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Two-dimensional model for the subsonic propagation of laser sparks

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Title:
Two-dimensional model for the subsonic propagation of laser sparks
Creator:
Batteh, Jad Hanna, 1947-
Publication Date:
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English
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xiii, 77 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Conduction ( jstor )
Conductive heat transfer ( jstor )
Electrons ( jstor )
Isotherms ( jstor )
Laser beams ( jstor )
Lasers ( jstor )
Phase velocity ( jstor )
Plasmas ( jstor )
Thermal radiation ( jstor )
Velocity ( jstor )
Dissertations, Academic -- Engineering Sciences -- UF
Engineering Sciences thesis Ph. D
Lasers ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 74-76.
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Typescript.
General Note:
Vita.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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TWO-DIMENSIONAL MODEL POE THE SUBSONIC
PROPAGATION OF LASER SPARKS








By

JAD HANNA BATTED












A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY






UNIVERSITY OF FLORIDA 1974



























To my wife, Jane,
and
to my parents












ACKNOWLEDGEMENTS


I would like to express my sincere gratitude to the members of my supervisory committee. In particular, my deepest appreciation is extended to the committee chairman Dr. Dennis R. Keefer, whose advice and guidance were essential to the completion of this endeavor.

I am indebted to Dr. Bruce Henriksen for suggesting

the problem and for his continued interest in this research.

My gratitude for the criticism, encouragement and technical assistance of Mr. John Allen is exceeded only by my regard for his personal friendship. Finally, I am especially grateful to my wife for her diligent preparation of this manuscript.






















lit













TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS . . . . . . . . iii

LIST OF TABLES . . . . . . . . v

LIST OF FIGURES . . . . . . . vi

NOMENCLATURE . . . . . . . . viii

ABSTRACT . . . . . . . . . xii

CHAPTER

I INTRODUCTION . . . . . . 1

II THEORETICAL DEVELOPMENT . . . . 7

III SOLUTION OF THE EQUATIONS . . . 16

IV SIMPLIFIED ONE-DIMENSIONAL
ANALYSIS . . . . . . . 24

V EXAMPLE CALCULATIONS . . . . 29 VI DISCUSSION OF RESULTS . . . . 37

vii CONCLUSIONS . . . . . . . 44

APPENDIX

I CONVERGENCE OF THE SERIES
SOLUTION . . . . . . . 61

II EFFECT OF RADIATION ON
SUBSONIC SPARKS . . . . . . 64

REFERENCES . . . . . . . . . ?4

BIOGRAPHICAL SKETCH . . . . . . 77





iv













LIST OF TABLES


Table Page

I Physical Properties of Laser Sparks
of .15 cm and .50 cm Radius . . . . 60









































v












LIST OF FIGURES


Figure Page

1. Co-ordinate System for the Steady State
Propagation of a Laser Spark in a Channel *. 47

2. Variation of c /7k, with Temperature
for Atmospheric-pressure Air . . . . 48

3. Variation of Thermal Conductivity and
Thermal Potential with Temperature for
Atmospheric-pressure Air . . . . . 49

4. Incident Laser Intensity Required to
Maintain a Spark as a Function of
Channel Radius .. ... .. .. .. .. 50

5. Isotherms for a Spark with R L 1 35 cm$
R 1. 25 R and u = 0 51

6. Isotherms for a Spark with R L =.1.5 cm,
R =l25 RL and u=20 cm/sec.o. ... o .52
7. Isotherms for a Spark with R L =.50 cm,
c a=l25 RL and u =0*0 .... 0 0 0 53
8. Isotherms for a Spark with R L =.50 cm, R =125 RL and u =20cm/sec . . . 54
9. Variation of the Thermal Conduction Parameter with R a R . . . . . . 55

10. Axial Variation of the Volumetric Energy
Loss Terms for RHL =.15 cm, Rc = 1.25 R L
and u = 0 # 0 0 . . 0 0 56





vi







Figure Page

11. Axial Variation of the Volumetric Energy Loss Terms for RL = .15 cm, R0 = 1.25 RL
and u = 20 cm/sec ... ......... . .. 57
12. Axial Variation of the Volumetric Energy Loss Terms for R = .50 cm, Rc = 1.25 RL
and u = 0 . . 8 . . . a 58
13. Axial Variation of the Volumetric Energy Loss Terms for R = .50 am, R. = 1.25 RL
and u = 20 cm/sec .............. 59

A.1 Isotherms for a Spark with RL = .50 cm,
Rc =2 RL, u = 1 m/sec and Ti = 6,000 OK 73































vii











NOMENCLATURE


a n coefficient of the series solution

A radial heat conduction parameter

A coefficient of the series solution for el defined
by Equation (3.8)

b n coefficient of the series solution

Bn coefficient of the series solution for 19, defined
by Equation (3.9)

n coefficient of the series solution

c p specific heat capacity, joule/gm OK

C n coefficient of the series solution for e9a defined
by Equation (3.10)

d spark length, cm

e electron

E i ionization potential, ev

f axial function for the solution of F n axial function for the solution of e

9 Gaunt factor

gn radial function for the solution of e, G n radial function for the solution of 9K

h Planck's constant, ev-sec

H1(r) Heaviside step function

1 0 zero order Bessel function of the first kind 1 1 first order Bessel function of the first kind viii








IV mean free path of radiation of frequency ) cm

L characteristic spark dimension, cm

m thermal radiation parameter, cm2

Mn terms of the series used for comparison in the
Welerstrass M-Test. n summation index

N e electron number density, cm-3

N+ positive ion number density, cm3

PA power absorbed by the spark, kW

PO incident laser power, kW

r radial co-ordinate, cm

R c channel radius, cm

R L laser beam radius, cm

S laser intensity, kW/cm2

S0 incident laser intensity, kW/cm2
So 0 incident laser intensity for a reduced ignition
temperature, kW/cm2

tn coefficient of the series solution

T temperature, OK

Te electron temperature, OK

Th heavy-particle temperature, OK

Ti ignition temperature, OK

T--m maximum value of the radially averaged temperatures, OK Tn (An/X on)Jl(XonRL/Rc)

u spark propagation velocity, cm/sec

Vr radial velocity component, cm/sec



ix








vx axial velocity component, cm/sec

x axial co-ordinate, cm
xM axial location of the maximum temperature, cm
th
xon nth zero of J
Y zero order Bessel function of the second kind
Z ionic charge



0(10
oCP 2 /2/1 ,m-1

separation constant, equal to xon2/R 2, cm-2












<424 exponential factors in the series solution, cm-1 g thermal potential, kW/cm
i ignition thermal potential, kW/cm

ei reduced ignition thermal potential, kW/cm
thermal potential in the region x < 0, kW/cm o thermal potential in the region x O0, kW/cm
eh solution of the homogeneous equation for kW/cm

Op particular solution of the equation for 49Z kW/cm
thermal potential averaged over the laser radius,
kW/cm


x








e, radially averaged thermal potential in the
region x < 0, kW/cm

e. radially averaged thermal potential in the
region x ? 0, kW/cm

em maximum of the radially averaged thermal
potentials, kW/cm

absorption coefficient for radiation of frequency
cm-1

Thermal conductivity, kW/cm OK radiative conduction coefficient, kW/cm OK

radiation frequency, see9density, gm/cm3

density of the cold gas ahead of the spark, gm/cm3

rate of energy loss by radiation, kW/cm3

4) separation constant, equal to Xon 2/R2, cm-2

























xi








Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy



TWO-DIMENSIONAL MODEL FOR THE SUBSONIC
PROPAGATION OF LASER SPARKS


By

Jad Hanna Batteh

August, 1974


Chairman: Dennis R. Keefer Major Department: Engineering Sciences



The properties of laser sparks are investigated by solving a simplified, two-dimensional energy equation describing the steady state, subsonic propagation of a spark in a channel. The propagation mechanism is assumed to be thermal conduction and thermal radiation is included as an effective optically thin emission. The solution, obtained In closed form, yields both axial and radial temperature profiles, as well as the relationship between the laser beam characteristics, the channel radius and the propagation velocity.

A solution of the radially integrated energy equation

is obtained by introducing the radial heat conduction parameter A suggested by Raizer. The parameter is evaluated by means of the solution to the two-dimensional energy equation.




xii








The model is applied to air sparks of .15 cm and

.50 cm radius propagating at velocities up to 20 cm/sec wonder the influence of 002 laser radiation. The results indicate a spark length on the order of 1 cm and temperatures near 16,000 OK. Approximately half the incident laser intensity is absorbed by the spark. Calculation of the radial heat conduction parameter verifies the value of 2.9 used by Raizer for an unbounded spark of .15 cm radius. Furthermore, A is found to depend on the spark radius.

The theory agrees with the experimental results for stationary sparks. However, it fails at predicting the spark properties for propagation velocities of several meters per second. It is postulated that, at these velocities, re-absorption of thermal radiation creates a nonequilibrium layer ahead of the spark front which enhances the absorption of laser radiation and increases the propagation velocity.

















xiii













CHAPTER I

INTRODUCTION


Interest in the interaction of matter with intense

beams of coherent radiation has increased with the development of high power lasers. In particular, extensive research is being devoted to the investigation of discharges maintained by the absorption of laser radiation. These discharges are commonly referred to as laser sparks.

Discharges are in wide demand for technological and

diagnostic applications and for laboratory studies of plasma processes. The current methods for the creation and maintenance of discharges are based on the release of electromagnetic energy in a gas. Until recently discharges, which are classified by the frequency of the electromagnetic field, ranged from the direct-current are discharge to the microwave discharge. Within this frequency range the transfer of electromagnetic energy requires special auxiliary equipment such as electrodes, conductor coils and wave guides. As a result the plasma is confined by boundaries which can erode and introduce contaminants.

The laser spark represents an extension of the frequency range of the generating field to infrared and visible frequencies. The electromagnetic energy is transported by

1






2

the laser beam thereby eliminating the need for special equipment. It is possible, therefore, to obtain pure plasmas burning in free space. These characteristics plus the high temperatures attainable in laser sparks extend the possible applications beyond those of lower frequency discharges. For example, a recent study 1 explored the feasibility of using laser sparks to increase the enthalpy of the flowing gas in a hypersonic wind tunnel. These laser-sustained discharges also offer the researcher interested in plasma physics a unique opportunity to study an unconfined plasma in the laboratory.

However, the occurrence of laser sparks Is often

undesirable. A laser beam of sufficiently high intensity can initiate a spark in a gas or by interaction with solid materials. The intensity of the laser radiation is attenuated in passing through the spark due to the conversion of radiation energy to thermal energy which takes place in the spark. Therefore, the intensity of laser radiation which can be transported to a target is limited by the threshold intensity for the creation of a spark. Exceeding this intensity results in the creation of a laser spark which absorbs a portion of the incident radiation ahead of the target.

Laser sparks generally fall into two categories

depending on whether the laser beam ignites the discharge or merely maintains an externally initiated discharge.






3

Ignition can occur in a gas which is usually transparent to laser radiation If the laser intensity creates sufficient ionization in a region to initiate absorption of the laser energy. 2-4 Due to the high intensities required, Ignition has been observed only with the focused output of pulsed lasers. The initial breakdown region quickly develops into a laser spark. Shook waves, driven by the absorption of laser radiation, propagate outward from the hot spark and heat the surrounding areas. In particular, the cold gas ahead of the spark and in the path of the laser beam is ionized and begins to absorb the incident radiation. Thus, a new absorbing layer is created and the spark front propagates in a direction opposite the laser flux due to a mechanism similar to that occurring in the detonation of explosives. Typical spark temperatures are 105 to 106 oK. As the spark propagates, it is observed to expand radially to fill the cone formed by the focused laser beam.
5
In 1969 Bunkin jjt al. demonstrated for the first

time that a laser Intensity lower than that required for breakdown could maintain an externally initiated discharge. They found that Nd-glass laser radiation at an intensity of 10 4 kW/cm2 could maintain an air spark which had been initially ignited by an electrical discharge. This intensity was two orders of magnitude less than that required to initiate breakdown. In this case the incident laser intensity is insufficient for generating a shock wave.






4~

Thermal heat conduction, aided by reabsorption of thermal radiation emitted from the hot spark, is the mechanism by which the cold gas ahead of the spark is ionized and becomes opaque to the incoming laser radiation. The spark propagates subsonically in a direction opposite the laser flux. Typical spark temperatures in the subsonic regime are 10, 000 25,000 O0This mode of propagation suggests the existence of a threshold laser intensity, such that the absorbed laser energy just compensates for the heat conduction and radiation losses, and the spark remains stationary. The existence of a threshold was demonstrated in 1970 by Generalov et al. 617 who used a continuous wave (ow) CO 2 laser to maintain a stationary plasma in high pressure argon andi xenon gases. Similar experiments have been conducted by Franzen,9 and recently Smith and Fowler8 have published data on the threshold intensity of 002 laser radiation in atmospheric-pressure air.

The modeling of subsonic laser sparks promotes

understanding of the creation and maintenance of these discharges. Unfortunately, only a limited amount of theoretical work has been done. Raizer, 10using a one-dimensional analysis and introducing a thermal conduction parameter for radial heat losses, derived threshold laser intensities and propagation velocities for sparks in atmospheric-pressure air under the influence of 00 2 and Nd-glass laser radiation. Hall, Maher and Wei.11






5

resorting to analytical approximations for the thermodynamic properties of air, also used a one-dimensional model to obtain the radially averaged spark temperatures and propagation velocity. In both References 10 and 11, thermal conduction was assumed to be the propagation mechanism. A recent one-dimensional study by Jackson and Nielsen 12 has focused on the role of radiation in the subsonic propagation of laser sparks.

The purpose of this investigation is to provide a

two-dimensional model for the subsonic propagation of laser sparks by explicitly including the radial heat conduction losses. This allows the calculation of both radial and axial temperature profiles as opposed to temperatures averaged over the radius. Furthermore, the solution is obtained for a spark propagating in a channel of arbitrary radius so that the effect of boundaries on spark propagation can be studied. The model is applied to air sparks at atmospheric pressure under the influence of C02 laser radiation.

The propagation mechanism is assumed to be thermal conduction with radiation serving only as an energy loss mechanism. Although the flow velocity Is restricted to one dimension, the two-dimensional nature of the temperature field is retained in the analysis. By using simple models for the temperature-dependent properties of the gas, a closed form solution is obtained which yields axial and








radial temperature profiles as well as the dependence of the propagation velocity on channel radius, spark radius and incident laser Intensity. The two-dimensional model also offers a method for evaluating the radial heat conduction parameter introduced by Raizer. 10












CHAPTER II

THEORETICAL DEVELOPMENT

In this analysis the steady state, subsonic

propagation of a laser spark in a gas is considered. The ignition process is not considered since its only purpose is to provide free electrons to initiate inverse bremsstrahlung absorption of the laser beam. Therefore, the characteristics of the spark are functions only of the laser-plasma interaction and not of the ignition process.

