1) The document describes the development, verification, and validation of a responsive boundary model. 2) Verification was performed using manufactured solutions and grid convergence studies to evaluate spatial and temporal discretization errors. 3) Validation compared model predictions of helium plume interferometry data to experimental measurements, analyzing sources of error and sensitivity. 4) Results showed good agreement between computation and experiment, demonstrating the model captured important physical phenomena.

1. Development, Verification, and Validation of the
Responsive Boundary Model
Weston Eldredge
University of Utah, Department of Chemical Engineering
May 4, 2011
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

2. Presentation Outline
1 Background and Motivation
2 Responsive Boundary Model Development
3 Verification of the Boundary Model
4 Validation: Consistency Analysis
5 Consistency Analysis with Helium Plume Data
6 Consistency Analysis with the Boundary Model
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

3. Sources of Error in Computation
Programmer Error (Bugs)
Finite-Precision/
Round-Off Error
Spatial and Temporal
Discretization Error
Boundary Condition Error
Verification: Is the
algorithim solving the
equations correctly?
Model Error - Continuum
Equations, Differential or
Integral equation
Capturing the Physics
Modeling Assumptions
and Simplifications
Boundary Condition Error
Validation: Am I solving the
correct equations?
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

4. Inlet Boundary for Pool Fire Simulation
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

5. Inlet Boundary for Pool Fire Simulation
Traditional inlet boundary for pool fire simulations is a source of error:
Assumes that evaporation rate is constant with time
Assumes evaporation rate is uniform over pool surface
Requires experimental data to determine the proper evaporation rate
Boundary condition can influence the behavior of the domain
solution
Responsive Boundary Model seeks to remedy this source of error:
There is a feedback relationship between the pool surface boundary
and the flame.
The pool is heated by the flame
As the pool gets warmer it vaporizes more quickly, feeding the flame
Different parts of the pool receive different amounts of heat input
The evaporation rate is determined by fuel properties and
surrounding conditions
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

6. Validation Hierarchy
Large Pool Fire
Boundary Condition:
Liquid Pool Model
Helium Plumes:
- Non-reactive
- Buoyancy driven
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

7. Boundary Model Requirements/Background
Capture enough physical phenomena to predict evaporation
rate from the pool surface
Simple enough to not burden ARCHES
Need to model fuel properties - single/multi-component
Evaporation rate is heat-transfer controlled (Hottel, 1958)
Most fuels heat to within a degree of their boiling point
(Blinov and Khudyakov, 1957)
Modes of heat transfer: conduction, convection, and radiation
Detail liquid flow patterns in the pool are not important
Liquid temperature is important
It is also important to model liquid regression
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

8. Conservation Equations
Conservation of Mass (Liquid Height Equation):
∂H
∂t =
−m evap
ρ
Conservation of Energy (Liquid Temperature):
∂T
∂t = α∂2T
∂z2 +
q rad +q conv −q evap
ρCP ∆z
Conservation of Species (For Multi-Compoment systems only):
dCi
dt =
−N i,evap
H
Interphase Mass Transfer Equation:
N evap = Ctkc ln 1−xbulk
1−xi
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

9. Boundary Model Domain
Figure: Physical Domain
Figure: Liquid Regression
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

10. Methods of Verification
Method of Manufactured Solutions (MMS)
Grid Convergence (Determine Order of Discretization Error):
p =
ln
“
f3−f2
f2−f1
”
ln (r)
Grid Convergence Index (GCI):
GCI = Fs
| |
rp−1 , = f2−f1
f1
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

11. MMS: Energy Equation:
T(z, t) = TNominal + (1 + 5e−t
) sin (2πz)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
295
296
297
298
299
300
301
Height (meters)
Temperature(kelvins)
Time = 1 second
Time = 2 seconds
Time = 5 seconds
Figure: Manufactured Solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−14
−13
−12
−11
−10
−9
−8
−7
−6
−5
Height (meters)
RelativeError(logplot)
Time = 1 second
Time = 3 seconds
Time = 5 seconds
Figure: Error in MMS
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

12. MMS: Height Equation: H(t) = 1 − ln (1 + αt), α = e−1
tend
0 5 10 15 20 25 30
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (seconds)
Height(meters)
Figure: Manufactured Solution
0 5 10 15 20 25 30
−19
−18
−17
−16
−15
−14
−13
Time (seconds)
RelativeErrorinModelOutput
Figure: Error in MMS
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

