Summary of Analytical and Computational Work Performed in Development of Thesis
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Transcript of Summary of Analytical and Computational Work Performed in Development of Thesis
Summary of Analytical and Computational Work Performed in
Development of Thesis
Johnathan R. Williams
1
General Overview• Objective of Thesis was to demonstrate the application of Monte
Carlo methods for Radiant Heat Exchange.• Three general problems were worked on to demonstrate this.• The first problem featured use of the Monte Carlo method to
determine the configuration factor for Radiation Heat transfer from an elemental area to a circular area.
• The second problem featured use of the Monte Carlo method to determine the configuration factor for Radiation Heat Transfer between two areas where shielding was present.
• The third problem involved application of the Monte Carlo method for Radiative Heat Exchange in a problem where conduction and convection were present at the boundaries.
2
Problem #1
Configuration Factor for Radiation Heat Transfer from an Elemental Area to a Circular Area•Published configuration factors already exist for a number of simple geometries.•This simple problem serves as initial validation for the application of the Monte Carlo method.•Configuration Factor was analytically determined using published configuration factor and compared to Monte Carlo solution.•Monte Carlo script was prepared using Visual Basic within Excel.
3
Problem Geometry & Analytical Solution
4
According to Thermal Radiation Heat Transfer, 4th Edition, by Robert Siegel and John Howell, the Configuration Factor is:
22
2
21 rh
rFd
It was arbitrarily assumed that r = 4 and h = 10 (units are unimportant as they will cancel, resulting in:
1379.116
16
410
422
2
21
dF
Monte Carlo Geometry
5
Development of Monte Carlo Code• For an elemental area, initial position is not a factor, thus we have two
parameters of concern ( and ).• is the angle at which the photon bundle is emitted relative to the
elemental areas normal. It can vary from 0 to /2.• is the cone angle. It can vary from 0 to 2.• For this problem is unimportant. As long as results in a hit, there will
be a hit anywhere within the cone angle.• The configuration factor is independent of surface properties. It only
depends on geometry. This allows for assumption of blackbody properties, which is computationally the simplest assumption.
• Random numbers between 0 and 1 are assigned to the cumulative density function for theta, R.
• For a blackbody (or a graybody) R = sin2.• is then calculated using the above relationship and compared to an
acceptance standard for a hit.
6
Acceptance Standard for
7
10
4
10
4
Acceptance standard, s is the solution of the following:
10
4tans
3805.s
Flowchart for Code Execution
8
User inputs number of trials (N), h
and r
Number of hits &
counter initially set to
0
Determine acceptance
standard based on r
and h
Is counter = N?
Randomly assign R and
calculate
Is less than standard?
Hits = Hits + 1
F = Hits/Ncounter =
counter + 1 No
No
Yes
Obtained Mean and Variability
9
Size 1000 10000 100000 1000000(VB) 1000000(CB)Run
1 0.124 0.1328 0.13917 0.137931 0.1378242 0.127 0.1385 0.13469 0.137702 0.1383963 0.138 0.139 0.13728 0.137578 0.1374714 0.125 0.1428 0.13876 0.138084 0.1382525 0.136 0.1359 0.13812 0.138079 0.1374436 0.138 0.1356 0.13856 0.138006 0.138617 0.15 0.1359 0.13846 0.138067 0.1379748 0.142 0.1337 0.13827 0.137788 0.1383449 0.155 0.1373 0.13876 0.137093 0.137651
10 0.147 0.1404 0.13923 0.137852 0.138808Mean 0.1382 0.13719 0.13813 0.137818 0.1380773Stand. Dev. 0.010644 0.003056668 0.001331457 0.000306907 0.00047696Variance 0.000113 9.34322E-06 1.77278E-06 9.4192E-08 2.2749E-07
Shown above are means, standard deviations, and variances obtained using 1000, 10000, 100000, and 1000000 trials. The solution for 1000000 trials was obtained using the visual basic macro and also using crystal ball. Recall that the predicted configuration factor was .1379. Note that the standard deviation/variance tends to get smaller with an increasing number of trials.
Obtained Distribution of for 10,000,000 Trials
10
Note that 13.76907% of the data has less than an angle of .3805. In other words, the simulated configuration factor is .1376907 for this case.
Fit of Data Trend Line
11
The statistics package in crystal ball reveals that the distribution of for 1,000,000 trials can be represented by a beta distribution with a min of -.01, a max of 1.58, an alpha of 2.21323 and a beta of 2.21116.
Summary of Data for Problem #1
• Data obtained from Monte Carlo simulation correlated well to predicted configuration factor.
• For 1,000,000 trials, the mean configuration factor for 10 observations was .137818 (a percent error of 0.0595%).
• The variance predictably drops with increasing number of trials. For 1,000,000 trials, the variance for 10 determinations of the configuration factor is only 9.4192*10-8.
