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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

(KftSA-CE- 182927) 3>HEOBETIOL STUDIES OP N88-23912

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
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 w

3

bJb 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- ~w

3

 - 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 axiaT dista^e.
References

RJ. DeYoung, G.H. Walker, M.D. Williams, G.L. Schuster, EJ. Conway, Preliminary 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 / 3

o yo

=<

We define and compute

//(a, a0) = - j exp(« cos 0 -

with

2 /

=^8 /

o 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*

H

M

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
Figur

e A.6(b)

19

ORIGINAL PAGE IS

OB EOOR
Figur

e A.7 <p'> vs S

Q Oil semi-lo

g scale.

20 QKIGCtfA

L PAGE tS

OE POOR QUALITY