The model consists of a uniform, monochromatic laser

beam, of intensity S 0 and constant radius R L9 passing through a channel of radius R c in which there is an absorbing plasma. Energy is absorbed from the laser beam by the plasma and is transferred to the cold gas through heat conduction. The channel wall is held at a fixed temperature T = 0 and is assumed to absorb all radiation incident upon it without re-emission. The spark moves in the channel without distortion with a constant velocity u relative to the cold gas, as shown in Figure 1. Since the flow velocities are subsonic the pressure gradients are small and are neglected in the analysis. Furthermore, the flow kinetic energy is neglected compared to the thermal energy,

7






8

since most of the absorbed radiation appears as a temperature increase as opposed to directed fluid motion. Viscous dissipation is also neglected since the loss due to dissipation Is less than the losses due to thermal radiation and heat conduction in the region of laser-plasma interaction. For this model the energy and continuity equations referred to a co-ordinate system fixed to the spark are

T T
%/X C P + P Vr C P
X

dT) + + 5 0 (2.1)

X )X r


(YVX) + 0 (2.2)
X r

In the above equations JP is the density, T is the temperature, vx and vr are the axial and radial velocity components, respectively, c p (T) is the specific heat capacity, ;[(T) is the thermal conductivity, S is the intensity of the laser beam, X. (T) is the absorption coefficient at frequency ) and 0 is the net energy lost by radiation per cm3 per see. The density is assumed to be related to the temperature and pressure by the perfect gas law.

In order to solve for both the velocity field and the temperature field, the two momentum equations must be included in the analysis. Clearly, the solution of these four, nonlinear, coupled partial differential equations is







9
a problem of considerable difficulty unless simplifying assumptions are made. Since the temperature field Is of primary interest in this analysis, its two-dimensional nature is retained. On the other hand, the velocity field is simplified by assuming vr = 0 and taking the axial mass flow rate independent of radius. This onedimensional approximation to the flow field is often used in spark propagation analysis. 10-12 The equations are then uncoupled. In fact, Equation (2.2) reduces to po v = const. u Puwhere is the density far ahead
x1
of the spark. Jackson and Nielsen,1 on the basis of preliminary results, predict little difference between the results obtained using the two-dimensional flow field and the one-dimensional flow field in radiationdominated propagation. For the model considered here, the error introduced by neglecting the radial deflection of the flow should diminish as the channel radius decreases. Since v r must vanish at r = 0 and r = Ec, the channel inhibits the radial velocity.

As a result of the one-dimensional flow approximation, the energy equation reduces to





e er + II r ) (2.3)
=

0( YOC P






10

where the thermal flux potential



0

has been used to replace T as the dependent variable.

Equation (2.3) can be linearized if suitable

approximations are made for the temperature dependence of ?fy, 0 A and c The first approximation is to

assume co / A equal to a constant which reduces a% to a constant. Figure 2 shows the variation of this ratio for atmospheric-pressure air plotted from the data of Reference 11. The definite temperature dependance, apparent in the figure, indicates that the assumption is a shortcoming in the theory. However, the assumption is a reasonable first approximation in the temperature range 10,000 25,000 OK which is of greatest Interest for subsonic laser sparks. In fact, it will be shown in Chapter VI that this approximation is valid for the velocities considered In this analysis. Although the authors of Reference 11 used analytical approximations for the temperature variations of enthalpy and thermal conductivity, their approximations are equivalent to assuming a constant c p/ X\

Un-ionized air is transparent to CO 2 laser radiation, which has a wavelength of 10.6 microns. However, absorption can occur by inverse bremsstrahlung (free-free transitions of the electrons) when air becomes ionized.









In that case the free electrons in the presence of scattering centers are accelerated by the electric field of the laser radiation. The electron and heavy-particle temperatures are then equilibrated by elastic collisions.

Although both neutral particles and ions can serve

as scattering centers, absorption in the field of neutral particles is important only in a very weakly ionized gas.13 At typical spark temperatures the ionization is appreciable so that only absorption due to electron-ion collisions is considered.

The inverse bremsstrahlung absorption coefficient for a high-temperature, ionized gas is given by the modified Kramers formula13


7,' 3. ( X/O cm (2.4)



where N+ is the number density of positive ions with charge Z, Ne is the number density of electrons, Te Is the electron temperature, ) is the radiation frequency and g is the Gaunt factor. In the single ionization temperature range, the absorption coefficient is approximately proportional to Ne2/Te*. Therefore, under equilibrium conditions, 7f, will exhibit a rapid increase at temperatures where the ionization becomes appreciable. For atmospheric-pressure air this temperature lies in the range 10,000 oK to 15,000 oK.14 In fact, equilibrium calculations10







12

of ?~iv for atmospheric-pressure air show a rapid increase near T = 12,000 'K.

The absorption coefficient of air for Nd-glass

laser radiation, which has a wavelength of 1.06 microns, consists of contributions from both free-free and boundfree electron transitions. It also exhibits a rapid increase at temperatures near 12,000 oK*10

Consequently, the solution domain of the energy equation is divided into two regions separated by the plane x = 0. For x4(0 the temperature is sufficiently low that there is negligible absorption of laser radiation, i.e. W, = 0. In the region x > 0, the absorption coefficient is assumed to be a constant so that


Sff = So HA(R-r) 71v P-e ?f



where S0is the incident laser intensity and H(R L r) is the Heaviside step function defined to be unity for
0 i- r! Land zero for r ',R The term "spark front" will be used to denote the plane x =0.

A complete treatment of the emission and re-absorption of thermal radiation would require the solution of a complicated integro-differential equation. It is included in this analysis through a simple phenomenological approximation. In the region x 4, 0, where the temperature is comparatively low and decreases rapidly with increasing







13
distance from the spark front, the net thermal emission is neglected. In the region x > 0, it is included as a simple volumetric loss term given by the linear relationship



The constant mn is chosen to give values of which agree with experimentally determined radiation losses for air at temperatures near 15,000 0 K

The analysis of Jackson and Nielsen12 for large radius sparks suggests that re-absorption of thermal radiation, rather than thermal conduction, is the dominant heating mechanism for the cold gas ahead of the spark. The effect of neglecting this heating, by assuming 0 for x < 0, and the role of radiation transport in spark propagation will be discussed in Chapter VI.

As a result of the approximations discussed in the

preceding paragraphs, e, which is the solution for the thermal potential in the region x <.0 must satisfy


+ -0-t 8~ 8 (2.5)

r )r r0


and 09Z which is the thermal potential in the region x > 0 must satisfy







14








(2.6)

= 5 o (R ,- & ) e I x



The boundary conditions at x = + 00 and at the channel

radius R are

e, (-oo, r)= e. (oo, r)= e, ( )=e (x,R=0 (2.7)

Since the temperature and the heat flux must be continuous at x = 0, the joining conditions are


e1 (or)= e (C, r) (2.8)


Ga (2.9)
\ X- x = 0 x Ix-=0




Equations (2.7) (2.9),together with the restriction that e be finite, completely determine the solution. There is, however, one further condition which must be met. Since x = 0 has been defined as the plane where appreciable absorption begins, the temperature at x = 0 as determined from the differential equations must







15

correspond to the temperature at which the plasma becomes opaque to the laser radiation. This consistency condition is satisfied by requiring the thermal flux potential, averaged over the laser radius at x = 0, to be equal to 49i the value of the potential at the ignition temperature T., i.e.



E)i RL r E), 0, (2.10)
RL 0


This equation results in a relationship between the propagation velocity and the required laser intensity.













CHAPTER III

SOLUTION OF THE EQUATIONS Equations (2.5) and (2.6) for the thermal potentials

can be solved in closed form by the separation of variables technique.

By assuming


91 (Xr) (X



the partial differential equation for 0, reduces to a set of ordinary differential equations














where the are separation constants.

The equation for gn is Bessel's equation, which has the solution








16






17

Jo and Y are the zero order Bessel functions of the first and second kind, respectively, and the an and bn are constants. Since Y o(0) = 00 bn must be zero for the solution to remain finite on the axis. The radial boundary condition in Equation (2.7) determines the separation constant. For e1 to vanish at r = Rc, PA must equal Xon/Rc where x on is the nth zero of J0.
The differential equation for fn has the solution


-I,.= c,, e-- ,,e


where

g~nzn--Z L C1" Z


and the cn and dn are constants. To satisfy the boundary condition 9, (- 0o ,r) = 0, dn must be zero.
The solution for G, can, therefore, be written as



61 = _.a --o x.,-) e"


(3.1)

+ ZXon



The constants an are determined by the joining conditions.






18
Equation (2.6) for eZ is a nonhomogeneous, linear partial differential equation. Its solution can be written as

e.= e+ et

where 1p is a particular solution, and is the
solution of the associated homogeneous equation

d h~ (l 0( d et,
) r )x(3.2)


-MG= 0


A particular solution of Equation (2.6) which
satisfies the boundary conditions at r = Rc and x = 00 is


= e nvx xo )- (33)


where the bn are constants. Substituting Equation (3.3) into Equation (2.6) yields


1- -0 (Xo. -) r So 7 H (R,- r)



4 rb 6 3. 4







19

Since the J (x onr/R ) form a complete, orthogonal set in the range 0 :. r 4 R., they can be used to form a unique series expansion of any function of r in that interval. In particular, the function So 74 H(RL r) can be expanded in terms of an infinite series. Therefore, for Equation (3.4) to be valid, the tn must be the coefficients of such a series expansion. The coefficients can be found by using the orthogonality property15 Rc


7Vr r(X0A 0 ) -( &
0


where m is the Kronecker delta and J is the first

order Bessel function of the first kind. Multiplying Equation (3.4) by rJ o(Xom r/R ), integrating from 0 to Rc and applying the orthogonality condition gives




= J r T.,(XftO )J

R, T, (X of)


The value of the integral, obtained with the aid of Equation (11.3.20) of Reference 15, is RcRLJl(xonRL/Rc)/Xon. The resulting equation for the constants bn of the particular solution is





20


~~ozrW RSo (XT 4 ()X(O* Rml
RCR



The solution of the homogeneous equation for eh is assumed to have the form

GV (x,r = F (x)
1'%A
Substitution into Equation (3.2) results in the set of ordinary differential equations


--AF -( + rm) F= 0
Aix' Aix


/ G Ald Gs j G=a
+ G = 0 = ,


where the are separation constants. The solutions
of these equations are obtained in a manner analagous to the procedure used for 9 Imposing the boundary conditions Fn( 00 ) = 0 and Gn(Rc) = 0 results in

E,,)o o .o~o) C )xr ee
r% T( ) e






21

where the cn are constants to be determined by the joining conditions.

Thus, the solution for e2 is given by



ea(x,r)= e- 3 X
,. Rc (3.5)

2,
+ s,3o(Xo,,- o) e ,."




Imposing the joining conditions given in Equations (2.8) and (2.9) results in a set of algebraic equations for the an and cn. Solution of these equations yields the following for and e?


E = so L -A (x+ e


R (3.6)



E) z- e0_. B, -T- ( xo,. %
Rc rt (3.7)





M RiC.



where the coefficients are given by






22

















C (+ i) AZ (3.10)




The relationship between the propagation velocity and the laser intensity is obtained by substituting e, from Equation (3.6) into Equation (2.10). The resulting equation is



E) oL oA A (3.11)


Since the A nare functions of OL the propagation velocity is contained in the summation. Therefore, the most convenient way to evaluate Equation (3.11) is to choose a value of u and solve for S 0. Equations (3.6)

and (3.7) can then be evaluated for el and (9Z

Equations (3.6) (3.11) are valid only if* each infinite series converges uniformly in the interval







23

Oo x 4' oo and 0 r 4 R c A proof of the

uniform convergence of these series is given in Appendix I.

The length of the spark d can be defined as the co-ordinate x where the temperature averaged over the laser radius falls below the ignition temperature T il Although the model does not terminate absorption of the laser radiation at x = d, the absorption of a real spark decreases rapidly for x ; d due to the rapid decrease in the degree of ionization. Therefore, the spark length is useful in approximating the power absorbed by the spark. Consistent with the exponential approximation for the absorption of laser radiation,

PP.-

PO I e (3.12)

where P A is the power absorbed by the spark and Po is the incident power.













CHAPTER IV

SIMPLIFIED ONE-DIMENSIONAL ANALYSIS


Although the solution in Equations (3.6) (3.10)

is straightforward, it is cumbersome due to the summation. In this chapter the equations are reduced to one dimension by introducing a radial heat conduction parameter. The parameter is in the form of an infinite series, but the spark properties are described by simple equations.

If Equations (2.5) and (2.6) are averaged over the laser radius, the resulting equations are




61 =0 A ix < 0 (4.1)

cxz AxRz



L)




where





0




24






25

In Equations (4.1) and (4.2), the radial heat conduction 10
term has been simplified, as suggested by Raizer, by

setting



RL R zrThe parameter A is assumed to be independent of x and r but it is a function of spark and channel properties.

For the one-dimensional equations, the boundary conditions are

,(-co )= 0( = 0


The joining conditions are given by





(0)&

X =O X O



The solutions of Equations (4.1) and (4.2) subject to these conditions are

-' ( I-i + 0 x
el (x) =ei e T (4.4)

7r (( x -'3) e2 (X) e e







26

where



4A (4.6)

oR





F (4.8)
0d 1
/ ( Y + A(4c.8)





L k X%('-0 ?~~



The relationship between the propagation velocity and the incident laser intensity is given by



k + cC'( () 4 ] (4).9


If the losses due to radial conduction are much
2
greater than the radiation losses, i.e. A >> mRL and if the absorption coefficient satisfies the relationship

2 0 (K o(( 6+ 1), then Equation (4.9) reduces to


o ) (4.10)




Raizer,10 neglecting thermal radiation losses and the attenuation of the incident laser beam, obtained







27

Equation (4.10) in his one-dimensional analysis. For large propagation velocities, I~ and e both approach

unity, if the variation of A with u is neglected. Then Raizer's result that the laser intensity is proportional to u 2 at large velocities is obtained.

For values of x greater than the point of maximum

temperature for CO 2 laser radiation, ez can be approximated by

? eX


so that d is the length of the spark.

