Description
Problem 3.
VAPOR PRESURE DATA REPRESENTATION BY POLYNOMIALS AND EQUATIONS
Regression of polynomials of various degrees. Linear regression of mathematical models with variable transformations. Nonlinear regression. Use of polynomials, a modified Clausius-Clapeyron equation, and the Antoine equation to model vapor pressure versus temperature data. Table presents data of vapor pressure versus temperature for benzene. Temperature T °C -36.7 -19.6 -11.5 -2.6 +7.6 15.4 26.1 42.2 60.6 80.1
Pressure, P Mm Hg 1 5 10 20 40 60 100 200 400 760
Some design calculations require these data to be accurately correlated by various algebraic expressions which provide P in mmHg as a function of T in °C. A simple polynomial is often used as an empirical modeling equation. This can be written in general form for this problem as 𝑃 = 𝑎 +𝑎 𝑇 +𝑎 𝑇 + 𝑎 𝑇 +⋯+ 𝑎 𝑇 (9) where a0... an are the parameters (coefficients) to be determined by regression and n is the degree of the polynomial. Typically the degree of the polynomial is selected which gives the best data representation when using a least-squares objective function. The Clausius-Clapeyron equation which is useful for the correlation of vapor pressure data is given by
log(𝑃) = 𝐴 −
.
(10)
where P is the vapor pressure in mmHg and T is the temperature in °C. Note that the denominator is just the absolute temperature in K. Both A and B are the parameters of the equation which are typically determined by regression. The Antoine equation which is widely used for the representation of vapor pressure data is given by log(𝑃) = 𝐴 − (11)
where typically P is the vapor pressure in mmHg and T is the temperature in °C. Note that this equation has parameters A, B, and C, which must be determined by nonlinear regression as it is not possible to linearize this equation. The Antoine equation is equivalent to the Clausius-Clapeyron equation when C = 273.15. → a) Regress the data with polynomials having the form of Equation (9). Determine the degree of polynomial which best represents the data. b) Regress the data using linear regression on Equation (10), the Clausius-Clapeyron equation. c) Regress the data using nonlinear regression on Equation (11), the Antoine equation.
Solution The spreadsheet was set up with the data for temperature and pressure entered in the indicated columns. Columns for Temp in Deg K, 1/Tk*1000 and Log P were developed Columns were then added for each of the curve fits that were to be carried out along with the parameters that would be manipulated to generate the calculated curves. These are shown above the calculated columns as parameters. Finally a row of sum of squares cells were added along the bottom. The contents were set up as array formulas and entered with Crtl+Shift+Enter. For example for the Calculated Clasius-Clapeyron column: in the sum of squares cell the following was entered: SUM((F16:F25-E16:E25)^2) followed by Crtl+Shift+Enter. The Solver program was then utilized for each of the curve fits. For example for the Clausius Clapeyron fit F5 and F6 (A and B) were manipulated until F28 (the sum of squares) was minimized. Forward differences were used to obtain the derivatives. For curve fitting the polynomials, the starting values were taken as the lower polynomials ending Problem 3 VAPOR PRESURE DATA REPRESENTATION BY POLYNOMIALS AND EQUATIONS
Parameters A and ao
8.75200927 5.764699914 64.40598719 0.581914345 24.45932364
24.6788 24.75426421
B and a1
2035.33113 676.2338056 5.890714249 2.067149475 1.198100707
1.6062 1.609016541
C and a2
153.7861354
0.086152483 0.039448075
D and a3
0.0360443 0.035605317
0.000744911 0.000413122 0.000412978
E and a4
3.96E-06
F and a5
4.23E-06 -2.51E-09
Temp. °C
Temp. °K
1/Tk*1000
Press, P (mm Hg)
Log P
Calculated ClausiusClap Log P
Calculated
Calculated
Calculated
Calculated
Calculated
Calculated
Polynomial
Polynomial
Polynomial
Polynomial
Polynomial
Antoine
Power 1
Power 2
Power3
Power 4
Power 5
Pressure
Pressure
Pressure
Log P
Pressure
Pressure
-36.7
236.45
4.229224
1
0
0.14414
-0.01082
-151.78323
39.592
-3.200
1.048
1.079
-19.6
253.55
3.943995
5
0.69897
0.72467
0.72518
-51.05201
-8.002
10.522
4.518
4.417
-11.5
261.65
3.821899
10
1
0.97318
1.01207
-3.33723
-12.960
14.765
10.415
10.406
-2.6
270.55
3.696174
20
1.30103
1.22907
1.29184
49.09013
-5.374
21.598
20.739
20.804
7.6
280.75
3.561888
40
1.60205999
1.50239
1.57454
109.17542
20.105
36.170
39.162
39.235
15.4
288.55
3.465604
60
1.77815125
1.69836
1.76772
155.12299
51.684
54.986
59.694
59.721
26.1
299.25
3.341688
100
2
1.95057
2.00547
218.15363
112.059
95.846
100.339
100.278
42.2
315.35
3.171080
200
2.30103
2.29781
2.31428
312.99413
240.076
201.251
200.265
200.165
60.6
333.75
2.996255
400
2.60205999
2.65364
2.61042
421.38327
441.070
407.708
399.768
399.869
80.1
353.25
2.830856
760
2.88081359
2.99028
2.87340
536.25220
717.752
756.353
760.052
760.026
Sum of Squares
0.060732441 0.002230624 118590.5562 8517.495122 204.7333672 1.989602091 1.943612782
Variance (sum of Squares/degrees of freedom)
0.007591555 0.000318661 14823.81952 1216.785017 34.12222786 0.397920418 0.485903196
values. To gain greater accuracy, derivatives were taken as central differences.
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