Objectives Uncertainty analysis Quality control Regression analysis

Objectives

• Uncertainty analysis

• Quality control

• Regression analysis

Example: Convective heat transfercalculation

h = q/(Tsurface – Tair)

22

_

_

T

Tδ

q

qδhh δ

convectivesurface

convectivesurface

22airsurface TTT

…….

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 18 20 22 24

Measured in present study

Awbi and Hatton (1999)

h = 1.823 T.0.293/H0.121 (SI)

h ro

om [W

/m2K

]

Troom

[C]

h = 1.74 T.33 (SI)

hro

om [B

TU

/h ft

2°F

]

Troom

[F]

2_2

_2

__ losswallradiativesurfacetotalsurfaceconvectivesurface qqqq

How about measurement variationor time or spatial averaging ?

• Sometimes quantities of interest fluctuate• Sometimes we are interested in slopes of

lines/regression equations• Often appropriate to use standard deviation or

standard error• Be careful and acknowledge when you are doing so

Example of uncertainty reporting

4

Repetition of experiments

5

6

General procedure

• Expend substantial effort on uncertainty• Consider all sources of error

• State best guess of reasonable “best guess”

• Always say something about uncertainty

• Clearly state your procedure in your Methodology

Experimental Design

• Huge topic (we will talk about this during whole course)

• Questions:• What do you need to know?• How well (i.e. with what confidence) do you need

to know it?• How much do you expect it to vary?• What factors are likely to influence/confound your

results?

• Always find fundamental parameters

8

Some Approaches

• Bench-scale experiments in laboratory, followed by large-scale laboratory tests, followed by field tests

• Field tests followed by controlled laboratory investigation

• Larger field studies

65

Example 1

• What is the average efficiency of AC units in Austin (average COP) ?

10

Quality Assurance (QA)Quality Control (QC)

How to incorporated QA/QC into your experimental study?

Experiment Design Phase:

• Define objective - What question are you trying to answer?

- How will you know you are finished?

• Choose - Factors of interest

- Parameters to measure

- Experiments control method(s)

- The data analysis techniques

11

How to incorporated QA/QC intoyour experimental study?

Experiment Design Phase:

For measured parametersconsider:- Range- Number of points- Number of repetitions

Create an experimental matrix

12

How to incorporated QA/QC intoyour experimental study?

Experimental matrix

Be real:

- Consider available time and funding

- Predict potential for failure - predict more experiments than minimum

- predict extra time for repetition

- Preliminary experiments help

13

How to incorporated QA/QC intoyour experimental study?

Measuring Phase:- Use measuring techniques that will meet the needs of your experiment

- Collect sufficient data (including repetition) to adequately characterize the measured parameter

- Record all available conditions/parameters (even those that are not in your matrix)

- Use experiment control methods

14

How to incorporated QA/QC intoyour experimental study?

Data Analysis Phase:

• Graphs & descriptive statistics first• Hypothesis testing • Regression next• Interpret the results• Draw conclusions

15

How to incorporated QA/QC intoyour experimental study?

Be ready to modify and/or go back and forth between phases

Example…..

Example from convection correlation development study

• Repetition and multiple measurements of important variables

• Repetition of experiments

• Independent development of data processing templates

• Comparison with previous research

• Energy balance check

• Large number of experiments

• Graphing to identify discrepancy

• …..

16

Regression analysis

17

Single variable:

Minimum number of points depends on number of variable in the function (3 for the function above).Using the data we can set the system of equation to find the coefficients.

Lagrange interpolation

18

Rewrite:

Find coefficients:

General form:

Model of complex system based on experimental data

19

Example: chiller modelTOA

water

Building users (cooling coil in AHU)

TCWR=11oCTCWS=5oC

T Condensation

Chiller model

20

EIRFPLCAPFTPP NOMINAL

OACWSOAOACWSCWS TTfTeTdTcTbaCAPTF 12

112

111

PLRcPLRbaEIRFPLR 333 NOMINALQ

QPLR

)(

Impact of temperatures:

Impact of capacity:

Two variable function fitting

What do you think

• Does correlation where R2=0.82 represent a good data modeling?

Mean:

Total sum of squares:

Sum of squares of residuals :

Coefficient of determination

Anscombe's quartet • Example of statistical misinterpretation of data

- all data have the same Mean (for x and y), Variance (for x and y)

- correlation R2: 0.816, linear regression: y=3.00+0.500·x

Anscombe's quartet • Example of statistical misinterpretation of data

- all curves have the same Mean (x, y), Variance (x, y)

- correlation R2: 0.816, linear regression

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

11

12

13

14

y

x

Data set A

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

11

12

13

14

y

x

Data set B

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

11

12

13

14

y

x

Data set C

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

11

12

13

14

y

x

Data set D

Moral of the story

Francis Anscombe (in 1973) demonstrated • the importance of graphing data before

analyzing it • the effect of outliers on statistical properties

25