By setting the derivative of 9,in Equation (4.5) equal to zero, the location of the maximum temperature is found to be






Raizer 10 uses a highly empirical method for calculating the thermal conduction parameter. lf the axial variations are neglected within a cylinder of radius R L with strongly cooled walls, the thermal potential will vary approximately as J0(x x1lr/R L). When substituted into Equation (4.3), this distribution results in the value A = x l2= 5.8. To take into account the fact that the temperature is still quite high at r = R L, Raizer reduces this value by j and uses an effective A = 2.9.






28

A more accurate determination of this parameter can be obtained from the two-dimensional solution presented in Chapter III. By using Equation (3.6) to calculate e and the radial derivative, substituting into Equation (4.3) and evaluating at x 0 the following equation for A is obtained:





where Tn = (An/Xon)Jl(xonHL/Rc). The convergence of the infinite series is demonstrated in Appendix I.













CHAPTER V

EXAMPLE CALCULATIONS


The equations derived in Chapters III and IV have been used to study the properties of sparks maintained in atmospheric-pressure air by CO2 laser radiation.

The variation of the thermal potential with temperature, obtained from the data of Reference 11, is shown in Figure 3. A value of .15 kW/cm was used for the value of the thermal potential at ignition. This value corresponds to the ignition temperature T i = 12,000 OK suggested by Raizer. 10

The absorption coefficient 71, for CO2 laser radiation in air increases rapidly for temperatures greater than 12,000 'OK, due to the rapidly increasing degree of Ionization. 10It reaches a maximum value of approximately .85 cm-1 at T 17,000 OK and decreases to a minimum of .38 cm-1 at T = 24.,000 OK. This behavior can be understood by considering the formula for ?i'v given in Equation (2.4f) Beyond 17,000 OK, first ionization is essentially complete. The electron density approaches a plateau while the temperature T e continues to increase. Therefore, WY' decreases. For temperatures greater than 24+,000 0K, the electron density again increases rapidly due to the



29







30

ionization of singly charged ions. Consequently, the absorption coefficient increases for temperatures greater than 24,000 OK. The value X = .74 cm-I was used in the calculations for x ), 0. It corresponds to an average value for the temperature range 15,000 OK to 20,000 OK, which are typical values for the temperature in the spark core.

The ratio cp/ A was set equal to 425 cm sec/gm. From Figure 1 this corresponds to a mean value in the temperature range 10,000 OK to 20,000 'K. The upstream density was taken as _P. = 1.3 x 10-3 gm/cm3.

The thermal emission parameter m was assumed to be 175 cm-2. The resulting losses are 44 kW/cm3 and 67 kW/cm3 for T = 15,000 K and 20,000 OK, respectively. Raizer's calculations0 indicate losses which vary from 48 kW/cm3 to 60 kW/cm3 over the same interval, with the maximum occurring at 18,000 OK. However, he uses only half of this emission in his calculations since he assumes half of the radiation is re-absorbed in the cold gas ahead of the spark and not really lost. The experimental and theoretical work of Hermann and Schade 1'17 on wall-stabilized nitrogen arcs at atmospheric pressure indicate radiation losses of 50 kW/cm3 at 20,000 OK. Therefore, the linear approximation with m = 175 cm-2 adequately represents the losses in the hot zone of the air spark.







31

With these values of the parameters, Equations (3.6) (3-11) were used to study the propagation velocities, threshold intensities and temperature profiles for air sparks of radius .15 and .50 cm. Calculations were carried out for propagation velocities from 0 to 20 cm/sec. The channel radius was varied from R c = R L to R. = 2RL* A computer program was written to evaluate the series. It was found that twenty terms were sufficient to insure convergence for the range of parameters studied.

The variation of laser intensity with channel radius for several propagation velocities is shown in Figure 4. For a spark of .15 cm radius, a decrease in channel radius results in a significant increase in the laser intensity required to maintain the spark. The presence of the channel boundary constricts the radial temperature profile. Therefore, the radial heat conduction loss from the spark is increased as R. decreases, and a higher laser intensity is required to overcome the increased losses.

The energy loss per unit volume due to thermal

c induction In the radial direction is proportional to R L-2 whereas the loss per unit volume due to optically thin radiation is independent of radius. Therefore, thermal radiation becomes the dominant loss mechanism for large radius sparks. In that case the presence of a boundary should have little effect on spark propagation. This is demonstrated in Figure 4, which shows that the required






32

intensity for RL 50 am is nearly independent of channel radius.

Figures 5 and 6 show the isotherms, calculated from Equations (3.6) (3-11), for a spark of .15 am radius in a channel with radius R a = 1.25 R L* Figure 5 depicts the threshold isotherms, and Figure 6 shows the isotherms for u = 20 am/see. The intense core and elongated structure agrees with the qualitative description of laser sparks given by Smith and Fowler.8

Figures 7 and 8 show the isotherms for a spark of

.50 am radius in a channel with radius R 0 = 1.25 R L* The isotherms of Figure 7 correspond to the threshold case while those of Figure 8 correspond to a propagation velocity of 20 cm/sec. The blunter isotherms, as compared to the isotherms of Figures 5 and 6, are due to the decreasing importance of radial heat conduction.

The plots of the Isotherms show a rapid increase in

temperature at the spark front followed by a slow decrease. The increase is due to the sudden deposition of energy, as a result of the rapid increase in the absorption coefficient at T = 129000 OK. The slow decline in temperature for the regions beyond the temperature maximum is a consequence of the small absorption coefficient for C02 laser radiation. The absorption length for C02 laser radiation in air is 11V 7f 1.4 am. Therefore,

there is little attenuation of the beam when it traverses







33

distances on the order of .10 As a result points

separated by this distance absorb almost the same energy from the laser beam.

Figures 5 8 show that the isotherms for propagating sparks are blunter at the spark front when compared to the isotherms for stationary sparks. This is a result of convective cooling. In the regions near x = 0, the convective cooling decreases with radius since a / )Y

decreases with r. This implies that the hotter regions near the axis are cooled more strongly than the outer, cooler regions. Thus, convection tends to decrease the radial variation of temperature near the spark front.

Representative values of the physical properties of

laser sparks, as calculated from Equations (3.6) (3.12)v are given in Table I. The symbol 5, denotes the maximum value of the radially averaged potentials, and Tm is the corresponding temperature. For the cases considered in this analysis the spark is on the order of a centimeter in length with average temperatures near 16,000 OK. Approximately half the incident laser intensity is absorbed by the spark. The higher intensity required to maintain a propagating spark, as opposed to a stationary spark, results in an increase in both the temperature and the spark length.

The variation of the thermal conduction parameter A, calculated for threshold conditions, with channel radius







34

is shown in Figure 9. The results indicate that A = 2.9 for a spark of .15 cm radius in a channel of .30 am radius.

As discussed in Chapter IV, Hatzer 10 obtained this same value by an approximate calculation. The agreement is only coincidental since his calculation indicates A is independent of R L* In fact, the authors of Reference 11 use A = 2.9 for sparks with R L varying from .028 cm to

1.67 am. Figure 9 shows, however, that A is a function of laser radius. For R 0 A L 21 the value of A for a spark of .50 am radius is nearly twice the value for a spark of .15 am radius.
Calculations of threshold laser intensities by

substituting A from Figure 9 into Equation (4.9) gave excellent agreement with the results of the two-dimensional analysis. Although A was found to vary with propagation velocity, an error of no more than 5% was incurred in using the value of A at threshold to calculate S 0 for propagation velocities up to 20 cm/sec,

The one-dimensional solution can be used to study

the relative importance of each loss process in subsonic spark propagation. Dividing each loss term in Equation (4.2) by the volumetric energy absorption 5 0 X. exp(- )Ix) results in nondimensional parameters for the losses due to axial heat conduction, convection, thermal radiation and radial heat conduction.







35

The variations of these nondimensional parameters with axial distance are shown in Figures 10 and 11 for R L = 15 cmt He = 1.25 RL and propagation velocities of O and 20 cm/sec. These are the same conditions used to obtain the isotherms in Figures 5 and 6. The axial location of the temperature maximum x m has been used to nondimensionalize the axial distance from the spark front. The values of x m are .132 and .176 cm for u = 0 and 20 cm/sec, respectively.

Figures 10 and 11 show that the losses due to radial heat conduction and to thermal radiation are approximately equal for R L = .15 cm. For x > x m they are the dominant loss mechanisms. Axial heat conduction is important only near x = 0, where the rapid increase in temperature occurs. Likewise, axial convection losses, for the velocities considered in this analysis, are significant only near the spark front, where de/c)X is large. Beyond xm, axial convection becomes an energy source due to the change in sign of the axial gradient of the temperature.

Figures 12 and 13 show the variations of the

nondimensional losses for a spark of radius .50 cm propagating at velocities of 0 and 20 cm/sec in a channel with Rc = 1.25 RL. These are the conditions for which the isotherms in Figures 7 and 8 were plotted. As in Figures 10 and 11, the axial location of the temperature







36

maximum has been used to nondimensionalize the axial distance. For Figures 12 and 13t xm equals .129 am and .231 am, respectively.

Figures 12 and 13 illustrate the dominant role of

thermal radiation in the dissipation of energy for sparks of large radius. Radial heat conduction accounts for only 10% of the energy loss. Again, axial heat conduction and convection, for the velocities considered here, are important only near the spark front.

It is apparent from Figures 10 13 that the laser

intensity for a propagating spark must exceed the threshold intensity so that the additional loss due to convection at the spark front can be overcome. However, near the temperature maximum convective losses are negligible so that the increased laser intensity results in a higher maximum temperature for a propagating spark than for a stationary one. Beyond xmt convection actually is a heating tem. The spark length increases with propagation velocity due to this additional source of energy and due to the increase of laser intensity with propagation velocity. These results are illustrated in Figures 5 8 and in Table I.













CHAPTER VI

DISCUSSION OF RESULTS


Previous theoretical studies of laser spark phenomena have not probed the effect of a channel boundary on subsonic spark propagation. In fact, aside from models based on Raizer's one-dimensional approximation, only the spherically symmetric case has been considered. 18 Consequently, the conclusions drawn from this analysis cannot be compared with previous theoretical results.

The technology of ow CO 2 lasers has only recently attained the power levels required to maintain subsonic laser sparks in air. As a result, experimental Investigations are scarce, and they have not focused on spark propagation in channels. However, the predictions for, R a/RL = 2 can be compared with the results of experiments on unbounded sparks since the channel boundary is far enough from the spark that Its influence is negligible. A spark with R c /R L = 2 will be denoted a "free" spark.

Fowler et al., 19 using a focused CO 2 laser beam in atmospheric-pressure air, studied threshold intensities and propagation velocities. They also determined the spatial variation of temperature and electron number density by a laser interferometric technique. Their


37







38

results indicate a threshold intensity near 100 kW/cm 2 for a .15 am radius spark. This value agrees well with the value of 106 kW/cm 2 obtained from Figure 4 for a free spark of .15 am radius.

The isotherms of Fowler et al. 19 for an incident power of 6.2 kW can be compared with the isotherms of Figure 5, which correspond to an incident power of 7.8 kW. The experimental results indicate an intense core, with a maximum temperature of l7vOOO OK, and an elongated structure. For example, the maximum radial extent of the 149000 ok isotherm is .10 am and its length is .70 am. From Figure 5. the values of the radial extent and length of the 14tooo OK isotherm are .12 am and .8o cm, respectively. The authors of Reference 19 found that thermal radiation was the dominant loss mechanism for R L > .25 am, in agreement with the results of Chapter V.

Recently Keefer et al. 20 studied the threshold

properties of laser sparks in air. They estimated the
2
threshold intensity to be 120 kW/cm for R L = .10 am. Using spectroscopic techniques, they found isotherms with a spatial variation similar to the variation reported by Fowler et al. 19 The maximum radial extent and length of the 149000 OK isotherm were found to be .15 am and 1.14 am, respectively.

In light of the available experimental results, the

model appears capable of predicting the threshold properties







39

of small radius sparks. It is worthwhile to consider the simplifying assumptions and the limitations they impose on the use of the model.

The temperature dependance of c p was neglected in the analysis. This approximation has no effect on threshold calculations (u = 0) since the ratio appears only in the convective loss term. For the propagation velocities considered in this analysis, convection is significant only near the spark front, as discussed in Chapter V. There the temperatures are not much different from the ignition temperature T,, Therefore, a p / A can be considered constant at a value typical of temperatures near 12,000 OK.

The absorption coefficient no was assumed to be zero for x < 0 and a constant equal to .74 em-1 for x > 0. The two regions are separated by the plane x = 0 where the temperature averaged over the laser radius equals the ignition temperature T .. Actually the absorption coefficient has a maximum value of approximately .85 em-1 at T = 179000 OK and decreases to .60 cm-1 at 201,000 ok. Therefore, temperatures in excess of 17,000 OK are more difficult to achieve than the model supposes.

Raizer 10 has calculated maximum temperatures for the one-dimensional case taking into account the variation of ?f, with temperature. His analysis shows that the maximum temperature increases with propagation velocity






40
'W

and that these temperatures do exceed 17,000 OK. These results support the temperature variation with velocity indicated in Table I and in Figures 5 8.

Hall et al. 11 compared the results for a two-step variation of No with a one-step model and concluded that the one-step model was a good approximation. Their solutions also indicate an increase in temperature associated with an increase in propagation velocity.

Therefore, the step-function formulation of 7f, used in this analysis should not Introduce appreciable error into the calculations of S of the required laser intensity, or if., the maximum of the radially averaged temperatures. However, an analysis including the actual temperature variation of 11'p would cause some suppression of temperatures greater than 17,000 OKI. For example, there would be a slight decrease in the area within the 20,000 0 K isotherm shown In Figure 6.

For values of x -,* d the temperature falls below 12,000 Ok and N. decreases rapidly. Since the model retains the value of .74 cm-1 the temperatures predicted by the model in that region exceed those which actually occur. The error introduced by this approximation is not severe since the spark attenuates the laser intensity to approximately half its initial value.

The analysis treats the transport of thermal radiation in an approximate manner. The assumption that the radiation







41

transport can be represented by an optically thin emission proportional to the thermal flux potential is an oversimplification. Actually, the net energy loss due to thermal radiation can only be found by solving the radiative transfer and energy equations simultaneously. As discussed in Chapter V. the role of radiation increases with spark radius. As a consequence, sparks of large radius may require a more careful treatment of the radiation transport.

