ASEE Southeast Section Conference  

INTEGRATING MODEL VALIDATION AND UNCERTAINTY ANALYSIS INTO AN

UNDERGRADUATE ENGINEERING LABORATORY

 W. G. Steele and J. A. Schneider

 Department of Mechanical Engineering

Bagley College of EngineeringMississippi State UniversityMississippi State, MS 39762

• INTRODUCTION

• MS STATE LABORATORY COURSES

• METHODOLOGY

• EXAMPLE

•CONCLUSION

INTRODUCTION

•Laboratories introduce the student to the use of various measurement devices along with the associated experimental uncertainties

• Theoretical engineering models are used to compare predicted outcome with the experimental results

• Usually no consideration of the uncertainty associated with the theoretical model calculations

• Concept of engineering model validation using uncertainty analysis is extension of verification and validation research for CFD and other computational design codes

MS STATE LABORATORY COURSES

• Experimental Orientationbasic measurementsdata acquisitionconcepts of uncertainty analysis

• Experimental Techniques Iexperiment design using uncertainty analysisexperiment operation

• Experimental Techniques IImodel, plan, design, construct, operate, and analyze results of an experiment including model validation

Consider a validation comparison:

E

Ur

m ± Um

X

ri

mi

m value from the model

r result from experiment

E comparison error

E = r - m = r - m

Validation Comparison of Model Results with Experimental Results

Reality (Mother Nature)

Simulation

Experiment

Model Result, m Um

Experimental Result, r Ur

Comparison Error, E E = r - m

METHODOLOGY

The comparison error

has an uncertainty

If is less than UE, the level of model validation is UE.

If is greater than UE, the level of model validation is .

mrE

2m

2rE UUU

E

E E

For the experimental result

the uncertainty is

where

br = systematic standard uncertainty

sr = random standard uncertainty

J21 X,...,X,Xrr

s+b 2= U 2r

2rr

The systematic standard uncertainty of the result is defined as 

where

and where Bi is the 95% confidence estimate (2bi) of the limits of the true systematic error for variable Xi.  

2

i2i

J

1=i

2r 2

B = b

θ

X

r=

ii

θ

The random standard uncertainty of the result is defined as 

s = s 2i

2i

J

1i=

2r θ

)Y,,Y,Y(fm K21

For the model result

the uncertainty is

21

K

1k

K

1k

2k

2k

2k

2km bs2U

θθ

EXAMPLE

Experiment result was the measured head loss in a pipe, hpr, over a range of flow rates.

52

25.0

m dghoCL8

fhpπ

ΔΔ

211.1

d d7.3Re9.6

log

3086.0f

ε

μπ

ρ

dhoC4

Re5.0

d

Δ

Engineering model was

where

and

Fluid Flow Test Facility

5

7

9

11

13

15

17

19

21

20,000 25,000 30,000 35,000 40,000 45,000 50,000

Reynolds Number

Pip

e H

ead

Lo

ss, i

nch

es o

f w

ater

Experimental Results

Model Predictions

Experimental Results vs. Model Predictions

 

Variable Value SystematicStandard

Uncertainty

RandomStandard

Uncertainty

pipe head loss (hpr) Variable 0.141 in 0.04 in

orifice head loss (ho) Variable 0.05 in 0.08 in

pipe length (L) 39.125 in 0.031 in -

pipe diameter (d) 0.697 in 0.00025 in -

flow coefficient (C) 11.45 in 2.5/sec 0.089 in 2.5/sec -

roughness () 3.6X10-6 in 50% -

water density () 999 kg/m3 0.3% -

water viscosity () 1.056X10-3 Nsec/m2

7% -

 

Uncertainty Estimates for Result and Model Variables

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

20,000 25,000 30,000 35,000 40,000 45,000 50,000

Reynolds Number

Co

mp

aris

on

Err

or,

inch

es o

f w

ater

Comparison Error

Uncertainty of Comparison Error

Comparison Error

Uncertainty Percentage Contributions

2E

2i

2i

2E

2i

2i

2/U

100s

2/U

100b θθ

  

Variable

Reynolds Number

22,623 48,279

SystematicStandard

Uncertainty

RandomStandard Uncertaint

y

SystematicStandard

Uncertainty

RandomStandard

Uncertaintypipe head loss (hpr) 42.6 3.4 15.1 1.2

orifice head loss (ho) 11.0 28.1 2.8 7.1

pipe length (L) 0.0 - 0.2 -

pipe diameter (d) 0.2 - 0.8 -

flow coefficient (C) 10.5 - 54.9 -

roughness () 0.0 - 0.0 -

water density () 0.0 - 0.0 -

water viscosity () 4.2 - 17.9 -

Sum = 100.0 Sum = 100.0

Uncertainty Percentage Contributions

0

10

20

30

40

50

60

b, pipe he

ad lo

ss

s, pipe he

ad lo

ss

b, orifi

ce hea

d loss

s, orifi

ce hea

d loss

b, pipe leng

th

b, pipe diam

eter

b, flow

coe

fficien

t

b, ro

ughn

ess

b, w

ater

den

sity

b, w

ater

visc

osity

Perc

enta

ge

Re = 22,623

Re = 48,279

CONCLUSION

• Understanding the limitations of physical models is key to the successful practice of engineering.

• The uncertainty of both the model and experiment results are used to assess the model validity.

• The validation process allows the identification of ranges where different or improved models are needed or shows that improved variable uncertainties are needed to reduce the validation uncertainty.