application of engineering models to fit experimental data
DESCRIPTION
Application of Engineering Models to Fit Experimental Data. Lecture 1. Extrapolate Data to Higher Temperatures and Compare to Experimental Data. Refine Model Using a 2 nd Order Polynomial. Extrapolate to Even Higher Temperatures. Conclusions. - PowerPoint PPT PresentationTRANSCRIPT
Application of Engineering Models to Fit Experimental Data
Lecture 1
Vapor Pressure of Water
y = 0.0281x - 1.8142R2 = 0.994
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
80 85 90 95 100 105
Temperature, oC
Vap
or P
ress
ure,
bar
Extrapolate Data to Higher Temperatures and Compare to
Experimental Data
Vapor Pressure of Water
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
80 90 100 110 120 130 140 150 160 170 180
Temperature, oC
Vap
or P
ress
ure,
bar
Refine Model Using a 2nd Order Polynomial
Vapor Pressure of Water
y = 0.0009x2 - 0.147x + 6.6246R2 = 0.999
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
80 90 100 110 120 130 140 150 160 170 180
Temperature, oC
Vap
or P
ress
ure,
bar
Extrapolate to Even Higher Temperatures
Vapor Pressure of Water
0.0
5.0
10.0
15.0
20.0
25.0
30.0
80 100 120 140 160 180 200 220 240
Temperature, oC
Vap
or P
ress
ure,
bar
Conclusions
• Trendline models are not to be extrapolated beyond the range of data that they were derived.
• All models are wrong, but some are useful.
As Engineers, How do we properly model data?
• Apply models based on physical principles• Test our models using statistical principles• Refine our models if there is disagreement
with data• If the model does not fit the data, throw out
the model, not the data
The Power of the Linear Model
y = mx + b• Does the model fit the data, or should a different model be used?• Are the values of m and b significantly different from zero?• Are the values of m and b significantly different from values obtained in a different experiment?
Before Applying Models to Data ….
• Graph the data to see if a linear model will produce a good fit or it the data exhibit curvature.
• See if there is excessive scatter in the data. Other variables may influence the results that are not accounted for.
• Test the results with statistics to determine if coefficients can be dropped out of the model.
Linear Regression
Go to Lecture1Example.xls
This can be more than one column but they need to be next to each other
Lower and upper confidence intervals for b and m
risk for stating b and m are not zero
tobs when comparing b and m to zero
Standard Errors of b and m
Values of b and m
Probability that the model fit the data by chance aloneSUMMARY OUTPUT
Regression StatisticsMultiple R 0.998011857R Square 0.996027667Adjusted R Square 0.995034584Standard Error 0.270977068Observations 6
ANOVAdf SS MS F Significance F
Regression 1 73.64628571 73.64629 1002.965 5.92514E-06Residual 4 0.293714286 0.073429Total 5 73.94
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 4.088571429 0.529883828 7.715977 0.001519 2.617375022 5.559768X Variable 1 1.025714286 0.032387954 31.66962 5.93E-06 0.935790722 1.115638
Linear Regression with Multiple Independent Variables
Go to Lecture1Example.xlsSheet 2
Non-Linear Equations
How to turn non-linear equations into linear equations
How do we get cos
Independent variable, Coefficients to be determined
z
y mx b
To look like this:
; cos
;
y z x
m b
In-Class Exercise
2
2
2
12 2
, to be determined, independent variable
sin 4
1
1
w
z w
zwz w
zw
z w w
Linearization of Non-Linear Equations
Go to Lecture1Example.xlsSheet 3
Review of Natural Logarithms
ln lnax a x ln lnae a e a
ln ln lnxy x y ln ln lna bx y a x b y
ln ln lnx x yy
ln ln ln
a
b
x a x b yy
Linearization of Equations
0 0; , are to be determinedBk A e A BT
0ln ln Bk AT
Linearization of Non-Linear Equations
Go to Lecture1Example.xlsSheet 4
Using Solver to Determine Coefficients in Non-Linear Equations
Minimize Error Between Model Predictions and Experimental Data
exp. modelError y y minumum
22
2
, ,1
0
exp. model
n
exp. i model ii
Error y y
y y