12
Problem #2
Configuration Factor for Radiation Heat Transfer from One Area to Another With Shielding Present•This is another problem type that can be easily solved analytically.•This serves as additional validation for the application of the Monte Carlo method.•Analytical Solution is via Hottel’s Crossed String Method.•Monte Carlo script was modified to account for additional parameters and hit acceptance criteria.
13
Problem Geometry
14
Looking for the fraction of thermal radiation leaving A1 that is incident on A2
Analytical Solution – Left Enclosure
15
c c/2
.7906c
.7906c
.7906c
.7906cSum of crossed lines = c*4*.7906Sum of crossed lines = 3.1624cSum of uncrossed lines = c*(1+.5+2*.7906)Sum of uncrossed lines = 3.0812c
Analytical Solution – Right Enclosure
16
cc/2
1.4142c
1.4142c
.3536c
.3536c Sum of crossed lines = c*2*1.4142Sum of crossed lines = 2.8284cSum of uncrossed lines = c*(1+.5+2*.3536)Sum of uncrossed lines = 2.2072c
Analytical Solution – Combined
17
A1*F1-2 = .5*(sum of crossed lines – sum of uncrossed lines)c*F1-2 = .5*(3.1624c + 2.8284c – 3.0812c – 2.2072c)c*F1-2 = .3512cF1-2 = .3512
Monte Carlo Geometry
18
Initial Position Randomly Chosen1
r
r
1
r2 = r*cos(1)
As a result of cone angle, photonnow collides with the shield
2
2r
Development of Monte Carlo Code• Now that there are two areas, initial position has become a factor. We now have three parameters
of concern (Initial Position, and ).• is the angle at which the photon bundle is emitted relative to the elemental areas normal. It can
vary from 0 to /2.• is the cone angle. It can vary from 0 to 2.• For this problem is now important. A non-zero will cause the distance traveled parallel to
A1/A2 for a photon bundle per distance traveled perpendicular to A1/A2 to be less (Reference the proceeding slide).
• The configuration factor is independent of surface properties. It only depends on geometry. This allows for assumption of blackbody properties, which is computationally the simplest assumption.
• Random numbers between 0 and 1 are assigned to the cumulative density function for initial position, Rpos.
• Initial position = c*Rpos.• The acceptance standard for is calculated based on initial position (the photon bundle can’t hit
the shield or leave the system boundary if a hit is to occur).• Random numbers between 0 and 1 are assigned to the cumulative density function for theta, R.• For a blackbody (or a graybody) R = sin2.• Random numbers between 0 and 1 are assigned to the cumulative density function for phi, R.• For a diffuse material, R = 2.
19
Flowchart for Code Execution
20
User inputs number of trials (N)
Number of hits &
counter initially set to
0
Determine acceptance
standard based initial
position
Is counter = N?
Calculate 1, 1 and 2
Is within the standard
range?
Hits = Hits + 1
F = Hits/N
counter = counter + 1
Randomly assign initial position, R
and R
No
Yes
No
Yes
Results for 1,000,000 trials
21
Size 1000000Run
1 0.3510912 0.3514223 0.3516294 0.3508815 0.3514166 0.3508647 0.3501418 0.3512279 0.351143
10 0.351133Mean 0.3510947Stand. Dev. 0.000411715Variance 1.6951E-07
Summary of Data for Problem #2
• Data obtained from Monte Carlo simulation correlated well to predicted configuration factor.
• For 1,000,000 trials, the mean configuration factor for 10 observations was .3510947 (a percent error of 0.0300%).
22
Problem #3Solution of a Problem with Conduction and Convection at the boundaries•Most problems feature combinations of all three modes of heat transfer to varying degrees.•A basic cylinder shape (similar to a combustor or many other flow problems) was analyzed.•A finite difference model was constructed in cylindrical coordinates with conduction and convection.•The finite difference model was validated via comparison to a similar model created in ANSYS.•Next, Radiation was introduced to the finite difference model.•For the first attempt, a macroscopic configuration factor was determined. It was assumed for the first attempt that radiation from each node would uniformly affect the other nodes.•After validating this model, a Monte Carlo solution for the configuration factor from each node to all other nodes was applied.•This is a computationally effective solution for opaque radiation heat transfer.
23
Problem #3 Parameters
24
Cylinder GeometryLength (L) = 3 ft = 36 in = 91.44 cmDiameter (D) = 1.5 ft = 18 in = 45.72 cmThickness (t) = 1 in = 2.54 cmMaterial PropertiesAssume that the material is a high temp Ni-AlloyDensity () = 8900 kg/m3
Specific Heat (cp) = 446 J/kg-KConductivity (k) = 91 W/m-KEmissivity () = 0.41Fluid PropertiesInlet Temperature (Tin) = 2500 F = 1644 KMass Flow Rate ( ) = 80 lbm/s = 36.288 kg/sDensity () = .2141 kg/m3
Dynamic Viscosity () = 57.4x10-6 Pa-sSpecific Heat (cp) = 1257 J/kg-KConductivity (k) = 10.25x10-2 W/m-K
m
Determination of Film Coefficient
25
1760577Re
104.57
4572./4.1032/2141.Re
/4.1032
16417./2141.
