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PHOTON USE
AUTOPLOT
file
PHI 5

cl;d 1 h1;d 1 ha;col3 1;blb4 2;redr
msg    temperature profile; press  to continue
pause
cl;d 1 radx;d 1 ra;col3 1;blb4 2;redr
msg    radiation-flux profile; press e to end
pause;end
ENDUSE

GROUP 1. Run title and other preliminaries

TITLE

DISPLAY
The problem considered is that of pure radiative heat
transfer in a 1d plane-parallel slab containing a uniformly
distributed heat source. The medium may absorb, emit and
scatter radiation and the boundaries of the slab are diffuse
emitters and reflecters kept at the same uniform temperature.
The medium is gray and because the wall temperatures are
equal, symmetry can be exploited.
ENDDIS

Since the energy transfer is by pure radiation the
energy equation is given by:

-d/dx (Qrad) + Qvol = 0

where Qrad is the radiative heat flux and Qvol is the uniform
volumetric heat generation rate in the medium. The equation
for the composite radiative heat flux is given by:

d/dx ( 1/(a+s) d/dx (Rx) ) + a (E - Rx) = 0

where a is the absorption coefficient, s is the scattering
coefficient, Rx is the composite radiation flux defined as
the average of the +ve and -ve radiation fluxes, and E is
the black-body emissive power. It may be noted that the
radiative heat flux is given by:

d/dx (Qrad) = 2a (E - Rx)

The black-body emissive power E=sig*T**4  where sig is the
Stefan-Boltzmann constant and T is the temperature of the
medium. The problem is the determination of the temperature
and composite radiative-flux distributions, as given by the
following analytical solutions:

Rx = Ew + 0.5*Qvol*L* [ 2/emw-1+0.5*(a+s)*L*{1-(x/L)**2} ]

E  = Ew + Qvol*L*[ 1./(2*a*L) + 1./emw - 0.5

+ 0.25*(a+s)*L*{1 - (x/L)**2} ]

where Ew is the emissive power at the wall, L is the slab
width from symmetry plane to wall, and emw is the emissivity
of the wall. For the case considered below, Qvol is taken to
be Qvol = Ew/L/(1./emw - 0.5).

The locally-defined parameters are as follows:
GSIGMA         Stefan-Boltzmann constant    {W/m**2/K**4}
SCAT           Scattering coefficient       {1/m}
ABSORB         Absorption coefficient       {1/m}
EMIW           emissivity of the wall
TWAL           hot wall temperature         { K }
QVOL           volumetric internal heat source {W/m**3}

CHAR(CH1);REAL(GSIGMA,SCAT,ABSORB,EMIW,TWAL,QVOL)
GSIGMA=5.6697E-8;SCAT=0.5;ABSORB=0.5;EMIW=1.0;TWAL=1000.0
GROUPS 3,4,5. X,Y,Z-direction grid specification
REAL(LENGTH);LENGTH=1.0;GRDPWR(X,50,LENGTH,1.0)
GROUP 7. Variables stored, solved & named

GROUP 7. Variables stored, solved & named
CP1=1.0
MESG( Enter required energy variable ? (TEM1 or H1)
IF(:CH1:.EQ.TEM1) THEN
+ MESG( TEM1 solution selected
ELSE
+ MESG( H1 solution selected
+ TMP1=LINH;TMP1B=1.0/CP1
ENDIF

CP1=1.0
MESG( Enter required energy variable ? (TEM1 or H1)
IF(:CH1:.EQ.' ') THEN
ENDIF
IF(:CH1:.EQ.TEM1) THEN
+ MESG( TEM1 solution selected
ELSE
+ MESG( H1 solution selected
+ TMP1=LINH;TMP1B=1.0/CP1
ENDIF
GROUP 8. Terms (in differential equations) & devices
** Deactive built-in sources, convection and conduction
TERMS(:CH1:,N,N,N,N,P,P)
GROUP 11. Initialization of variable or porosity fields
** Define Qvol = Ew/L/ (1./emw - 0.5)
** analytical solution
STORE(HA,RA);INTEGER(JJM1)
BET=0.5*(ABSORB+SCAT)*XULAST;ACON=1.0/(2.*ABSORB*XULAST)+0.5*ALF
DO JJ=1,NX
+PATCH(IN:JJ:,INIVAL,JJ,JJ,1,NY,1,NZ,1,1)
+GX=0.5*XFRAC(JJ)
IF(JJ.NE.1) THEN
+JJM1=JJ-1;GX=XFRAC(JJM1)+0.5*(XFRAC(JJ)-XFRAC(JJM1))
ENDIF
+GX=GX*XULAST;GXRAT=(GX/XULAST)**2
+EG=EW+QVOL*XULAST*(ACON+0.5*BET*(1.-GXRAT))
+RAN=EW+GAM*(ALF+BET*(1.0-GXRAT));TA=(EG/GSIGMA)**0.25
+INIT(IN:JJ:,RA,ZERO,RAN);INIT(IN:JJ:,HA,ZERO,TA)
ENDDO
GROUP 13. Boundary conditions and special sources
** Net radiation flux from wall
PATCH(WALLR,EAST,NX,NX,1,NY,1,NZ,1,1)
** uniformly-distributed volumetric heat source
PATCH(QHEAT,VOLUME,1,NX,1,NY,1,NZ,1,1)
COVAL(QHEAT,:CH1:,FIXFLU,QVOL)
GROUP 15. Termination of sweeps
LSWEEP=50
GROUP 16. Termination of iterations