The analysis presented here has been limited to propagation velocities of 20 cm/sec or less. However, experiments5',8'19 indicate that the combustion front velocities can exceed several meters per second. The propagation of sparks at such high velocities by thermal conduction alone leads to excessively high temperatures. For example, the solution of Equations (3.6) (3.11) for a free spark of radius .50 cm propagating at 1 in/sec yields a required laser intensity of 740 kW/cin2 This intensity results in a value of Ym exceeding 40,000 0k, which is a factor of two greater than experiments indicate.5'8'19

Due to the thermal expansion of the spark, the

velocity in the laboratory frame can exceed u which is the velocity measured relative to the cold gas ahead of the spark.5 'l The magnitude of the difference can only be obtained by solving the full two-dimensional flow equations. It is doubtful, however, that thermal expansion could account for the large discrepancy.









The discrepancy between observed and predicted

velocities has led to a closer examination of the role of thermal radiation in spark propagation. Several researchersl1012918 have noted that radiation emitted by the hot spark can be re-absorbed by the cold gas ahead of the spark. Jackson and Nesn2have shown that, for large radius sparks, heating of the cold gas by reabsorption can far exceed the heating due to thermal conduction. Their results predict propagation velocities of several meters per second with temperatures T. of approximately 20,000 OK. However, the velocities they obtain are still a factor of five less than the experimental results they use for comparison.

The experiment of Keefer et al.2 suggests that highly nonequilibrium processes may be occurring in the cooler plasma regions. They found that the dominant radiation in the sheath surrounding the hot plasma core came from the 1t negative system of N2. Furthermore, radiation

from the 2nd positive system of N2, usually present in high-temperature air, was conspicuously absent. The anomalous N2 +radiation Indicates that nonequilibrium processes, such as photolonization, may be occurring.

This result suggests another explanation for the high propagation velocities. The thermal radiation escaping the plasma core, rather than raising the temperature of the cold gas as a whole as assumed in Reference 1?,






43

could lead instead, through photoionization and inverse bremsstrahlung absorption, to a nonequilibrium region ahead of the spark. A discussion of this phenomenon and its effects on spark propagation is given in Appendix II.













CHAPTER VII

CONCLUSIONS


The two-dimensionalenergy equation describing spark propagation in a channel has been used to investigate the properties of subsonic laser sparks. The closed-form solution yields both axial and radial temperature profiles, as well as the dependance of the propagation velocity on the incident laser intensity, laser radius and channel radius.

The model was applied to .15 cm and .50 cm radius

sparks driven by CO2 laser radiation at velocities up to 20 cm/sec In atmospheric-pressure air. The results indicate the channel has a significant influence on spark propagation for the .15 cm spark when the channel radius is less than twice the spark radius. For sparks of .50 cm radius, the effect of the channel is negligible.

For the cases considered, the spark length ranges

from .34 cm to 1.26 cm. The maximum value of the radially averaged temperature varies from 13,000 0k to 17,000 OK. The ratio of the power absorbed by the spark to the incident power ranges from 22% to 61%. For a fixed R L and Re these quantities increase with increasing laser Intensity.

44







45

For the .15 Om spark, radial heat conduction and

thermal radiation losses are approximately equal. However, for the .50 cm spark, radiation is the dominant loss mechanism. In both cases, axial convection and axial heat conduction are significant only near the spark front.

The predicted threshold intensity and isotherm

variations for the.15 em spark agree with the available experimental data. However, the model fails to predict the high propagation velocities experimentally observed. In order to maintain such high velocities, thermal radiation must play a significant role in the heating of the cold gas ahead of the spark.

An equation for the thermal conduction parameter A, Introduced by Raizer, has been obtained from the twodimensional solution. The evaluation of A for a free spark of .15 em radius confirms the value of 2.9 used by Raizer. Furthermore, A was found to depend on channel and laser radius.

In summary, the two-dimensional model presented In this analysis can be used as a reasonable approximation for the properties of laser sparks at threshold and at low propagation velocities. In those cases, the assumption that thermal radiation serves only as a loss mechanism is adequate. For high propagation velocities, a more careful treatment of the radiation transfer is required.







46

In spite of the rapidly expanding theoretical and experimental efforts, understanding of subsonic spark phenomena is'far from complete. In particular, the ,observed propagation velocities have eluded a satisfactory explanation. Further analysis in two areas would provide greater insight into the physical processes occurring in subsonic sparks. A detailed study of the radiation transport, including nonequilibrium effects, would provide a more complete knowledge of the propagation mechanism. And a study of the two-dimensional flow field around the spark would lead to a more accurate determination of the role convection plays in the energy balance.

















R L








T =0



Figure 1. Co-ordinate System for the Steady State Propagation of a Laser
Spark in a Channel.















103



IO
E









2





102 I I I I,
0 10 20 30
T, 103 K Figure 2. Variation of Cp/ A with Temperature
for Atmospheric-pressure Air.







.6 A -.04
e A


( -%) (C:OK)

.3 -.02






0 10 20 30
T, 10' K
Figure 3. Variation of Thermal Conductivity and Thermal Potential e
with Temperature for Atmospheric-pressure Air.






50

3










E




2

0












0 L
1.0 1.5 2.0
R~/

Figure 4.* Incident Laser Intensity Required to Maintain
a Spark as a Function of Channel Radius.
Solid curves are for R L = .15 am and broken
curves are for R L = .50 cm. Curves 1, 2 and3
correspond to velocities of 0, 10 and
20 cm/see, respectively.










31

RL









I I I I I I I
0 .3 .6 .9
x, cm
Figure 5. Isotherms for a Spark with RL = .15 cm, Rc = 1.25 RL and u = 0.
(1) T = 0, (2) T = 10,000 OK, (3) T = 14,000 OK, (4) T = 16,000 OK.











2

RL









I I I I I I I
0 .3 .6 .9
x, cm
Figure 6. Isotherms for a Spark with RL = .15 cm, Ro = 1.25 RL and u = 20 cm/sec.
(1) T = 0, (2) T = 12,000 OK, (3) T = 16,000 OK, (4) T = 20,000 OK.






53








21



RL












I I t I I I I I

-.5 0 .5 1.5 2.0
X,cm

Figure 7. Isotherms for a Spark with R = .50 cm,
R0 = 1.25 RL and u = 0. (1) T = 0, (2) T = 8,000 K, (3) T = 10,000 OKI
(4) T = 13,000 OK.






54







1



3 2


RL4












SI I II I 1 III

0 .5 1.0 1.5 2.0
X,cm

Figure 8. Isotherms for a Spark with RL = .50 cm,
Rc = 1.25 RL and u = 20 cm/sec. (1) T = 0,
(2) T = 10,000 OK, (3) T = 14,000 oK,
(4) T = 17,000 K.






55









12








6







0
1.0 1.5 2.0
R OR L Figure 9. Variation of the Thermal Conduction Parameter
with R a /H L' Solid curve is for RL = .15 cm
and broken curve is for R L = .50 cm.







56




.6



.52 Energy Loss 4



.3








.22





X/Xm


Figure 10. Axial Variation off the Volumetric Energy
Loss Terms for RL = .15 cm, Re = 1.25 RL and u = 0. (1) Radial heat conduction,
(2) Thermal radiation, (3) Axial heat
conduction.






57



.6



.5


.4
Energy Loss
.3


.2


.1



0

1 2
-. 1 X/Xm


Figure 11. Axial Variation of the Volumetric Energy Loss Terms for RL = .15 cm, Rc = 1.25 RL and
u = 20 cm/sec. (1) Radial heat conduction,
(2) Thermal radiation, (3) Axial heat
conduction, (4) Convection.






58






1.0







Energy






.22



0 1 2
X/Xm



Figure 12. Axial Variation of' the Volumetric Energy
Loss Terms for R L = 50 cm, RB0 = 1.25 R L
and u = 0. (1) Thermal radiation,
(2) Radial heat conduction, (3) Axial
heat conduction.





59



1.0


.8


.6
Energy
Loss .4 '



.2



_1 2
X/Xm


Figure 13. Axial Variation of the Volumetric Energy
Loss Terms for RL =.50 cm, Rc = 1.25 RL
and u = 20 cm/sec. (1) Thermal radiation, (2) Radial heat conduction, (3) Axial heat
conduction, (4) Convection.






60
Table I
Physical Properties of Laser Sparks of .15 cm and .50 cm Radius

R c u Tm d P)A/Po

cm cm/sec kW/cm 103 OK cm

RL = .15 cm

.15 0. .24 15. .83 .46
.15 10. .28 16. 1.02 .53
.15 20. .32 17. 1.24 .60
.30 0. .21 14. .66 .39
.30 10. .25 15. .94 .50
.30 20. .31 17. 1.26 .61



R L =.50 cm

.50 0. .19 13.5 .50 .31
.50 10. .23 14.5 .84 .46
.50 20. .30 16.5 1.26 .61
1.00 0. .17 13. .34 .22
1.00 10. .22 14. .74 .42
1.00 20. .29 16. 1.21 .59












APPENDIX I

CONVERGENCE OF THE SERIES SOLUTION

The two-dimensional energy equation describing subsonic spark propagation was solved in Chapter III in terms of infinite series. For the solution to be valid, the series must be shown to converge uniformly for every point (x,r) in the solution domain 00 x & o
0 r R .

The following properties of Bessel functions and
their zeroes will be used in proving the convergence of the series:21

i J(z) 1 (Al.1)
J1(z) j 1 (Al.2)

Jl(xon) 0 (Al.3)
lim Xon (n J) 1 (A1.4)
n -vo
lim J1(z) = .798 z" cos(z 3 ~' /4) (Al.5)
z-* o

lim Xon J 12(xon) = .637 (Al.6)
n-1 on
The uniform convergence of a series of function can
be proved by the Weierstrass M-Test.22 It states that the series Z_ f (x,r) converges uniformly in an interval of the variables x and r if Ifn(x,r) 4 Mn in that interval and Z Mn converges.

61







62

In both Equations (3.6) and (3.7), the exponential term and Bessel function which multiply the Any Bn and Cn are bounded, in absolute value, by one. For the series in Equation (3.6), Mn can be taken as



= [L + X Ot T]since I Ani 1 Mn. From the definitions of S and t

and Equation (Al.4),


=, = v
n -Therefore,

lim Mn = Dn2


where D is a positive, finite constant for R < oO

Since the series I n-2 converges and since 1-2
lim Mn/n-2 = D 4 oo nn
then Mn converges by comparison.22Cosuetyte


infinite series in Equation (3.6) converges uniformly. The convergence of Equation (3.11) follows by comparison with the series AnAt first glance, the two series in Equation (3.7)






63

appear to have singularities at values of velocity such that


CA. 2 7f,


However, when this condition is satisfied, the exponential factors for the two series are equal. The two series can then be written as one series with coefficients Dn = Bn + Cn. When this is done, the singularity disappears.

By the Weierstrass M-Test, the series in Equation (3.7) are uniformly convergent if the series IBnl and

I jCn converge. The convergence is easily shown by comparison. Since

li IBnl / IAI =2
M% --00o

lir ICnI / Ani =1


and 2 JAn converges, then IBnland ICnI

converge.
The infinite series Xon2 Tn in Equation (4.11)
can be shown to converge by comparison with the series L n-3/2. By applying Equations (Al.4) (Al.6) it can t%
be shown that

3 z R. 3Xo,, T R C ,c5 (Xo "RC

3/2 R%. (\0o /Z


=0O


Therefore, the series 1r Xon 2 Tn converges.
V%













APPENDIX II

EFFECT~ OF RADIATION ON SUBSONIC SPARKS


It was determined in Chapter VI that spark propagation at velocities of several meters per second by thermal conduction results in temperatures which exceed the experimentally observed temperatures. Therefore, it has been postulated that re-absorption of thermal radiation plays a dominant role in the propagation of laser sparks at high velocities.

Jackson and Nielsen obtained propagation velocities of several meters per second by Including radiative transfer in a one-dimensional model. Their solution was based on calculations of the radiative emission and absorption for equilibrium air. The recent results of Keefer et al. 20 suggest, however, that highly nonequilibrium processes are occurring in the cooler plasma regions.

Except for two limiting cases, the solution of the

energy equation including radiative transfer poses a problem of considerable difficulty. In the optically thick limit, where the mean free path of the thermal radiation L,, is much less than some characteristic dimension of the spark L, the radiative energy flux can be written as Cv, T, where

J1r (T) is the radiative analogue of the thermal conductivity.231 In that case, the effect of radiative transport is merely an



64e







65

increase in the thermal conduction coefficient. In the optically thin limit, where 1,, >> L, the plasma acts as a volume radiator, and the rate of' energy loss per unmit volume can be written as a function of' temperature. 23Both limiting cases are valid only where local thermodynamic equilibrium exists.

The model described in Chapter II assumes that radiation can be treated as an effective, optically thin emission. This assumption is adequate for small radius sparks propagating at low velocities. In that case, energy transport by thermal radiation is no greater than energy transport by thermal conduction.

For laser sparks in which radiation dominates the

energy transport, the radiative transfer, unfortunately, is neither optically thick nor optically thin. The radiation escaping the hot plasma core lies in the vacuum ultraviolet (vuv). 12In the core region, where there is a high degree of' ionization, the plasma is optically thin to vuv radiation. However, in the cooler regions this radiation is strongly absorbed by the neutral species.

Jackson and Nielsen12 found that most of the emission from the plasma core lies in the energy range from 14.5 to 20 ev, due mainly to electron recombination with N + and

0 + ions. Photons within that range possess energy exceeding the ionization potentials E i of' the neutral species of' air, Ei being 15.6 ev, 12.1 ev, 14.5 ev and 13.6 ev for N 2, 02, N and 0f respectively.2 The photoionization cross sections







66

for photons whose energy exceeds E i are high. For N 2 and these cross sections are on the order of 20 Mb. where a megabarn (Mb) equals 10l8 em. For N and 0 the cross sections are approximately 8 Mb.24-26

The absorption coefficient for the photons equals the sum, over the absorbing species present, of the product of the cross section for photoionization and the number density. Equilibrium, atmospheric-pressure air at 12,000 O'K is composed primarily of neutral nitrogen and oxygen atoms.l14 The combined neutral atom densities are on the order of

5 x1017 cm-3. A cross section of 8 Mb results in a absorption coefficient for the photoionizing radiation of 4 cm-1 and a photon mean free path of ty = .25 cm. Consequently, the radiation is strongly absorbed in air at temperatures near 12,000 OK.

