A second computer program was developed for the simulation of an n - C3F7I iodine laser. The computer program is given in Appendix A and a typical output from the computer program is illustrated in Appendix B

Heinbockel, John H.

English

The previously constructed one dimensional model for the simulated operation of an iodine laser assumed that the perfluoroalkyl iodide gas n-C3F7I was incompressible. The present study removes this simplifying assumption and considers n-C3F7I as a compressible fluid

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DEPARTMENT OF MATHEMATICAL SCIENCES
'COLLEGE OF SCIENCES
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
THEORETICAL STUDIES OF SOLAR LASERS AND CONVERTERS
By
John H. Heinbockel, Principal Investigator
Progress Report
For the period January 1988 to July 1988
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Grant NAG-1-757
Dr. Robert C. Costen, Technical Monitor
SSD-High Energy Science Branch
N88-23912(KftSA-CE- 182927) 3>HEOBETIOL STUDIES OP
SClAft IASEBS £NB CCNVEEX'EBS Progress Bepprt,
Jan. i- Jul. 1S88 (Old: DominicB DBiv.) 22 p
CSC1 20E Uuclas
G3/36 0116^26
June 1988
https://ntrs.nasa.gov/search.jsp?R=19880014588 2020-03-20T07:00:55+00:00Z
DEPARTMENT OF MATHEMATICAL SCIENCES
COLLEGE OF SCIENCES
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
THEORETICAL STUDIES OF SOLAR LASERS AND CONVERTERS
By
John H. Heinbockel, Principal Investigator
Progress Report
For the period January 1988 to July 1988
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Grant NAG-1-757
Dr. Robert C. Costen, Technical Monitor
SSD-High Energy Science Branch
Submitted by the
Old Dominion University Research Foundation
P.O. Box 6369
Norfolk, Virginia 23508-0369
June 1988
Progress Report Grant NAG-1-757
For the period January 1988 to July 1988
THEORETICAL STUDIES OF SOLAR LASERS AND CONVERTERS
ABSTRACT.
The previously constructed one dimensional model for the simulated operation of an
iodine laser assumed that the perflouroalkyJ iodide gas n - C^F7I was incompressible.
The present study removes this simplifying assumption and considers n - C^FjI as a
compressible fluid.
Introduction.
The iodine laser being modeled uses the gas n - C^FjI flowing in a cylindrical tube.
As a first approximation to the chemical kinetics and flow it was assumed that the gas
was incompressible and operating at a constant temperature. To remove these simplifying
assumptions we consider the effects of fluid density variation as a function of both pressure
and temperature in the flow. In particular we have added to our previous model:
(i) An equation of state.
We assume the equation of state of an ideal gas and write
P = i)RT (1)
where P is the gas pressure [Pa], r) is the gas density [Kg/m3], T is the temperature [° K],
and R = R*/296[J/Kg°K\t and R* is the gas constant, R* = 8.317 [J/moU °K], For
n - C$FiI we use the value of 296 Kg/mole to convert R* to R.
(ii) Continuity equation.
The continuity equation expressing the conservation of mass flow can be represented
+ (to („) = <> (2)
where r\ is the fluid density [Kg/m3], and $ is the fluid velocity [m/s]. For steady state
conditions and a flow in the axial direction we let ^ = uk and reduce the continuity
equation to the form
^vv-,-0 (3)
An integration of this equation gives
ijw = cj (4)
where ci is a constant of integration.
1
(iii) Momentum equation.
The momentum equation for a control volume having a mass i)dr, where dr is an
element of volume, is given by
ti = fjjfr,dT (5)
From Newton's second law we have
DM D
where D/Di'is the material derivative. We have
#-//'(*••)+»///'•* -m
where dff is an element of surface area. The surface integral term in (7) represents the efflux
of momentum through the control volume and the volume integral term in (7) represents
the change in momentum inside the control volume. Using the Gauss-divergence theorem
we can change the surface integral to a volume integral and write
V • , dr (8)
Then the momentum equation becomes
