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