This strong absorption indicates the radiation transport in the region x < 0 might be adequately represented by the diffusion approximation AIrl T. Jackson and Nielsen 12 found, however, that this approach yielded values of the energy deposition by re-absorption which were a factor of forty smaller than their numerical calculations. The large discrepancy is due to the rapid temperature variation, near the spark front, over distances on the order of 1 As a result the radiation field is highly anisotropic and the diffusion approximation is invalid.

The effect of the absorption at temperatures near

12,000 OK is, initially, an increase in the concentrations







67

of N + 1 0 + and electrons above their equilibrium values. If the relaxation time is short enough, the concentrations will equilibrate to values characteristic of some higher temperature. The net effect of the radiation is then an equilibrium heating of the cold gas, as assumed by Jackson and Nielsen. 12 One reaction which is essential for equilibration is the electron-recombination reaction


N + + e + M 0 N + M (A2.1)


where M is an atomic or molecular species. At electron temperatures near 12,000 OK, the reaction rate constants for such reactions are on the order of 10-29 cm 6 /see. 27

There is, however, the competing reaction


N + + N + M ;0 N 2 + + M (A2.2)


for the excess N + The constant for this reaction has not been accurately determined. An estimate for temperatures near 12,000 OK is 10-30 + 4 cm 6 /see. 27 For equilibrium, atmospheric-pressure air at 12,000 0 K the nitrogen atom number density Is approximately six times the electron number density,,14 so that reaction (A2.2) may proceed at a much faster rate than reaction (A2.1). Consequently, a nonequilibrium concentration of N 2 + may exist outside the hot plasma core. In fact, investigators 8120 have reported strong N 2 + radiation outside the core region. The role of reaction (A2.1) In the relaxation scheme increases with increasing electron number density. The relative







68

importance of reactions (A2.1) and *(A2.2) may change, therefore, as the departure from equilibrium increases.

At 6,000 OK equilibrium air consists mostly of N2, N and 0, in approximately equal concentrations.l4 Therefore, if the distance in which the temperature drops from 12,000 OK to 6,000 OK is on the order of 4' = .25 cm or less, some radiation can penetrate into the cooler regions and directly photoionize N2 by the reaction

N2 + hO -- N2+ + e, ht Z 15.6 ev (A2.3)


It is interesting to note that the molecular ions produced by reaction (A2.3) for h d > 18.7 ev are almost equally distributed among the ground state and the first two excited states, A 2ITu and B 2 u +.24 It is the

transition from B 2 u+ to the ground state X 1 g+ which produces the Ist negative band of N2+observed experimentally in References 8 and 20.

A careful study of the reactions which occur subsequent to photoionization is necessary for determining the conditions at the spark front. Reactions related to electron loss or gain are especially important since the absorption of laser radiation is directly related to the electron number density. A major electron loss process is the dissociative recombination of N2+

N2+ +e -" 2N (A2.4)







69

Its rate constant is on the order of' 107 "cm3/sec.2 Three-body recombination

X+ e + M1 X + M1 (A2.5)


where X is N. 0 or N2, also plays a significant role. The reaction rate constant for (A2.5) at electron temperatures near 6,000 0K is typically 5 x 10-2 cm 6/sec when M1 is an atomic or molecular species and

5 x 10-2 CM6 /sec when M Is an electron. 27 Electron attatchment to oxygen atoms and molecules may also be important since the negative ions can react with other species to form stable compounds.29'30

If the recombination reactions are fast enough, the

electron number density will relax to its equilibrium value before the electrons can be appreciably heated by the laser radiation. The assumption of' rapid relaxation is made implicitly in the analysis of' Jackson and Nielsen. 12 if' chemical relaxation occurs too slowly, a nonequilibrium layer will form which will enhance the absorption of' laser radiation.

The development of' the nonequilibrium layer occurs due to the combined absorption of' thermal radiation and laser radiation. The absorption of thermal radiation induces appreciable ionization ahead of the spark analagous to the effect of' precursor radiation in shock waves.31 The free electrons are then selectively heated by inverse







70

bremsstrahlung absorption of laser radiation and electron thermal conduction. The electrons quickly relax to a Maxwell-Boltzmann distribution at a temperature Te as a result of the high efficiency of electron-electron collisions in transferring energy. Since the transfer of energy to the heavy particles by electron collisions is inefficient, a two-temperature plasma, with electron temperature exceeding the heavy-particle temperature, develops. Further ionization is then caused by collisions of heavy particles with high-energy electrons. Consequently, an appreciable electron density can exist in a layer ahead of the spark where the temperature of the heavy particles is still quite low.

The inverse bremsstrahlung absorption coefficient, given by Equation (2.4) in Chapter II, Is proportional to the square of the electron number density. Therefore, the development of a nonequilibrium layer with a high electron number density ahead of the spark means that absorption of the laser radiation and consequent heating

-occurs in regions where the heavy-particle temperatures are less than 12,000 'K.

Although this is a nonequilibrium process, the

equilibrium solution obtained in Chapter III can be used to illustrate the effects of the anomalous absorption on spark propagation. Absorption of laser radiation in regions with temperatures less than the equilibrium ignition







71

temperature can be simulated by lowering the ignition thermal potential to ei= 0. < .15 kW/cm. The

required laser intensity for a given propagation velocity Is then reduced, according to Equation (3.11), to



50 so



where S 0 is the intensity calculated using 19i = .15 kW/cm. Equations (3.6) (3.10) indicate that the spark thermal potentials are reduced by the same ratio. Therefore, an absorption wave preceding the temperature rise lowers the required laser intensity and the maximum spark temperature for a given propagation velocity.

For example, initiating the absorption at temperatures near 6,000 OK ( E0 .025 kW/cm) results in laser intensities and thermal potentials reduced by a factor of six. Therefore, a free spark of radius .50 cm propagating at

1 rn/sec would require only 123 kW/cm 2and would result in a value of T. = 17,000 %K. For an Ignition temperature of 12,000 OKI S would be 74f0 kW/cm 2 and TM would exceed 4i0,000O K.

Figure A-i shows the isotherms for a spark of radius .50 cm propagating at 1 rn/sec, assuming 4~ = .025 kW/cm. The effects of convection can be seen in the blunt spark front and elongated isotherms. It is interesting to note that the 12,000 OK isotherm crosses the x-axis at x = .10 cm.







72

Since the 6,000 'K isotherm crosses near x = 0, the width of' the zone in which the temperature falls from 12,000 OK to 6,000 OK is .10 am. This distance is less than the absorption length of .25 cm for the vuv radiation emitted from the core. Therefore, the model, though simple, is at least consistent In that the radiation can penetrate to regions where T < 6,000 OK.
The intention of' the discussion presented above was

to describe, in a qualitative way, the effect re-absorption of' thermal radiation would have on spark propagation. A future investigation keying on the processes occurring in the spark front would be useful in understanding the propagation mechanism. In particular, a one-dimensional model of' a nonequilibrium gas under the simultaneous influence of two radiation fields would be a logical extension to this analysis.











I2

RL 3
4









I I I lII II I IlI
0 .5 1.0 1.5 2.0 2.5 3.0
x, cm

Figure A-1. Isotherms for a Spark with RL = .50 cm, R = 2 RL, u = 1 m/sec
and Ti = 6,000 OK. (1) T = 8,000 OK, (2) T = 12,000 OK,
(3) T = 16,000 OK, (4) T = 17,000 OK.
-,,3
kJ3












REFERENCES

1. Buonadonna, V. R., Knight, C. J. and Hertzberg A., "The
Laser Heated Hypersonic Wind Tunnel," Aerospace Research
Laboratories Report, ARL 73-0074, 1973.
2. Raizer, Y. P., "Breakdown and Heating of Gases under the
Influence of a Laser Beam," Soy. Phys.-Usp., 8, 650 (1966).

3. George, E. V., Bekefi, G. and Yaakobi, B., "Structure of
a Plasma Fireball Produced by a 002 Laser," Phys. Fluids,
14, 2708 (1971).

4. Ready, J. F., Effects of High Power Laser Radiation,
New York: Academic Press, 1971.

5. Bunkin, F. V., Konov, V. I., Prokhorov, A. M. and Fedorov, V. B., "Laser Spark in the 'Slow Combustion'
Regime," JETP-Lett., 2, 371 (1969).
6. Generalov, N. A., Zimakov, V. P., Kozlov, G. I., Masyukov,
V. A. and Raizer, Y. P., "Continuous Optical Discharge,"
JETP-Lett., 11, 302 (1970).
7. Generalov, N. A., Zimakov, V. P., Kozlov, G. I., Masyukov,
V. A. and Raizer, Y. P., "Experimental Investigation of a
Continuous Optical Discharge," Soy. Phys.-JETP, 34, 763
(1972).
8. Smith, D. C. and Fowler, M. C., "Ignition and Maintenance
of a CW Plasma in Atmospheric-pressure Air with C02 Laser Radiation," App1. Phys. Lett., 22, 500 (1973).
9. Franzen, D. L., "Continuous Laser Sustained Plasmas,"
J. Appl. Phys., 44, 1727 (1973).
10. Raizer, Y. P., "Subsonic Propagation of a Light Spark
and Threshold Conditions for the Maintenance of Plasma
by Radiation," Soy. Phys.-JETP, 31, 1148 (1970).
11. Hall, R. B., Maher, W. E. and Wei, P. S. P., "An
Investigation of Laser-Supported Detonation Waves,"
Air Force Weapons Laboratory Technical Report
AFWL-TR-73-28, 1973.



74







75

12. Jackson, J. P. and Nielsen, P. E., 'Role of Radiative
Transport in the Propagation of Laser Supported Combustion Waves," AIAA Paper No. 74-228, 1974.

13. Zeldovich, Y. B. and Raizer, Y. P., Physics of Shock
Waves and High Temperature Hydrodynamic Phenomena Vol. 1,
edited by W. D. Hayes and R. F. Probstein, New York:
Academic Press, 1966.

14. Wei, P. S. P. and Hall, R. B., 'Emission Spectra of
Laser Supported Detonation Waves," J. Appl. Phys., 4
2311 (1973).

15. Luke, Y. L., 'Integrals of Bessel Functions, Handbook
of Mathematical Functions, edited by M. Abramowitz and
I. A. Stegun, Washington, D.C.: U.S. Government
Printing Office, 1966.
16. Hermann, W. and Schad E., "Transportfunktionen von
Stickstoff bis 26000 "K,' Z. Physik, 233, 333 (1970).

17. Hermann, W. and Schade, E., "Radiative Energy Balance
in Cylindrical Nitrogen Arcs," J. Quant. Spectrosc.
Radiat. Transfer, 12 1257 (1972).

18. Raizer, Y. P., "Propagation of Discharges and Maintenance
of a Dense Plasma by Electromagnetic Fields,' Soy. Phys.USy., 15, 688 (1973).

19. Fowler, M. C., Smith, D. C., Brown C. 0. and Radley, R. J.,
"Laser Supported Absorption Waves,& United Aircraft
Research Laboratories Report No. N921716-9, 1974.

20. Keefer, D. R., Henriksen, B. B. and Braerman, W. F.,
"Experimental Study of a Stationary Laser Produced
Air Plasma," to be published.

21. Over, F. W. J., "Bessel Functions of Integer Order,"
Handbook of Mathematical Functions, edited by
M. Abramowitz and I. A. Stegun, Washington, D.C.: U.S.
Government Printing Office, 1966.

22. Widder, D. V., Advanced Calculus (2nd Edition),
Englewood Cliffs, N.J.: Prentice Hall, Inc., 1961.

23. Penner, S. S. and Olfe, D. B., Radiation and Reentry,
New York: Academic Press, 1968.

24. Huffman, R. E., "Photochemical Processes: Cross-Section
Data," Reaction Rate Handbook (2nd Edition), edited by
M. H. Bortner and T. Baurer, Santa Barbara, Calif.: DASIAC, DoD Nuclear Information and Analysis Center,
1972.






76

25. Wilson, K. H. and Nicolet, W. E., "Spectral Absorption
Coefficients of Carbon, Nitrogen and Oxygen Atoms,"
J. Quant. Spectrose. Radiat. Transfer, 12, 1257 (1972).
26. Samson, J. A. R. and Cairns, R. B., "Absorption and
Photolonization Cross Sections of 02 and N2 at Intense
Solar Lines," J, Geophys. Res., 69, 4583 (1964).
27. Bortner, M. H., Kummler R. H. and Baurer, T., "Summary
of Suggested Rate Constants," Reaction Rate Handbook (2nd Edition), edited by M. H. Bortner and T. Baurer,
Santa Barbara, Calif.: DASIAC, DoD Nuclear Information
and Analysis Center, 1972.
28. Bardsley, J. N. and Biondi, M. A., "Dissociative
Recombination," Advances in Atomic and Molecular Physics
Volume 6, edited by D. R. Bates and I. Esterman,
New York: Academic Press, 1970.

29. Phelps, A. V., "Electron Attachment and Detachment
Processes," Reaction Rate Handbook (2nd Edition),
edited by M. H. Bortner and T. Baurer, Santa Barbara, Calif.: DASIAC, DoD Nuclear Information and Analysis
Center, 1972.

30. Ferguson, E., "Ion-Neutral Reactions A. Thermal Processes,"
Reaction Rate Handbook (2nd Edition),,edited by M. H.
Bortner and T. Baurer, Santa Barbara, Calif.: DASIAC,
DoD Nuclear Information and Analysis Center, 1972.

31. Ferrari, C. and Clarke, J. H., "On Photoionization
Ahead of a Strong Shock Wave," Supersonic Flow,
Chemical Processes and Radiative Transfer, edited by
D. B. Olfe and V. Zakkay, New York: Pergamon
Press, 1964.













BIOGRAPHICAL SKETCH


Jad Hanna Batteh was born in Ramallah, Jordan, on January 5, 19147. After graduating from Ribault Senior High School of Jacksonville, Florida, in June, 1964, he enrolled at the University of Florida. As an undergraduate, he was elected to membership in Phi Eta Sigma, Sigma Tau and Tau Beta Pi honorary fraternities. He received the degree of Bachelor of Science in Aerospace Engineering with High Honors in December, 1967.

Beginning July, 1968,, he was employed for two and a half years as an aircraft structures engineer at the Lockheed-Georgia Company in Marietta, Georgia. During that time, he attended the Georgia Institute of Technology from which he received the degree of Master of Science in Aerospace Engineering in December, 1970.

Since the fall of 1971, he has been enrolled in the Graduate School of the University of Florida. During his tenure, he has held an NDEA Fellowship and a Graduate School Fellowship.

He is married to the former Jane Joseph Bateh of Jacksonville, Florida.