where is the dyadic w3bJb for = w, and
(10)
We also make the substitution / = /// fdr where / is the average force per unit
volume
as this allows the momentum equation to be written in the form
Neglecting viscosity and using / = -VP as the average force per unit volume which
is due to the fluid pressure P , we have
From equation (4) we use 170; = ci and examine the steady state form of the equation (12)
to obtain
Integrating this equation gives
P + CiW = cfc (14)
where c2 is a constant of integration.
(hr) Energy Equation.
In terms of the specific enthalpy h per unit mass, the energy equation for the fluid
flow is
where P is the pressure, T is the temperature, K is the thermal conductivity and q = q(z]
is the radiation heat flux. In one dimension this can be written
dh dh 3P dP
The specific enthalpy h can be expressed in terms of the specific heat at constant
pressure and
h = CpT (17)
Using the relation C;-C; = R* . (is)
where R* = 8.317 [J/mole - °K\ is the universal gas constant.
The specific heat at constant pressure C* can be expressed in terms of the specific
heat at constant volume C*. We use the numerical data
Temperature °K C* for CzF7I[J/mole°K\
298.15 146 . .
400.00 169 ( '
500.00 186
and assume that C£ can be obtained from the relation
Ct* = C; (T) = a, expQS, (T - 300)), 298.15 < T < 500 (20)
By applying a least square's fit of the data with the exponential curve hi (20) we obtain
the best fit parameter values of
o, = 147.23 ft =0.0012 (21)
This gives the steady state energy equation
(22)
where Cv(7) is the correct dimensional form of C*,
Having neglected the viscosity terms, we also neglect the effect of thermal conductivity
and integrate the equation
Ci[C,(T) + R\dT + wci<iw - q(z)dz = 0 (23)
to obtain
4 / C, (T)dT + d R(T - 300) + ^  -Q(z) = c< (24)
where
and 04 is a constant of integration.
Implementation of Model*
In order to compare and verify the computer model with laboratory data it is assumed
that we are given the boundary conditions as z - 0 and z-L. We use the subscript 0 to
denote values at z = 0 and subscript L to denote the values at z = L. By setting 2 = 0 we
obtain:
from (4) ci = ifowo
from (14) <£ = PQ + Ciw0
from (24) we have
c4 = c, I" R(TQ - 300) + / C,(T) dT\ + ^
Assuming that g = q(z} can be represented by a constant, at z = L we obtain from the
equation (24) the relation
?o = - (c, \R(TL - 300) + / C9(T) dT\ + £u£ - c*)zi i [ j j * }
Knowing the values of the above constants the temperature, density, pressure and flow
rate as a function of distance z can be calculated as follows:
From the equation of state we can write
and multiplication by w gives
Pu = qwRT = ciRT
Using the relation (14) in the above equation gives
or
r-es^ (25,
We also use the scale factors
where applicable to insure all equations balance dimensionally.
For
0 < 2 < z,
zi <«< L
We may then solve for w = w(z) such that the energy equation
F(w) = ci j C,(T)dT+ CiR(T - 300) 4- ~w3 - Q(z) - c4 = 0 (28)
is satisfied where 7 is determined by the relation (25). Since this is a nonlinear equation,
the method of interval halving is used. The value of w is changed until F(u) changes sign
and then interval halving is used until the desired solution accurracy is obtained. The
temperature 7 is then determined from (25), the density is obtained from the relation (4)
and the pressure is obtained from the relation (1). We then have the flow rate, temperature,
pressure and density as functions of distance 2.
Reaction Rates.
The reaction rate coefficients
, AS, fo,
which occur in the chemical kinetics associated with the iodine laser can be assumed
to vary with temperature. Various assumptions can be made concerning the effects of
temperature on these rate coefficients. Various Arrenhis type temperature variation effects
can be modeled using one or more of the following expressions:
K =
Since the coefficients in these expressions are not known, as a first approximation to
temperature on the rate coefficients, we will keep most of the rate coefficients constant.
