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 CV T a0 a1T a2T 2 a3T 3 Temp K 300 600 900 1200 CV 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 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 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 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 CV T a0 a1T CV a2T 2 1 a3T a2T 2 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 T 1000 The polynomial would be T CV a0 a1 a2 2 a3 3 where 1000 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 nearsingular 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 CV 6 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 4 5 Separator Reactor 1 2 3 6 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 system is always adjusted to maintain the concentration of acid in the recycle stream at 10 00 mass For these given inputs form the system of equations to solve for the composition of all the streams Is there a unique solution to the problem If so calculate the flow rates in the recycle stream for these sets of inputs If not why not a …