77








I certify that I have read this study andi that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate,, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

/ ,2
-X~w 'Iz/
Dennis R. Keefer, Chai/x'man Associate Professor of
Engineering Sciences

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and3 quality, as a dissertation for the degree of Doctor of Philosophy.




/Bernard M. Leadon
Professor of Engineering
Sciences
I certify that I have read this study and that 4n my opinion it conforms to acceptable stk-andards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Ulrich Hi. KurzweFV
Associate Professor of
Engineering Sciences

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Robert G. Blake
Associate Professor of
Mathematics








I certify that I have read this study and that In my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate,, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



z,,,Joh W.Fowr
/I Professor of Physics C'x and Astronomy

This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as a partial fulfillment of the requirements for the degree of Doctor of Philosophy.

August, 1974




/xVean, Collegig of Engineering




Dean, Graduate School




















girt




Full Text
CHAPTER VII
CONCLUSIONS
The two-dimensionalenergy equation describing spark
propagation in a channel has been used to investigate the
properties of subsonic laser sparks. The closed-form
solution yields both axial and radial temperature profiles,
as well as the dependance of the propagation velocity on
the incident laser intensity, laser radius and channel
radius.
The model was applied to .15 cm and .50 cm radius
sparks driven by C02 laser radiation at velocities up to
20 cm/sec in atmospheric-pressure air. The results
indicate the channel has a significant influence on spark
propagation for the .15 cm spark when the channel radius
is less than twice the spark radius. For sparks of .50 cm
radius, the effect of the channel is negligible.
For the cases considered, the spark length ranges
from .34 cm to 1.26 cm. The maximum value of the radially
averaged temperature varies from 13,000 K to 17,000 K.
The ratio of the power absorbed by the spark to the
incident power ranges from 22% to 61%. For a fixed RL
and Rc, these quantities increase with increasing laser
intensity.
44


Figure
Page
11.
Axial Variation of the Volumetric Energy
Loss Terms for RL = .15 cm, Rc = 1.25 RL
and u = 20 cm/sec
12.
Axial Variation of the Volumetric Energy
Loss Terms for R^ = .50 cm, Rc = 1.25 RL
and u=0
. 58
13.
Axial Variation of the Volumetric Energy
Loss Terms for RL = .50 cm, Rc = 1.25 RL
and u = 20 cm/sec
. 59
A. 1
Isotherms for a Spark with RL = .50 cm,
Rc = 2 Rl, u = 1 m/sec and = 6,000 K
. 73
vil


71
temperature can be simulated by lowering the ignition
thermal potential to QL Q. < .15 kW/cm. The
required laser intensity for a given propagation velocity
is then reduced, according to Equation (3.11), to
/
where SQ is the intensity calculated using - = .15 kW/cm.
Equations (3.6) (3.10) indicate that the spark thermal
potentials are reduced by the same ratio. Therefore, an
absorption wave preceding the temperature rise lowers the
required laser intensity and the maximum spark temperature
for a given propagation velocity.
For example, initiating the absorption at temperatures
near 6,000 K ( cr .025 kW/cm) results in laser inten
sities and thermal potentials reduced by a factor of six.
Therefore, a free spark of radius .50 cm propagating at
1 m/sec would require only 123 kW/cm and would result in
a value of Tm = 17,000 K. For an ignition temperature
of 12,000 K, SQ would be 740 kW/cm'* and Tm would exceed
40,000 K.
Figure A-l shows the isotherms for a spark of radius
.50 cm propagating at 1 m/sec, assuming Q = .025 kW/cm.
The effects of convection can be seen in the blunt spark
front and elongated isotherms. It is interesting to note
that the 12,000 K isotherm crosses the x-axis at x = .10 cm.


66
for photons whose energy exceeds E^^ are high. For N2 and
02 these cross sections are on the order of 20 Mb, where a
megabarn (Mb) equals 10"1 cm2. For N and 0 the cross sec-
nji p L
tions are approximately 8 Mb.
The absorption coefficient for the photons equals the
sum, over the absorbing species present, of the product of
the cross section for photoionization and the number density.
Equilibrium, atmospheric-pressure air at 12,000 K is com-
14
posed primarily of neutral nitrogen and oxygen atoms.
The combined neutral atom densities are on the order of
5 x 101? cm"-^. A cross section of 8 Mb results in a absorp
tion coefficient for the photoionizing radiation of 4 cm"1
and a photon mean free path of iv .25 cm. Consequently,
the radiation is strongly absorbed in air at temperatures
near 12,000 K.
This strong absorption indicates the radiation transport
in the region x < 0 might be adequately represented by
v i p
the diffusion approximation r* T. Jackson and Nielsen
found, however, that this approach yielded values of the
energy deposition by re-absorption which were a factor of
forty smaller than their numerical calculations. The large
discrepancy is due to the rapid temperature variation, near
the spark front, over distances on the order of .
As a result the radiation field is highly anisotropic and
the diffusion approximation is invalid.
The effect of the absorption at temperatures near
12,000 K is, initially, an increase in the concentrations


41
transport can be represented by an optically thin emission
proportional to the thermal flux potential is an oversim
plification. Actually, the net energy loss due to thermal
radiation can only be found by solving the radiative transfer
and energy equations simultaneously. As discussed in
Chapter V, the role of radiation increases with spark radius.
As a consequence, sparks of large radius may require a more
careful treatment of the radiation transport.
The analysis presented here has been limited to prop
agation velocities of 20 cm/sec or less. However,
experimentg58,19 indicate that the combustion front
velocities can exceed several meters per second. The prop
agation of sparks at such high velocities by thermal
conduction alone leads to excessively high temperatures.
For example, the solution of Equations (3.6) (3.11) for
a free spark of radius .50 cm propagating at 1 m/sec
o
yields a required laser intensity of 740 kW/cm This
intensity results in a value of Tm exceeding 40,000 K,
which is a factor of two greater than experiments
indicate.'
Due to the thermal expansion of the spark, the
velocity in the laboratory frame can exceed u which is the
velocity measured relative to the cold gas ahead of the
spark.10,18 The magnitude of the difference can only be
obtained by solving the full two-dimensional flow equations.
It is doubtful, however, that thermal expansion could
account for the large discrepancy.


T = 0
T = 0
Figure 1. Co-ordinate System for the Steady State Propagation of a Laser
Spark in a Channel.


68
importance of reactions (A2.1) and (A2.2) may change,
therefore, as the departure from equilibrium increases.
At 6,000 K equilibrium air consists mostly of N2,
14
N and 0, in approximately equal concentrations.
Therefore, if the distance in which the temperature drops
from 12,000 K to 6,000 K is on the order of iy = .25 cm
or less, some radiation can penetrate into the cooler
regions and directly photoionize N2 by the reaction
N2 + h*> * N2+ + e, htS 2. 15.6 ev (A2.3)
It is interesting to note that the molecular ions
produced by reaction (A2.3) for h J > 18.7 ev are almost
equally distributed among the ground state and the first
two excited states, A and B u It is the
transition from B 2 £ + to the ground state X 1 ^ +
W o
which produces the ls^ negative band of N2+ observed
experimentally in References 8 and 20.
A careful study of the reactions which occur sub
sequent to photoionization is necessary for determining
the conditions at the spark front. Reactions related to
electron loss or gain are especially important since the
absorption of laser radiation is directly related to the
electron number density. A major electron loss process
is the dissociative recombination of N2+
N2+ + e 2N (A2.4)


15
correspond to the temperature at which the plasma becomes
opaque to the laser radiation. This consistency condition
is satisfied by requiring the thermal flux potential,
averaged over the laser radius at x = 0, to be equal to
Q¡ the value of the potential at the ignition
temperature i.e.
(2.10)
This equation results in a relationship between the
propagation velocity and the required laser intensity.


Thermal heat conduction, aided by reabsorption of thermal
radiation emitted from the hot spark, is the mechanism by
which the cold gas ahead of the spark is ionized and
becomes opaque to the incoming laser radiation. The spark
propagates subsonically in a direction opposite the laser
flux. Typical spark temperatures in the subsonic regime
are 10,000 25,000 K.5"8
This mode of propagation suggests the existence of a
threshold laser intensity, such that the absorbed laser
energy just compensates for the heat conduction and
radiation losses, and the spark remains stationary. The
existence of a threshold was demonstrated in 1970 by
7
Generalov et al. *' who used a continuous wave (cw) C02
laser to maintain a stationary plasma in high pressure
argon and xenon gases. Similar experiments have been
o 8
conducted by Franzen,7 and recently Smith and Fowler have
published data on the threshold intensity of CC>2 laser
radiation in atmospheric-pressure air.
The modeling of subsonic laser sparks promotes
understanding of the creation and maintenance of these
discharges. Unfortunately, only a limited amount of
theoretical work has been done. Raizer,10 using a
one-dimensional analysis and introducing a thermal
conduction parameter for radial heat losses, derived
threshold laser intensities and propagation velocities for
sparks in atmospheric-pressure air under the influence
of C02 and Nd-glass laser radiation. Hall, Maher and Wei,


18
Equation (2.6) for 02 is a nonhomogeneous, linear
partial differential equation. Its solution can be
written as
= 0P + 0k
where p is a particular solution, and ^ is the
solution of the associated homogeneous equation
x2 r 3r 3 r / } x
- m y, = O
A particular solution of Equation (2.6) which
satisfies the boundary conditions at r = Rc and x = CO is
ep = e-*"x T0(x.£j (3.3)
where the bn are constants. Substituting Equation (3.3)
into Equation (2.6) yields
1 T0 (x. ~ ) = So H ( Ru r )
a rs<.
K ( K + <*) + rr\
(3.4)
ba


62
In both Equations (3.6) and (3.7), the exponential term
and Bessel function which multiply the An, Bn and Cn are
bounded, in absolute value, by one. For the series in
Equation (3.6), Mn can be taken as
<** a' (*+ )
l
since \ Anl Mn. From the definitions of and
and Equation (A1.4),
2 1T fl
\\try. = |;#n =
ft * oo n * o* otRc
Therefore,
lim Mn = Dn"2
ft -* oo
where D is a positive, finite constant for R < &o .
c
Since the series 2 n2 converges and since
t\
lim Mn/n"2 = D < 70
ft ~9 oo
< 22
then Mn converges by comparison. Consequently, the
infinite series in Equation (3-6) converges uniformly. The
convergence of Equation (3.11) follows by comparison with
the series Z A .
a. n
At first glance, the two series in Equation (3.7)


57
Figure 11. Axial Variation of the Volumetric Energy Loss
Terms for RL .15 cm, R = 1.25 RL and
u = 20 cm/sec. (1) Radial heat conduction,
(2) Thermal radiation, (3) Axial heat
conduction, (4) Convection.


55
RA
Figure 9. Variation of the Thermal Conduction Parameter
with Rc/Rl. Solid curve is for R^ = .15 cm
and broken curve is for RT = .50 cm.


radial temperature profiles as well as the dependence of
the propagation velocity on channel radius, spark radius
and incident laser intensity. The two-dimensional model
also offers a method for evaluating the radial heat
conduction parameter introduced by Raizer.*0


35
The variations of these nondimensional parameters
with axial distance are shown in Figures 10 and 11 for
RT = .15 cm, R 1.25 Rt and propagation velocities of
L C Lj
0 and 20 cm/sec. These are the same conditions used to
obtain the isotherms in Figures 5 and 6. The axial
location of the temperature maximum xm has been used to
nondimensionalize the axial distance from the spark front.
The values of xffl are .132 and .176 cm for u = 0 and
20 cm/sec, respectively.
Figures 10 and 11 show that the losses due to radial
heat conduction and to thermal radiation are approximately
equal for Rr = .15 cm. For x > x they are the dominant
loss mechanisms. Axial heat conduction is important
only near x = 0, where the rapid increase in temperature
occurs. Likewise, axial convection losses, for the
velocities considered in this analysis, are significant
only near the spark front, where dQ/d* is large.
Beyond xm, axial convection becomes an energy source due
to the change in sign of the axial gradient of the
temperature.
Figures 12 and 13 show the variations of the
nondimensional losses for a spark of radius .50 cm prop
agating at velocities of 0 and 20 cm/sec in a channel
with Rc = 1.25 Rl. These are the conditions for which
the isotherms in Figures 7 and 8 were plotted. As in
Figures 10 and 11, the axial location of the temperature


CHAPTER I
INTRODUCTION
Interest in the interaction of matter with intense
beams of coherent radiation has increased with the develop
ment of high power lasers. In particular, extensive
research is being devoted to the investigation of dis
charges maintained by the absorption of laser radiation.
These discharges are commonly referred to as laser sparks.
Discharges are in wide demand for technological and
diagnostic applications and for laboratory studies of plasma
processes. The current methods for the creation and main
tenance of discharges are based on the release of electro
magnetic energy in a gas. Until recently discharges, which
are classified by the frequency of the electromagnetic
field, ranged from the direct-current arc discharge to the
microwave discharge. Within this frequency range the
transfer of electromagnetic energy requires special
auxiliary equipment such as electrodes, conductor coils
and wave guides. As a result the plasma is confined by
boundaries which can erode and introduce contaminants.
The laser spark represents an extension of the frequency
range of the generating field to infrared and visible
frequencies. The electromagnetic energy is transported by
1


38
2
results indicate a threshold intensity near 100 kW/cm
for a .15 cm radius spark. This value agrees well with
the value of 106 kW/cm2 obtained from Figure 4 for a
free spark of .15 cm radius.
The isotherms of Fowler et al.^ for an incident
power of 6.2 kW can be compared with the isotherms of
Figure 5 which correspond to an incident power of 7.8 kW.
The experimental results indicate an intense core, with a
maximum temperature of 17,000 K, and an elongated struc
ture. For example, the maximum radial extent of the
14,000 K isotherm is .10 cm and its length is .70 cm.
From Figure 5, the values of the radial extent and length
of the 14,000 K isotherm are .12 cm and .80 cm, respectively.
The authors of Reference 19 found that thermal radiation
was the dominant loss mechanism for R^ > .25 cm, in
agreement with the results of Chapter V.
20
Recently Keefer et al. studied the threshold
properties of laser sparks in air. They estimated the
threshold intensity to be 120 kW/cm for R^ = .10 cm. Using
spectroscopic techniques, they found isotherms with a
spatial variation similar to the variation reported by
19
Fowler et al. 7 The maximum radial extent and length of the
14,000 K isotherm were found to be .15 cm and 1.14 cm,
respectively.
In light of the available experimental results, the
model appears capable of predicting the threshold properties