Incompressible flow model.
The incompressible flow model was presented in the previous progress report (reference
2). Nominal results from that model are illustrated in the figures 1 through 5 for various
values of the flow rate w. Values of w used in these graphs are:
w = 750., 1750., 2750., 5500. cm/a
while other parameters were held constant. These other parameters were:
P = 23.5 Torr,
RI = .95848 reflectivity at left end of the tube,
#2 = .81634 reflectivity at right end of the tube,
concentration = 1267 (peak concentration in solar constants.)
Both the incompressible and compressible flow models are being modified to include
temperature effects. Also an analysis by B.C. Costen of the pumping photon density is
being studied and incorporated into the model. This analysis is given in the Appendix A
along with figures of the results therein.
LOG CC3F7D
ro
X
t—t
D
D
o
41 I I I I I II I I I II
INFOPLT 1
Figure 1. Log (C.^ ) vs axial distance.
6
IhFDPLT 2.
Figure 2. Log (I2) vs axial distance.
7
LOG ci*:
INFOPLT 3.
Figure 3. Log (I*) vs axial distance.
8
LOG HID
^ i
INFOPLT t.
Figure 4. Log (I) vs axial distance.
9
INFOPLT 5.
Figure 5. ±Log|l*-£l| vs ax iaT dista^e.
References
RJ. DeYoung, G.H. Walker, M.D. Williams, G.L. Schuster, EJ. Conway, Prelimi-
nary Design and Cost of a 1-Megawatt Solar Pumped Iodide Laser Space- to-Space
Transmission Station, NASA TM 4002, September 1987.
J.H. Heinbockel, Theoretical Studies of Solar Lasers and Converters, Progress Report.
Research Grant NAG-1-757, Old Dominion University Research Foundation, January
1988.
11
Appendix A
Pumping Photon Density within Absorbing Lasant Gas
by R.C. Costen
Assumptions: There is a 2-D cylindrical geometry which is treated as a black body
cavity.
A. Consider the relation of 2-D surface brightness Bi(W/m?) to the 2-D surface
flux F<2(W/m?). Consider a one meter depth of a cylindrical cavity of arbitrary shape as
illustrated in the figure. A.I
Figure A.1 Cylindrical geometry
We have
•L*
-*
W per m of perimeter entering the cavity per m of depth.
12
B. Determination of the pumping photon density at an interior point resulting from
a length dL of perimeter.
Figure A.2 Density at an interior point
The power in dB due to a length dL is given by
W in d9 per unit depth. This power (J/*) is due to the photons moving at the velocity of
light c (m/a) The energy density in dti per m of depth is
(J/m)\ i *
Using dB-^ and noting that the photon energy density dfo due to dL is given by
per m depth. For an absorbing gas we have
(J/m2)
per m depth and where 6, (m) is the absorption length.
13
G. 2-D Circular cylinder with absorbing gas and uniform incident flux Fa.
Figure A.3 Cylindrical geometry
At the point P we have due to dL that
per m of depth. Using
we can write
or
Q o
per m of depth. Using the law of cosines
COS0
—
r = -r t cos $ + rg - r? sin
we have
Changing r\ to r the pumping photon density is given by:
p(r\r0)6) = T «p (n - r? ain3
in J/ma per m of depth.
14
Figure A.4 Geometry
D. Average pumping photon density
The average pumping photon density IB given by
Letting a = r/6 where SQ = W$ i* *ne ^aaer *BDe radius per absorption length, we have
= />(«, «o)
and
2 r*°
= -j /3o yo
=<
We define and compute
//(a, a0) = - j exp(« cos 0 -
with
2 /
= ^ /8o Jo
which is the average pumping photon density.
Then
/"
/
/O
15
Note:
< /f \ A -^ \ ^^ ^— I IT Jl f\O I {| fk I ^r -^—• 1 1* j* / \ 1A j|
*°~*° J0-*0 °' ~" C
and at the center a = 0 we have
and on the wall s = so we have
16
ft > .SCALED AVERAGE PUMPING PHOTON DENSITY
I
i
e*
HM
Figure A.5 < p ' > vs SQ.
17 OF .POOR
10.0
i
CD
5
o
CO
1.0
0.1
0.01
P001
0.0 0.2 Q.4 Q.6 0.8 1-0
, NORMALIZED RADIUS
Figure A.6(a) p1 vs S/S .
ORIGBSTAD PAGE !3
OF POOR QUALITY
18
Figure A.6(b) 19
ORIGINAL PAGE IS
OB EOOR
Figure A.7 < p ' > vs SQ Oil semi-log scale.
20 QKIGCtfAL PAGE tS
OE POOR QUALITY


Theoretical studies of solar lasers and converters

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