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/moleK) 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 (http://ocw.mit.edu),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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View Full DocumentOften 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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 Fall '04
 ClarkColton
 Thermodynamics, Numerical Analysis, pH, Mass flow rate, Numerical Methods Applied

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