements are good to 3 significant figures. Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].To ensure material compatibility, the system is always adjusted to maintain the concentration of acid in the recycle stream at 10.00 mass%. Also the separator extracts 90% of ester from stream 3 into stream 6. Write a Matlab function which takes as its input a vector containing the 7 measured numbers (in order Ac1, Al1, F3, F5, F6, r3, r4) and constructs and solves the linear system of equations to compute the composition of all the streams. 1. Use your function to compute the composition of the recycle stream for the following cases: a) Ac1 = 50.0 Kg/sec, Al1 = 10.0 Kg/sec, F3 = 1.20 Kmol/sec, F5 = 0.212 Kmol/sec, F6 = 0.662 Kmol/sec, r3 = 1.93 and r4 = 0.157. Is there a unique solution to the problem? Also estimate the uncertainty in your answers due to the conditioning of the system of equations and the uncertainty in the measurements. 2. Now instead of the above measurements suppose someone made measurements of Mass flow rate of all the components in stream 1 and stream 6 (Ac1, Al1, W1, E6, Ac6, W6). The total mass flow rate of the recycle stream is provided (M4). The total molar flow rate of stream 5 is given F5. To ensure material compatibility, the