13. Grid Convergence with the Manufactured Solutions
10
−3
10
−2
10
−1
10
−10
10
−9
10
−8
10
−7
10
−6
10
−5
dz
ErrorinTemperature
Error in Temperature
Second Order Slope
First Order Slope
10
−2
10
−1
10
0
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
dt
ErrorinTemperature
Error in Temperature
Fourth Order Slope
Third Order Slope
Energy equation with spatial step size Energy equation with temporal step size
10
−2
10
−1
10
0
10
−14
10
−13
10
−12
10
−11
10
−10
10
−9
10
−8
dt
ErrorinHeight
Error in Height
Fourth Order Slope
Third Order Slope
Height equation with temporal step size
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

14. Grid Convergence with the Full Model
Table: Grid convergence of mass burn rate with spatial step size
Case Number ∆z (meters) Mass Burn Rate ( kg
m2−sec
) Computed Order Error Band (±)
6 1.87 × 10−3 0.07450803
5 9.33 × 10−4 0.07939743
4 4.67 × 10−4 0.08217099 P654 0.81792 GCI4 5.531%
3 2.33 × 10−4 0.08323433 P543 1.38313 GCI3 0.993%
2 1.17 × 10−4 0.08351618 P432 1.91561 GCI2 0.158%
1 5.83 × 10−5 0.08358657 P321 2.00142 GCI1 0.035%
Table: Grid convergence for mass burn rate with time step size
Case Number ∆t (seconds) Mass Burn Rate ( kg
m2−sec
) Computed Order Error Band (±)
6 0.500000 0.0633
5 0.250000 0.0644
4 0.125000 0.0678 P654 1.10895 GCI4 2.292%
3 0.062500 0.0685 P543 1.02887 GCI3 1.236%
2 0.031250 0.0689 P432 1.01340 GCI2 0.622%
1 0.015625 0.0690 P321 1.00690 GCI1 0.312%
Table: Grid convergence of mass burn rate without interpolation
Case Number ∆t (seconds) Mass Burn Rate ( kg
m2−sec
) Computed Order Error Band (±)
6 1.000000 0.000562592164344898
5 0.500000 0.000562592296427578
4 0.250000 0.000562592304129631 P654 4.10005 GCI4 1.06 × 10−9
3 0.125000 0.000562592304594782 P543 4.04947 GCI3 6.64 × 10−11
2 0.062500 0.000562592304623373 P432 4.02405 GCI2 4.16 × 10−12
1 0.031250 0.000562592304623373 P321 4.02336 GCI1 2.56 × 10−13
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

15. Consistency Analysis: Terminology
ε - Denotes an experimental data space
Ye - Denotes the ”true” value of a quantity of interest
ye - The experimental measure of Ye
ue and le - Upper and lower uncertainty bounds of ye
ym(x) - Model prediction for Ye as a function of model input
parameter space x = {x1, x2, ...}
Dataset unit - {ye, ue, le, ym(x)}
Dataset - A collection of dataset units
Consistency - Based on a positive value for quantity Cε which
is the largest value for quantity γ for which the following
holds:
(1 − γ) ue ≥ | ym(x) − ye | ≥ le (1 − γ) , for each e ∈ ε.
βi ≥ xi ≥ αi , for i = 1, ..., n
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

16. Sensitivity Analysis
Active Variables -
”Experinece shows, that for an individual experiment e, only a
small subset of model parameters has a measurable influence
on the property Ye.” - Feeley et al. (2004)
Sensitivity analysis - When a dataset is found to be
inconsistent a further analysis shows which uncertainty
parameters the consistency measure, Cε, is most sensitive to:
∆Cε ≤
Pn
j=1
“
λj
(α)
∆αj + λj
(β)
∆βj
”
+
P
e ∈ ε
“
λe
(l)
∆le + λe
(u)
∆ue
”
If an uncertainty bound has a much higher Lagrangian
multiplier, λ, than the others, this could indicate a problem
with the parameter itself or the evaluation of it’s error bounds.
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

17. Holographic Interferometry with Small Helium Plumes
Figure: Holographic Image
centimeters
centimeters
0 2 4 6 8 10 12
0
2
4
6
8
10
12
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
Figure: Computational Reproduction
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