42
The discrepancy between observed and predicted
velocities has led to a closer examination of the role
of thermal radiation in spark propagation. Several
researchers10"12*18 have noted that radiation emitted by
the hot spark can be re-absorbed by the cold gas ahead of
12
the spark. Jackson and Nielsen have shown that, for
large radius sparks, heating of the cold gas by re-
absorption can far exceed the heating due to thermal
conduction. Their results predict propagation velocities
of several meters per second with temperatures Tffl of
approximately 20,000 K. However, the velocities they
obtain are still a factor of five less than the experi
mental results they use for comparison.
20
The experiment of Keefer et al. suggests that highly
nonequilibrium processes may be occurring in the cooler
plasma regions. They found that the dominant radiation
in the sheath surrounding the hot plasma core came from
S t 4*
the 1 negative system of N2 Furthermore, radiation
from the 2n positive system of N2, usually present in
high-temperature air, was conspicuously absent. The
anomalous N2+ radiation indicates that nonequilibrium
processes, such as photoionization, may be occurring.
This result suggests another explanation for the
high propagation velocities. The thermal radiation
escaping the plasma core, rather than raising the temper
ature of the cold gas as a whole as assumed in Reference 12,


13
distance from the spark front, the net thermal emission
is neglected. In the region x > 0, it is included as a
simple volumetric loss term given by the linear relation
ship
(p = rr\
The constant m is chosen to give values of

agree with experimentally determined radiation losses
for air at temperatures near 15,000 K.
.12
The analysis of Jackson and Nielsen for large
radius sparks suggests that re-absorption of thermal
radiation, rather than thermal conduction, is the dominant
heating mechanism for the cold gas ahead of the spark.
The effect of neglecting this heating, by assuming for x < 0, and the role of radiation transport in spark
propagation will be discussed in Chapter VI.
As a result of the approximations discussed in the
preceding paragraphs, which is the solution for the
thermal potential in the region x < 0 must satisfy
1 r(r ) = 0
rjcV jr' 3 x
3 e, 2 e,
(2.5)
and 6i which is the thermal potential in the region
x > 0 must satisfy


23
-CO £ x £ oo and 0 £ r £ R A proof of the
c
uniform convergence of these series is given in Appendix I.
The length of the spark d can be defined as the
co-ordinate x where the temperature averaged over the
laser radius falls below the ignition temperature T^.
Although the model does not terminate absorption of the
laser radiation at x = d, the absorption of a real spark
decreases rapidly for x > d due to the rapid decrease in
the degree of ionization. Therefore, the spark length is
useful in approximating the power absorbed by the spark.
Consistent with the exponential approximation for the
absorption of laser radiation,
e
(3.12)
where PA is the power absorbed by the spark and PQ
is the incident power.


bremsstrahlung absorption of laser radiation and electron
thermal conduction. The electrons quickly relax to a
Maxwe11-Boltzmann distribution at a temperature T0 as a
result of the high efficiency of electron-electron
collisions in transferring energy. Since the transfer of
energy to the heavy particles by electron collisions is
inefficient, a two-temperature plasma, with electron
temperature exceeding the heavy-particle temperature,
develops. Further ionization is then caused by collisions
of heavy particles with high-energy electrons. Consequently,
an appreciable electron density can exist in a layer
ahead of the spark where the temperature of the heavy
particles is still quite low.
The inverse bremsstrahlung absorption coefficient,
given by Equation (2.4) in Chapter II, is proportional
to the square of the electron number density. Therefore,
the development of a nonequilibrium layer with a high
electron number density ahead of the spark means that
absorption of the laser radiation and consequent heating
occurs in regions where the heavy-particle temperatures
are less than 12,000 K.
Although this is a nonequilibrium process, the
equilibrium solution obtained in Chapter III can be used
to illustrate the effects of the anomalous absorption on
spark propagation. Absorption of laser radiation in regions
with temperatures less than the equilibrium ignition


75
12. Jackson, J. P. and Nielsen, P. E., "Role of Radiative
Transport in the Propagation of Laser Supported
Combustion Waves," AIAA Paper No. 74-228, 1974.
13. Zeldovich, Y. B. and Raizer, Y. P., Physics of Shock
Waves and High Temperature Hydrodynamic Phenomena Yol, 1.
edited by W. D. Hayes and R. P. Probstein, New York:
Academic Press, 1966.
14. Wei, P. S. P. and Hall, R. B., "Emission Spectra of
Laser Supported Detonation Waves," J Appl. Phys.. 44.
2311 (1973).
15. Luke, Y. L., "Integrals of Bessel Functions," Handbook
of Mathematical Functions, edited by M. Abramowitz and
I. A. Stegun, Washington, D.C.: U.S. Government
Printing Office, 1966.
16. Hermann, W. and Schade, E., "Transportfunktionen von
Stickstoff bis 26000 K," Z. Physik. 233. 333 (1970).
17. Hermann, W. and Schade, E., "Radiative Energy Balance
in Cylindrical Nitrogen Arcs," J. Quant. Spectrosc.
Radiat, Transfer. 12 1257 (197271
18. Raizer, Y. P., "Propagation of Discharges and Maintenance
of a Dense Plasma by Electromagnetic Fields," Sov. Phys.-
Usp.. 1, 688 (1973).
19. Fowler, M. C., Smith, D. C., Brown, C. 0. and Radley, R. J.,
"Laser Supported Absorption Waves," United Aircraft
Research Laboratories Report No. N921716-9, 1974.
20. Keefer, D. R., Henriksen, B. B. and Braerman, W. F.,
"Experimental Study of a Stationary Laser Produced
Air Plasma," to be published.
21. Olver, F. W. J., "Bessel Functions of Integer Order,"
Handbook of Mathematical Functions, edited by
M. Abramowitz and I. A. Stegun, Washington, D.C.: U.S.
Government Printing Office, 1966.
22. Widder, D. V., Advanced Calculus (2nd Edition).
Englewood Cliffs, N.J.: Prentice Hall, Inc., 1961.
23. Penner, S. S. and Olfe, D. B., Radiation and Reentry.
New York: Academic Press, 1968.
24. Huffman, R. E., "Photochemical Processes: Cross-Section
Data," Reaction Rate Handbook (2nd Edition), edited by
M. H. Bortner and T. Baurer, Santa Barbara, Calif.:
DASIAC, DoD Nuclear Information and Analysis Center,
1972.


resorting to analytical approximations for the thermodynamic
properties of air, also used a one-dimensional model to
obtain the radially averaged spark temperatures and
propagation velocity. In both References 10 and 11, thermal
conduction was assumed to be the propagation mechanism. A
12
recent one-dimensional study by Jackson and Nielsen has
focused on the role of radiation in the subsonic propagation
of laser sparks.
The purpose of this investigation is to provide a
two-dimensional model for the subsonic propagation of laser
sparks by explicitly including the radial heat conduction
losses. This allows the calculation of both radial and
axial temperature profiles as opposed to temperatures
averaged over the radius. Furthermore, the solution is
obtained for a spark propagating in a channel of arbitrary
radius so that the effect of boundaries on spark propagation
can be studied. The model is applied to air sparks at
atmospheric pressure under the influence of C02 laser
radiation.
The propagation mechanism is assumed to be thermal
conduction with radiation serving only as an energy loss
mechanism. Although the flow velocity is restricted to
one dimension, the two-dimensional nature of the temperature
field is retained in the analysis. By using simple models
for the temperature-dependent properties of the gas, a
closed form solution is obtained which yields axial and


26
where
y'=/l+ ^ (mRl + A)
oc C ^ + X' )
cUaf'-O Z*
(4.6)
(4.7)
(4.8)
The relationship between the propagation velocity and
the incident laser intensity is given by
* (y + ar7)
(X C if '+ I) ** 2 ^)u
(4.9)
If the losses due to radial conduction are much
2
greater than the radiation losses, i.e. A >> mRL and
if the absorption coefficient satisfies the relationship
2 << oC( + l), then Equation (4.9) reduces to
5 =
A
c*L Y C #+ 0
(4.10)
Raizer,lw neglecting thermal radiation losses and the
attenuation of the incident laser beam, obtained


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
and Astronomy
This dissertation was submitted to the Dean of the College
of Engineering and to the Graduate Council, and was accepted
as a partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
August, 197^
^^Dean, College of Engineering
Dean, Graduate School


ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to the
members of my supervisory committee. In particular, my
deepest appreciation is extended to the committee chairman
Dr. Dennis R. Keefer, whose advice and guidance were
essential to the completion of this endeavor.
I am indebted to Dr. Bruce Henriksen for suggesting
the problem and for his continued interest in this research.
My gratitude for the criticism, encouragement and
technical assistance of Mr. John Allen is exceeded only
by my regard for his personal friendship. Finally, I am
especially grateful to my wife for her diligent preparation
of this manuscript.
iii


33
distances on the order of .10 y As a result points
separated by this distance absorb almost the same energy
from the laser beam.
Figures 5-8 show that the isotherms for propagating
sparks are blunter at the spark front when compared to
the isotherms for stationary sparks. This is a result of
convective cooling. In the regions near x = 0, the
convective cooling decreases with radius since ^ *
decreases with r. This implies that the hotter regions
near the axis are cooled more strongly than the outer,
cooler regions. Thus, convection tends to decrease the
radial variation of temperature near the spark front.
Representative values of the physical properties of
laser sparks, as calculated from Equations (3.6) (3.12),
are given in Table I. The symbol Qm denotes the maximum
value of the radially averaged potentials, and Tm is the
corresponding temperature. For the cases considered in
this analysis the spark is on the order of a centimeter
in length with average temperatures near 16,000 K.
Approximately half the incident laser intensity is
absorbed by the spark. The higher intensity required to
maintain a propagating spark, as opposed to a stationary
spark, results in an increase in both the temperature
and the spark length.
The variation of the thermal conduction parameter A,
calculated for threshold conditions, with channel radius


In spite of the rapidly expanding theoretical and
experimental efforts, understanding of subsonic spark
46
phenomena is' far from complete. In particular, the
observed propagation velocities have eluded a satisfactory
explanation. Further analysis in two areas would provide
greater insight into the physical processes occurring
in subsonic sparks. A detailed study of the radiation
transport, including nonequilibrium effects, would provide
a more complete knowledge of the propagation mechanism.
And a study of the two-dimensional flow field around the
spark would lead to a more accurate determination of the
role convection plays in the energy balance.


34
is shown in Figure 9. The results indicate that A = 2.9
for a spark of .15 cm radius in a channel of .30 cm radius.
As discussed in Chapter IV, Raizer1^ obtained this
same value by an approximate calculation. The agreement
is only coincidental since his calculation indicates A is
independent of R^. In fact, the authors of Reference 11
use A = 2.9 for sparks with R^ varying from .028 cm to
1.67 cm. Figure 9 shows, however, that A is a function
of laser radius. For R /Rt = 2, the value of A for a
spark of .50 cm radius is nearly twice the value for a spark
of .15 cm radius.
Calculations of threshold laser intensities by
substituting A from Figure 9 into Equation (4.9) gave
excellent agreement with the results of the two-dimensional
analysis. Although A was found to vary with propagation
velocity, an error of no more than % was incurred in
using the value of A at threshold to calculate SQ for
propagation velocities up to 20 cm/sec,
The one-dimensional solution can be used to study
the relative importance of each loss process in subsonic
spark propagation. Dividing each loss term in Equation (4.2)
by the volumetric energy absorption SQ exp(- fy,x)
results in nondimensional parameters for the losses due
to axial heat conduction, convection, thermal radiation
and radial heat conduction.


The model is applied to air sparks of .15 cm and
.50 cm radius propagating at velocities up to 20 cm/sec
under the influence of C02 laser radiation. The results
indicate a spark length on the order of 1 cm and tempera
tures near 16,000 K. Approximately half the incident
laser intensity is absorbed by the spark. Calculation
of the radial heat conduction parameter verifies the
value of 2.9 used by Raizer for an unbounded spark of
.15 cm radius. Furthermore, A is found to depend on the
Y
spark radius.
The theory agrees with the experimental results for
stationary sparks. However, it fails at predicting the
spark properties for propagation velocities of several
meters per second. It is postulated that, at these
velocities, re-absorption of thermal radiation creates a
nonequilibrium layer ahead of the spark front which en
hances the absorption of laser radiation and increases
the propagation velocity.
xiii


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TWO-DIMENSIONAL MODEL FOR THE SUBSONIC
PROPAGATION OF LASER SPARKS
By
Jad Hanna Batteh
August, 197^
Chairman: Dennis R. Keefer
Major Department: Engineering Sciences
The properties of laser sparks are investigated by
solving a simplified, two-dimensional energy equation
describing the steady state, subsonic propagation of a spark
in a channel. The propagation mechanism is assumed to be
thermal conduction and thermal radiation is included as an
effective optically thin emission. The solution, obtained
in closed form, yields both axial and radial temperature
profiles, as well as the relationship between the laser
beam characteristics, the channel radius and the propagation
velocity.
A solution of the radially integrated energy equation
is obtained by introducing the radial heat conduction param
eter A suggested by Raizer. The parameter is evaluated by
means of the solution to the two-dimensional energy equation.
xii


22
O xoaT42(Xoj+W<1
(3.8)
(A ( Xa + ^<0
ot()C 0 2^
(3.9)
c(. ( \ + ^<0 + 2
ca= A*
*(< -o ***
(3.10)
The relationship between the propagation velocity
and the laser intensity is obtained by substituting ,
from Equation (3.6) into Equation (2.10). The resulting
equation is
e = ifa n, 50 2
a
(3.1D
Since the are functions of & the propagation
velocity is contained in the summation. Therefore, the
most convenient way to evaluate Equation (3.11) is to
choose a value of u and solve for SQ. Equations (3.6)
and (3.7) can then be evaluated for ( and 02 .
Equations (3.6) (3.11) are valid only if each
infinite series converges uniformly in the interval