/288.36
16417.4
4572.
4
6
3
23
222
sPa
msmmkgvD
smv
mmkg
skg
A
mv
mmD
A
Avm
Flow is Turbulent
Determination of Film Coefficient (Cont.)
26
Km
Wh
m
KmW
D
Nukh
k
hDNu
Nu
Nu
Nu
KmW
KkgJsPa
k
cp
2
2
8.05.0
8.05.0
2
6
51.410
4572.
/1025.101.1831
1.1831
176057770392.0022.0
RePr022.0
70392.0Pr/1025.10
/1257104.57Pr
Governing Equations
27
Without Radiation, the governing equations are:
q” = 0 at the left boundaryq” = -500,000 W/m2 at the right boundaryq” = 0 at the lower boundary
At the upper boundary:
ThAy
TkA
qq cky
""
Through the interior:
02 TThe right boundary condition is a little bit odd, but was intentionally chosen for future model validation, a relatively easy convective solution, and it still allows for a suitable demonstration of the application of the Monte Carlo method.
Solution for Temperatures
28
After discretizing the governing equations, the following system of equations is obtained:
[A][T] = [C]
Where:[A] is a coefficient matrix for nodal temperatures[T] is the matrix containing the nodal temperatures[C] is the results matrix (mainly populated with zeros or heat flux where appropriate)
This system of equations is linear and can readily be solved either explicitly or implicitly.
Results of Finite Difference Model
29
Axial Distance (in)
Rad
ial D
ista
nce
(in)
Finite Difference Prediction for Solid (No Radiation, Temp in K)
0 6 12 18 24 30 36
0.00
0.25
0.50
0.75
1.00
1250
1300
1350
1400
1450
1500
1550
1600
Results of Finite Difference Model
30
0 6 12 18 24 30 361643.55
1643.6
1643.65
1643.7
1643.75
1643.8
1643.85
1643.9
1643.95
1644
1644.05
Axial Distance (in)
Flu
id T
empe
ratu
re (
Kel
vin)
Finite Difference Prediction for Fluid (No Radiation)
Validation of Finite Difference Model
31
Results of Finite Difference Model (English Units)
32
Axial Distance (in)
Rad
ial D
ista
nce
(in)
Finite Difference Prediction for Solid (No Radiation, Temp in F)
0 6 12 18 24 30 36
0.00
0.25
0.50
0.75
1.00
1800
1900
2000
2100
2200
2300
2400
Results of Finite Difference Model (English Units)
33
0 6 12 18 24 30 362499
2499.1
2499.2
2499.3
2499.4
2499.5
2499.6
2499.7
2499.8
2499.9
Axial Distance (in)
Flu
id T
empe
ratu
re (
Fah
renh
eit)
Finite Difference Prediction for Fluid (No Radiation)
Extension of Radiation to Model
34
• For initial extension of Radiation Heat Transfer to finite difference model, it is assumed that Radiative Heat Transfer only involves opaque surface to surface exchange.
• A macroscopic configuration factor for exchange along the entire cylindrical area to the same cylindrical area is determined analytically.
• The configuration factor is applied to each node and it is assumed that each node will uniformly distribute thermal radiation to all other nodes (including itself).
• This is used to construct a coefficient matrix for Radiation Heat Transfer.
• Because Radiation Heat Transfer is a function of the difference of temperature to the fourth power, the governing equation for the finite difference model is now non-linear and requires an iterative solution. A successive under-relaxation technique is employed.
Determination of Configuration Factor
35
Definitions: R = r/a; X = (2R2+1)/R2
055728.0
320185.418182
1
180625.
125.1
25.
125.2
25.36
9
42
1
21
5.2/1221
2
2
21221
F
F
X
R
XXF
Everything else leaving surface 1 impacts the cylinder wall!Calling the cylinder wall surface 3F1-3 = 1-F1-2 = 1-0.055728 = 0.944272
Determination of Configuration Factor (Cont.)
36
A1F1-3 = A3F3-1
F3-1 = (A1F1-3)/A3
A1 = D2/4 = *(18in)2/4 = 254.47in2
A3 = DL = *(18in)*(36in) = 2035.8in2
F3-1 = (254.47in2*0.944272)/2035.8in2 = .11803F3-2 = F3-1 = .11803F3-3 = 1-F3-1-F3-2 = 1-2*.11803 = .76394For this model, there are 144 axial divisions (145 nodes)Assuming that each node radiates uniformly to each other node, the configuration factor for node x to node y is:Fx-y = .76394/145 = .005269
Governing Equations
37
The same boundary conditions and governing equations that were applied for the No Radiation case are applied. The only difference is that along the upper boundary, we also have a Radiation term.