18. Experimental Setup
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

19. Important Relations
Gladstone-Dale Equation:
nm(x, y, t) = 3
2
1
ρm(x,y,t)
P
i
¯γi (x,y) N0
i
P
i
¯γi (x,y) + n0
Optical Path Length Difference Equation:
S (x, y, t) = ∆Φ(x,y,t)
λ = 1
λ [nm (x, y, t) − n0] dz
Interference fringes appear for S-values of -0.5, -1.5, -2.5, . . .
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

20. Sources of Error
Experimental Error:
Variation in fringe location:
Determined from repeated
statistical analysis of samples of
processed interferograms
Uncertainty in image processing
algorithms: A Box-Behnkin
experiment tests the range of
algorithm parameters thought to
be most influencial on the output
Bias error: Comparison of data
with an analytic solution for the
base of the plume where the
composition is pure helium
Computational Error:
Spatial discretization of the
helium domain
Uncertainty in scenario
parameters thought to be the
active variables:
1. System Temperature: 295.15
K - 305.15 K
2. Helium Inlet Flow Rate:
0.1215 m
sec - 0.1485 m
sec
3. Air Co-Flow (Approximating
Boundary Condition): 0.0135 m
sec
- 0.0675 m
sec
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

21. Evaluation of Discretization Error
−0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002
−8
−7
−6
−5
−4
−3
−2
−1
0
Radial Location (meters)
InterferenceOrder
Resolution: 803
Resolution: 1003
Resolution 1253
Figure: 3cm above the helium port
−0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 0
−7
−6
−5
−4
−3
−2
−1
0
Radial Location (meters)
InterferenceOrder
Resolution: 803
Resolution: 1003
Resolution 1253
Figure: 5cm above the helium port
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

22. Table: Box-Behnkin design for the helium plume simulations
System Temperature Air Co-Flow Helium Inlet Flow
(Kelvins) ( m
sec) ( m
sec)
Case-Base 305.15 0.0405 0.1350
Case-001 305.15 0.0135 0.1215
Case-002 305.15 0.0675 0.1215
Case-003 305.15 0.0135 0.1485
Case-004 305.15 0.0675 0.1485
Case-005 295.15 0.0405 0.1215
Case-006 315.15 0.0405 0.1215
Case-007 295.15 0.0405 0.1485
Case-008 315.15 0.0405 0.1485
Case-009 295.15 0.0135 0.1350
Case-010 315.15 0.0135 0.1350
Case-011 295.15 0.0675 0.1350
Case-012 315.15 0.0675 0.1350
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

23. Experiment Versus Computation
−0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002
−14
−12
−10
−8
−6
−4
−2
0
Radius, m
Inferencefringeorder
Exp 3 cm
Sim 3 cm
−0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 0
−14
−12
−10
−8
−6
−4
−2
0
Radius, m
Inferencefringeorder
Exp 5 cm
Sim 5 cm
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

24. Pair-Wise Consistency Analysis
1 2 3 4 5 6 7 8 9 10 11
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Dataset Unit
ConsistencyMeasure
12 13 14 15 16 17 18 19 20 21 22
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Dataset Unit
ConsistencyMeasure
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

25. Sensitivity Analysis: Lagrangian Multipliers
0 10 20 30 40 50 60
−0.5
0
0.5
1
1.5
2
Lagrangian Multiplier Number
LagrangianMultiplierValue
0 5 10 15 20 25 30 35 40 45 50
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Lagrangian Multiplier Number
LagrangianMultiplierValue
Original dataset Dataset after 1 data point removed
0 5 10 15 20 25 30 35 40 45 50
−0.5
0
0.5
1
1.5
2
Lagrangian Multiplier Number
LagrangianMultiplierValue
0 5 10 15 20 25 30 35 40 45
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Lagrangian Multiplier Number
LagrangianMultiplierValue
Dataset after 2 data points removed Dataset after 3 data points removed
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

26. Experiment: Klassen and Gore (1992)
Figure: Heptane
Figure: Methanol
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

27. Experiment: Klassen and Gore (1992)
30-centimeter diameter pool fires (Heptane and Methanol)
Fuel replenished from reservoir to maintain pool level
Load cells with reservoir track mass loss of fuel
Nitrogen-purged Gardon type gauges track incident radiation
to pool surface
Optical properties of flame also studied
Simulations use 90cm x 90cm x 90cm domain
Pool inlet situated at the bottom-center of domain
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

