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)