The governing equation for Radiative Exchange between node x and node y is:
q”r = Fx-y(Tx4 – Ty
4)
Where:
= Stefan-Boltzman constant (5.67x10-8 W/m2K4) = EmissivityFx-y = The configuration factor for node x to y
Solution for Temperatures
38
After discretizing, the system of equations now becomes:[A][T] + [B][T4] = [C]The difference being a T4 term and its associated coefficient matrix
To solve this, the system can be rewritten as:[A*][T] = [C]
Where:[A*] = [A + BT3]
This system of equations is no longer linear and must be solved implicitly. A SUR subroutine is employed.
Results of Finite Difference Model (English Units)
39
Axial Distance (in)
Rad
ial D
ista
nce
(in)
Finite Difference Prediction for Solid (Uniform config. factor, Temp. in F)
0 6 12 18 24 30 36
0.00
0.25
0.50
0.75
1.00
1800
1900
2000
2100
2200
2300
2400
Results of Finite Difference Model (English Units)
40
0 6 12 18 24 30 362498.9
2499
2499.1
2499.2
2499.3
2499.4
2499.5
2499.6
2499.7
2499.8
2499.9
Axial Distance (in)
Flu
id T
empe
ratu
re (
Fah
renh
eit)
Finite Difference Prediction for Fluid (Uniform config. factor)
Development of Monte Carlo Code• We again have three parameters of concern (Initial Position, and ).• is the angle at which the photon bundle is emitted relative to the elemental areas normal. It can
vary from 0 to /2.• is the cone angle. It can vary from 0 to 2.• The configuration factor is independent of surface properties. It only depends on geometry. This
allows for assumption of blackbody properties, which is computationally the simplest assumption.• Random numbers between 0 and 1 are assigned to the cumulative density function for initial
position, Rpos for an interval around each node.• Initial position = c*Rpos + the Initial Position of the Node Interval.• Random numbers between 0 and 1 are assigned to the cumulative density function for theta, R.• For a blackbody (or a graybody) R = sin2.• Random numbers between 0 and 1 are assigned to the cumulative density function for phi, R.• For a diffuse material, R = 2.• What makes this problem tricky is that the cylinder wall has curvature (it’s circular).• We know that if is 90 degrees and is 0 or 180 degrees, a photon bundle will pass without hitting
the cylinder. Otherwise, if is 90 degrees, it will immediately impact the cylinder wall (at the node location of emission). If is 0 degrees, it will impact the cylinder wall (directly across from the initial position at the node location of emission). If is 90 degrees or 270 degrees, it will impact the node location of emission.
41
Determination of Impact Location• Unique results for special angles of and were discussed
on the proceeding slide.• For all other angles, the photon impact location must be
identified.• The location of photon emission can be treated as bottom
dead center.• 1 determines the “height” of photon travel per unit length
traveled.• 1 determines the distance the photon bundle travels
radially per unit length traveled.• The photon impacts the cylinder wall when the radial
location of the photon meets the radial location of the cylinder wall at that height.
42
Determination of Impact Location
43
11
190 h
l
1
1
tan
tan
hlh
l
1
l
x
11
1
1
tantan
tan
tan
hx
lxl
x
Emission Location & Origin
Equation of circle is:22
2
22
DDhx
We now have two simultaneous equations to solve:
1
12
12
12
12
2
12
122
21
21
22
222
12
122
222
11
tan
tantan1tantan12
0tantan1
0tantan
44tantan
22tantan
hl
DDDh
Dhh
Dhhh
DDDhhh
DDhh
From l, we know the node that is hit!!From the fraction of photon bundles leaving node x and impacting node y, we have the configuration factor for x to y.
This permits integration into the finite difference model.
Results of Finite Difference Model (English Units)
44
Axial Distance (in)
Rad
ial D
ista
nce
(in)
Finite Difference Prediction for Solid (Monte Carlo determined config. factors, Temp. in F)
0 6 12 18 24 30 36
0.00
0.25
0.50
0.75
1.00 1700
1800
1900
2000
2100
2200
2300
2400
Results of Finite Difference Model (English Units)
45
0 6 12 18 24 30 362498.9
2499
2499.1
2499.2
2499.3
2499.4
2499.5
2499.6
2499.7
2499.8
2499.9
Axial Distance (in)
Flu
id T
empe
ratu
re (
Fah
renh
eit)
Finite Difference Prediction for Fluid (Monte Carlo determined config. factors)