28. Computational Domain
Figure: Simple diagram of the pool domain
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

29. Pool Reaction Acceleration
0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (seconds)
OverallMassBurnRate(kg/m2
−sec)
Figure: Normal operation
0 2 4 6 8 10 12
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (seconds)OverallMassBurnRate(kg/m2
−sec)
Figure: Accelerated
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

30. Fuel Mass Flux from Pool Surface
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

31. Gas-Phase Fuel Composition Near Pool Surface
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

32. Liquid Surface Temperature (Boiling Point = 371.57 K)
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

33. Radiative Heat Flux to Pool Surface
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

34. Convective Heat Flux to Pool Surface
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

35. Flame Temperature over Pool Surface
Figure: Transient phase Figure: Steady phase
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

36. Grid Convergence: Heptane
0 2 4 6 8 10 12 14 16 18
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (Seconds)
AverageMassBurnFlux(kg/m2
−s)
Resolution: 643
Resolution: 803
Resolution: 1003
Resolution: 1253
Resoluiton: 1503
Resolution: 1953
Resolution: 2453
0 2 4 6 8 10 12 14 16 18
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
4
Time (Seconds)
AverageRadiativeHeatFlux(W/m2
)
Resolution: 643
Resolution: 803
Resolution: 1003
Resolution: 1253
Resoluiton: 1503
Resolution: 1953
Resolution: 2453
Mass flux Incident radiation
0 2 4 6 8 10 12 14 16 18
−0.5
0
0.5
1
1.5
2
2.5
3
x 10
4
Time (Seconds)
AverageConvectiveHeatFlux(W/m2
)
Resolution: 643
Resolution: 803
Resolution: 1003
Resolution: 1253
Resoluiton: 1503
Resolution: 1953
Resolution: 2453
Incident convection
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

37. Grid Convergence: Methanol
0 1 2 3 4 5 6 7 8 9 10
0
0.005
0.01
0.015
0.02
0.025
Time (Seconds)
AverageMassBurnFlux(kg/m2
−s)
Resolution: 803
Resolution: 1003
Resolution: 1253
Resolution: 1573
Resolution: 1963
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
x 10
4
Time (Seconds)
AverageRadiativeHeatFlux(W/m2
)
Resolution: 803
Resolution: 1003
Resolution: 1253
Resolution: 1573
Resolution: 1963
Mass flux Incident radiation
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Time (Seconds)
AverageConvectiveHeatFlux(W/m2
)
Resolution: 803
Resolution: 1003
Resolution: 1253
Resolution: 1573
Resolution: 1963
Incident convection
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

38. Sources of Error
Computational Sources:
Estimation of fuel
thermophysical properties
Mass transfer model
ARCHES empirical soot
model
Two primary variables
chosen:
1- Mass transfer
coefficients (±15%)
2- Empirical soot
coefficient (0.1 - 10.0)
Experimental Sources:
Measurement of mass flux
(author’s ”estimate”
±5%)
Measurements of incident
radiation (estimate of
±10%)
Gardon-type gauges used
in buoyant fires are known
to have error as high as
±40% (Nakos, 2005)
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

39. Experiment/Simulation Data Comparison
2.46 2.48 2.5 2.52 2.54 2.56 2.58 2.6 2.62 2.64 2.66
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
MassBurnRate(kg/m2
−sec)
Experiment
Computation
Figure: Mass flux
−2 0 2 4 6 8 10 12 14 16
10
15
20
25
30
35
40
Pool Radius (centimeters)
IncidentRadiation(kW/m2
)
Experiment
Computation
Figure: Radiative flux
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

40. Pair-Wise Consistency Analysis
1 2 3 4 5 6 7 8 9 10
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1ConsistencyMeasure
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

41. Consistency Analysis
To find a region of consistency for this dataset an error of
±40% must be assumed for the radiation readings, and an
error of ±15% must be assumed for the mass flux
measurements
There are at least two possibilities to expalin the
inconsistency:
1. Both ARCHES and the Responsive Boundary model have
error issues
2. The radiation measurements have an instrumental bias
that is causing under-measurement
Condensation on the sensing element could effect the flux
readings
Liquid fuel spillage on the sensing element would lower the
measured flux readings
Issues with the flux meter’s view angle and the calibration
process have been known to cause error in the flux meter’s
readings especially near the edges of the pool.
Weston Eldredge Development, Verification, and Validation of the Responsive Boun

42. Questions or Comments?
Weston Eldredge Development, Verification, and Validation of the Responsive Boun