# 978-0123869449 Chapter 8 Part 1

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Homework Help
Book Title
Authors
Michael F. Modest CHAPTER 8
8.1 Prepare a little Monte Carlo code that integrates I(z)=Rb
af(z,x)dx. Apply your code to a few simple integrals,
plus
si(z)=Zπ/2
0
ezcos xcos(zsin x)dx =Si(z)π
2.
Note: Si(1) =0.94608.
A=0.
B=1.570796
WRITE(*,*) ’INPUT # OF BUNDLES’
FINT=0.
DO I=1,NBUNDLS
X=A+(B-A)*RAN1(ISEED)
END
0 1000 2000 3000 4000 5000
0.92
0.94
0.95
0.97 8.2 In a Monte Carlo simulation involving the plate of Problem 3.9 but of finite width w, a photon bundle is to be
emitted from the plate with a wavelength of λ=2µm. Find the emission point and direction of this photon
bundle in terms of random numbers.
0.(cos4θ)
0 CHAPTER 8 243
8.3
w
w
A triangular, isothermal surface as shown has the following spectral emittance:
ǫλ=
0.1, λ < 2µm; θ60
0.6, λ > 2µm; θ60
0.0,all λ;θ > 60
For a Monte Carlo simulation
(a) find a point of bundle emission in terms of random numbers,
(b) find a wavelength of bundle emission in terms of random numbers,
(c) find a direction of bundle emission in terms of random numbers. =0.6f(λT)0.5f(λcT)λ>λc
0.60.5f(λcT), λ < λc,
0.60.5f(λcT), λ > λc, CHAPTER 8 245
8.4 A semicircular disk as shown has a temperature distribution given by T(r)=
T0/1+(r/R)2, and its emittance is gray and nondiuse with
ǫ=ǫ
λ(λ, θ, ψ)=(0.6, 0 θ30,
0, θ > 30.
For a Monte Carlo simulation
(a) find a point of emission in terms of random numbers,
(b) find a direction of emission in terms of random numbers. You may leave your answer in simple implicit
form. 8.5 A light pipe with direct solar irradiation is to be investigated via a Monte Carlo method. Such a device consists
of a straight or curved tube covered with a highly-reﬂective material to pipe light into a room. At visible
wavelengths the reﬂectance from the pipe wall is ρ
λ(θout)=1.5ρλcos θout, with reﬂection angle θout measured
from the local surface normal, and visible light intensity due to direct sunshine may be approximated by
Lλ=KλIλ,sun =Cexp[A2(λλ0)2], λ0=0.56µm,A=20m.
(a) Find the pertinent relationship to determine wavelengths of emission as a function of random number.
(b) Find an expression for reﬂection angle vs. random number.
(a) CHAPTER 8 247
8.6 At the Aaronsburg (Pennsylvania) Apple Fest you have won a large piece of el-
derberry pie (yumh!) as shown. The wheels in the oven must have been spinning,
because it appears that the number of elderberries per unit area increases lin-
early proportional with radius! If there are 1000 elderberries otherwise randomly
distributed on the slice, make a scatter plot of elderberries on the pie slice.
x
x 8.7 Consider a black disk 0 rRwith temperature distribution T4(r)=T4
0eC(r/R)2. Develop the random
number relations for points of emission; draw random numbers for 1000 emission points and draw them in
a scattergram for the cases of C=0 and C=5. Use R=10 cm.
CE=1.-exp(-C)
DO i=1,NB
!
! point of emission
!
IF(CE>0.) THEN 8.8 A disk of radius Ris opposed by a square plate (sides of length R) parallel to it, and a distance Raway. Find
the view factor from disk to square plate. Use 100,000 bundles, plotting updated results after every 5,000
bundles.
F=0.
!
! break up into 20 samples
!
nout=NB/20
DO i=1,20
DO j=1,nout
!
sy=sint*sin(psi)
sz=sqrt(1-sint2)
!
! photon distance to square, check x,y-coordinates there
!
d=R/sz  CHAPTER 8 251
8.9 Consider two infinitely long parallel plates of width wspaced a distance hapart (see Configuration 32) in
Appendix D).
(a) Calculate F12via Monte Carlo for the case that the top plate is horizontally displaced by a distance L. Use
L=h=w.
(b) Calculate Fs
12via Monte Carlo for the case that both plates are specular (with identical reﬂectances
ρs
1=ρs
2=0.5), but not horizontally displaced. Use L=0; h=w.
Prepare a figure similar to Fig. 8-2, also including analytical results for comparison.
PI=4.*ATAN(1.)
! SET LENGTHS, EMISSIVITIES
EPS2=0.5 ! assumed diffuse
WW=1. ! width of plates
HH=1. ! distance between plates
!
! EXACT VIEW FACTOR (FOR ABOVE AL,WW,HH)
!F12EX=(.5*(SQRT((AL+WW)**2+HH**2)+SQRT((AL-WW)**2+HH**2))-SQRT(AL**2+HH**2))/WW
!WRITE(*,*) ’INPUT # OF BUNDLES’ PSI=2.*PI*RAN1(ISEED) ! azimuthal angle of emission
COSPS=COS(PSI)
SINPS=SIN(PSI)
SINSQTH=RAN1(ISEED) ! sinˆ2 of polar angle of emission
NNOW=100.*K
WRITE(LU,2000) NNOW,F12/NNOW,F12EX
1000 CONTINUE
2000 FORMAT(I6,2E12.4)
STOP
END
EPS1=0.5 ! assumed specular
AL=0. ! horizontal displacement of plates (here = 0)
HH=1. ! distance between plates
!
! EXACT VIEW FACTORS (FOR ABOVE AL,WW,HH)
!N=0
10 N=N+1
F(N)=SQRT(1.+N*N)-N

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