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Appendix II Model Theory

The proceeding sections give details on how various physical characteristics are handled by GunRad.

Bundle Energy (per unit time)

Ebundle - Specified in the input file, it is constant for the entire run.

Number of Bundles to Send per Wall or Media Element (per unit time)

[ ADD EQUATIONS] A - element area
e - emissivity
Ke - extinction coefficient
q - heat flux
V - element volume
w - single scattering albedo
s - Stefan-Boltzmann constant
T - Temperature

Note: Ka = (1-w)Ke & Ke = Ks + Ka

Location to Send a Bundle

x = xi + Random1 x deltax
y = yi + Random2 x deltay
z = zi + Random3 x deltaz

Randomn - A random number between zero and one inclusively. 'n' is to indicate that a random number is only used once. xi - 'x' location of the ith element
yi - 'y' location of the ith element
zi - 'z' location of the ith element
deltax - 'x' length of the element
deltay - 'y' length of the element
deltax - 'z' length of the element

Only x & y are needed for the Wall elements and they are in local coordinates. X, y, & z are required for the Media and they are in global/ box coordinates.

Direction to Send Bundle

Wall (in local coordinates): q = sin-1([Random1]0.5) j = 2p x Random2
Polar and Azimuthal angles

ni = sin(q)cos(j) nj = sin(q)sin(j) ni = cos(q)
Direction vector components

Media (in global/ box coordinates): q = cos-1(1-2 x Random1) j = 2p x Random2
Polar and Azimuthal angles

ni = sin(q)cos(j) nj = sin(q)sin(j) ni = cos(q)
Direction vector components

Absorption with the Wall or Media

Wall: Absorbed if Random < e , otherwise reflected.
Media: Absorbed if Random > w, otherwise it is scattered.

Bundle Direction When Reflected Off Of The Wall

Diffuse reflectivity: A new random direction is calculated as stated in the "Direction to Send Bundle" section.

Specular reflective: The bundles direction is reflected off the wall.

Bundle Direction When Scattered by the Media

Isotropic scattering: A new random direction is calculated as stated in the "Direction to Send Bundle"section.

Bundle Interaction

tint = - log(Random)
twall = lwall x Ke
lwall - Length to the intersecting wall
tint - Optical length to media interaction
twall - Optical length to the intersecting wall

If tint is less than or equal to twall, the bundle will interact with the media
If tint is greater than twall, the bundle will interact with the wall

When a bundle has an interaction with the media or the wall and it is not absorbed, a new tint, twall, and direction are calculated.

Wall Element Temperature

T = [(Ebundle x nreceived)/(Aes)]0.25 nreceived - the number of bundles received

Media Element Temperature

T = [(Ebundle x nreceived)/(4VKe(1-w)s)]0.25 nreceived - the number of bundles received

Wall & Media Element Incoming Flux

q = Ebundle x nreceived

Unknown Wall or Media Temperature

The following steps are used to determine an unknown wall or media temperature:
  1. Initially, all of the elements of the unknown wall and/or media are set to send zero bundles.
  2. The model is ran and the number of bundles received per element is recorded.
  3. The percent error of an element based on net flux is calculated.
    Qnet = qfinal x A
    Qsent = nsent x A
    Qreceived = nreceived x A
    Percent = 100 x ABS(ABS(Qsent - Qnet)/Qreceived - 1)
    ABS - Absolute value
  4. The number of bundles to send for the next iteration is calculated.
    nsendi+1 = nsendi +FI (nreceived - nsendi) + (Aqi/Ebundle)
    Wall element

    nsendi+1 = nsendi +FI (nreceived - nsendi) + (Vqi/Ebundle)
    Media element

    FI - iteration factor (default value is one)
    i - iteration number
  5. The model is ran again.
  6. Steps three through five are repeated until one of the following conditions is true:
    1. The percent error for all of the elements is less than the maximum specified value.
    2. The iteration number is equal or greater than maximum specified value.
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