CHAPTER II
THEORETICAL DEVELOPMENT
In this analysis the steady state, subsonic
propagation of a laser spark in a gas is considered.
The ignition process is not considered since its only
purpose is to provide free electrons to initiate inverse
bremsstrahlung absorption of the laser beam. Therefore,
the characteristics of the spark are functions only of
the laser-plasma interaction and not of the ignition
process.
The model consists of a uniform, monochromatic laser
beam, of intensity SQ and constant radius R^, passing through
a channel of radius Rq in which there is an absorbing plasma.
Energy is absorbed from the laser beam by the plasma and
is transferred to the cold gas through heat conduction.
The channel wall is held at a fixed temperature T = 0 and
is assumed to absorb all radiation incident upon it
without re-emission. The spark moves in the channel
without distortion with a constant velocity u relative to
the cold gas, as shown in Figure 1. Since the flow
velocities are subsonic the pressure gradients are small
and are neglected in the analysis. Furthermore, the flow
kinetic energy is neglected compared to the thermal energy,
7


appear to have singularities at values of velocity such that
However, when this condition is satisfied, the exponential
factors for the two series are equal. The two series can
then be written as one series with coefficients
D = B + C When this is done, the singularity disappears,
n n n
By the Weierstrass M-Test, the series in Equation (3.7)
are uniformly convergent if the series Z |Bn| and
^ |Cn | converSe* The convergence is easily shown by
comparison. Since
lim
n -* oo
I Bn I
I / I
lAn I
lim
n co
N
I 7 I
M
and Z ¡An | converges, then Z J Bn | Z |cn J
converge.
o
The infinite series Z xon Tn Equation (4.11)
can be shown to converge by comparison with the series
Z_ n-^2. By applying Equations (A1.4) (A1.6) it can
be shown that
l i m
n 0O
pj *
if
T*
Rc
cos (,X0/V Rt
- 4 )
- 3/z
8 lr
CO
'/z
= 0
Z xon2 Tn converges
Therefore, the series


CHAPTER III
SOLUTION OF THE EQUATIONS
Equations (2.5) and (2.6) for the thermal potentials
can be solved in closed form by the separation of variables
technique.
By assuming
,(*.--) = lO*'1 Vco
n
the partial differential equation for 0, reduces to a
set of ordinary differential equations
2
where the are separation constants.
The equation for gn is Bessel's equation, which has
the solution
To (|30 + bn Y0 ( r)
16


21
where the cn are constants to be determined by the Joining
conditions.
Thus, the solution for 02 is given by
-
ez(x.O=e Z To (X0 -T )
n. c
(3.5)
V -v / r n f C l ~ ) *
n C
Imposing the Joining conditions given in Equations
(2.8) and (2.9) results in a set of algebraic equations
for the an and cn. Solution of these equations yields
the following for , and :
e,= 8 K 50-^ Z AT0(x0-^ ) e
Rc n c
Tl+Wx
(3.6)
2 5
Rc
Rc
- tfx
e v Zbt0(xm^)
(3.7)
-Zc,
To(X-it)e-ll"<)X
where the coefficients are given by


To my wife, Jane,
and
to my parents


32
intensity for RL = .50 cm is nearly independent of channel
radius.
Figures 5 and 6 show the isotherms, calculated from
Equations (3.6) (3.11), for a spark of .15 cm radius
in a channel with radius R = 1.25 RT. Figure 5 depicts
the threshold isotherms, and Figure 6 shows the isotherms
for u = 20 cm/sec. The intense core and elongated struc
ture agrees with the qualitative description of laser
O
sparks given by Smith and Fowler.
Figures 7 and 8 show the isotherms for a spark of
.50 cm radius in a channel with radius Rc = 1.25 R^. The
isotherms of Figure 7 correspond to the threshold case
while those of Figure 8 correspond to a propagation
velocity of 20 cm/sec. The blunter isotherms, as compared
to the isotherms of Figures 5 and 6, are due to the
decreasing importance of radial heat conduction.
The plots of the isotherms show a rapid increase in
temperature at the spark front followed by a slow decrease.
The increase is due to the sudden deposition of energy, as
a result of the rapid increase in the absorption coef
ficient at T = 12,000 K. The slow decline in temperature
for the regions beyond the temperature maximum is a
consequence of the small absorption coefficient for C02
laser radiation. The absorption length for C02 laser
radiation in air is ty = ^iy = 1.4 cm. Therefore,
there is little attenuation of the beam when it traverses


TWO-DIMENSIONAL MODEL FOB THE SUBSONIC
PROPAGATION OP LASER SPARKS
By
JAD HANNA BATTEH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OP
THE UNIVERSITY OP FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974


CHAPTER IV
SIMPLIFIED ONE-DIMENSIONAL ANALYSIS
Although the solution in Equations (3.6) (3.10)
is straightforward, it is cumbersome due to the summation.
In this chapter the equations are reduced to one dimension
by introducing a radial heat conduction parameter. The
parameter is in the form of an infinite series, but the
spark properties are described by simple equations.
If Equations (2.5) and (2.6) are averaged over the
laser radius, the resulting equations are
J 0,
de,
A 0,
(X
X < o
dx2
x
d'e.
A 2
{ A \ -
- j m H ) 0
d x2
X
v r y 2
+ So
X e-^x = 0
X >0
)
where
Q(x)= i r 0UjO Ac
R* J
o
(4.1)
(4.2)
24


53
1
-.5 O .5 1.5 2.0
X cm
Figure 7. Isotherms for a Spark with = .50 cm,
Rc = 1.25 Rl and u = 0. (1) T = 0,
(2) T = 8,000 K, (3) T = 10,000 K,
(4) T = 13,000 K.


17
J and are the zero order Bessel functions of the first
o o
and second kind, respectively, and the an and bn are
constants. Since YQ(0) = 0 t>n must be zero for the
solution to remain finite on the axis. The radial boundary
condition in Equation (2.7) determines the separation
constant. For , to vanish at r = R, must equal
where xrtv1 is the n1' zero of J .
on c on o
The differential equation for fn has the solution
<5,n S2n x
Cn e + Jo e
where
and the cn and dn are constants. To satisfy the boundary
condition ,(-oo,r) = 0, dn must be zero.
The solution for O, can, therefore, be written as
e
(3.1)
The constants an are determined by the joining conditions.


BIOGRAPHICAL SKETCH
Jad Hanna Batteh was born in Ramallah, Jordan, on
January 5, 1947. After graduating from Ribault Senior
High School of Jacksonville, Florida, in June, 1964, he
enrolled at the University of Florida. As an undergraduate,
he was elected to membership in Phi Eta Sigma, Sigma Tau
and Tau Beta Pi honorary fraternities. He received the
degree of Bachelor of Science in Aerospace Engineering with
High Honors in December, 1967.
Beginning July, 1968, he was employed for two and a
half years as an aircraft structures engineer at the
Lockheed-Georgia Company in Marietta, Georgia. During that
time, he attended the Georgia Institute of Technology from
which he received the degree of Master of Science in
Aerospace Engineering in December, 1970.
Since the fall of 1971, he has been enrolled in the
Graduate School of the University of Florida. During his
tenure, he has held an NDEA Fellowship and a Graduate
School Fellowship.
He is married to the former Jane Joseph Bateh of
Jacksonville, Florida.
77


e,
s*
em
*,
A
*r
j
?
?o
*
, z
radially averaged thermal potential in the
region x < 0, kW/cm
radially averaged thermal potential in the
region x > 0, kW/cm
maximum of the radially averaged thermal
potentials, kW/cm
absorption coefficient for radiation of frequency
v> cm1
thermal conductivity, kW/cm K
radiative conduction coefficient, kW/cm K
radiation frequency, sec1
density, gm/cmJ
density of the cold gas ahead of the spark, gm/ca?
rate of energy loss by radiation, kW/cnP
2 2 2
separation constant, equal to xQn /Rc cm
xi


65
increase in the thermal conduction coefficient. In the
optically thin limit, where iv L, the plasma acts as
a volume radiator, and the rate of energy loss per unit
23
volume can be written as a function of temperature. v Both
limiting cases are valid only where local thermodynamic
equilibrium exists.
The model described in Chapter II assumes that radia
tion can be treated as an effective, optically thin emission.
This assumption is adequate for small radius sparks propaga
ting at low velocities. In that case, energy transport by
thermal radiation is no greater than energy transport by
thermal conduction.
For laser sparks in which radiation dominates the
energy transport, the radiative transfer, unfortunately, is
neither optically thick nor optically thin. The radiation
escaping the hot plasma core lies in the vacuum ultra-
12
violet (vuv). In the core region, where there is a high
degree of ionization, the plasma is optically thin to vuv
radiation. However, in the cooler regions this radiation is
strongly absorbed by the neutral species.
12
Jackson and Nielsen found that most of the emission
from the plasma core lies in the energy range from 14.5 to
20 ev, due mainly to electron recombination with N+ and
0+ ions. Photons within that range possess energy exceeding
the ionization potentials of the neutral species of air,
E^ being 15.6 ev, 12.1 ev, 14.5 ev and 13.6 ev for N2, 02,
24
N and 0, respectively. The photoionization cross sections


LIST OP FIGURES
Figure
Page
1.
Co-ordinate System for the Steady State
Propagation of a Laser Spark in a Channel .
47
2.
Variation of c / X with Temperature
for Atmospheric-pressure Air
48
3.
Variation of Thermal Conductivity and
Thermal Potential with Temperature for
Atmospheric-pressure Air
49
4.
Incident Laser Intensity Required to
Maintain a Spark as a Function of
Channel Radius
50
5.
Isotherms for a Spark with R^ = .15 cm,
R_ = 1.25 Rt and u = 0
C Lt
51
6.
Isotherms for a Spark with R^ = .15 cm,
R^ = 1.25 Rt and u =20 cm/sec
52
7.
Isotherms for a Spark with RL = .50 cm,
R = 1.25 R^ and u=0
53
8.
Isotherms for a Spark with R^ = .50 cm,
RQ = 1.25 and u = 20 cm/sec
54
9.
Variation of the Thermal Conduction
Parameter with RC/RL
55
10.
Axial Variation of the Volumetric Energy
Loss Terms for RL = .15 cm, Rc = 1.25 RL
and u=0
56
vi


CHAPTER VI
DISCUSSION OF RESULTS
Previous theoretical studies of laser spark phenomena
have not probed the effect of a channel boundary on
subsonic spark propagation. In fact, aside from models
based on Raizer's one-dimensional approximation, only the
18
spherically symmetric case has been considered. Con
sequently, the conclusions drawn from this analysis cannot
be compared with previous theoretical results.
The technology of cw C02 lasers has only recently
attained the power levels required to maintain subsonic
laser sparks in air. As a result, experimental investi
gations are scarce, and they have not focused on spark
propagation in channels. However, the predictions for
R /Rt = 2 can be compared with the results of experiments
on unbounded sparks since the channel boundary is far
enough from the spark that its influence is negligible.
A spark with Rc/Rl 2 will be denoted a "free" spark.
19
Fowler et al., using a focused C02 laser beam in
atmospheric-pressure air, studied threshold intensities
and propagation velocities. They also determined the
spatial variation of temperature and electron number
density by a laser interferometric technique. Their
37


59
Figure 13. Axial Variation of the Volumetric Energy
Loss Terms for RL = .50 cm, Rc = 1.25 RL
and u = 20 cm/sec. (1) Thermal radiation,
(2) Radial heat conduction, (3) Axial heat
conduction, (4) Convection.


9
a problem of considerable difficulty unless simplifying
assumptions are made. Since the temperature field is of
primary interest in this analysis, its two-dimensional
nature is retained. On the other hand, the velocity
field is simplified by assuming vr = 0 and taking the
axial mass flow rate independent of radius. This one
dimensional approximation to the flow field is often used
in spark propagation analysis.1012 The equations are
then uncoupled. In fact, Equation (2.2) reduces to
p vx = const. = fi0 u where j>Q is the density far ahead
12
of the spark. Jackson and Nielsen, on the basis of
preliminary results, predict little difference between
the results obtained using the two-dimensional flow
field and the one-dimensional flow field in radiation-
dominated propagation. For the model considered here,
the error introduced by neglecting the radial deflection
of the flow should diminish as the channel radius decreases.
Since v must vanish at r = 0 and r = E the channel
r c *
inhibits the radial velocity.
As a result of the one-dimensional flow approximation,
the energy equation reduces to
?o cP u
a
O


1
o
I 1 1 I L
.3 .6 .9
x, cm
Figure 6. Isotherms for a Spark with RL = .15 cm, Rc = 1.25 RL and u = 20 cm/sec.
(1) T = 0, (2) T = 12,000 K, (3) T 16,000 K, (4) T = 20,000 K.
V_n


ionization of singly charged ions. Consequently, the
absorption coefficient increases for temperatures greater
than 24,000 K. The value = .74 cm1 was used in
the calculations for x > 0. It corresponds to an average
value for the temperature range 15,000 K to 20,000 K,
which are typical values for the temperature in the
spark core.
The ratio c / A was set equal to 425 cm sec/gm.
From Figure 1 this corresponds to a mean value in the
temperature range 10,000 K to 20,000 K. The upstream
density was taken as PQ 1.3 x 10-^ gm/cnP.
The thermal emission parameter m was assumed to be
175 cm2. The resulting losses are 44 kW/cnP and
67 kW/cnP for T = 15,000 K and 20,000 K, respectively.
Raizer's calculations10 indicate losses which vary from
48 kW/cm-^ to 60 kW/cm^ over the same interval, with the
maximum occurring at 18,000 K. However, he uses only
half of this emission in his calculations since he
assumes half of the radiation is re-absorbed in the cold
gas ahead of the spark and not really lost. The experi-
1 6 17
mental and theoretical work of Hermann and Schade on
wall-stabilized nitrogen arcs at atmospheric pressure
indicate radiation losses of 50 kW/cm-^ at 20,000 K.
_2
Therefore, the linear approximation with m = 175 cm
adequately represents the losses in the hot zone of the
air spark.


45
For the .15 cm spark, radial heat conduction and
thermal radiation losses are approximately equal. However,
for the .50 cm spark, radiation is the dominant loss
mechanism. In both cases, axial convection and axial heat
conduction are significant only near the spark front.
The predicted threshold intensity and isotherm
variations for the.15 cm spark agree with the available
experimental data. However, the model fails to predict
the high propagation velocities experimentally observed.
In order to maintain such high velocities, thermal radi
ation must play a significant role in the heating of the
cold gas ahead of the spark.
An equation for the thermal conduction parameter A,
introduced by Raizer, has been obtained from the two-
dimensional solution. The evaluation of A for a free spark
of .15 cm radius confirms the value of 2.9 used by Raizer.
Furthermore, A was found to depend on channel and laser
%
radius.
In summary, the two-dimensional model presented in
this analysis can be used as a reasonable approximation
for the properties of laser sparks at threshold and at
low propagation velocities. In those cases, the assumption
that thermal radiation serves only as a loss mechanism
is adequate. For high propagation velocities, a more
careful treatment of the radiation transfer is required.


CHAPTER V
EXAMPLE CALCULATIONS
The equations derived in Chapters III and IV have
been used to study the properties of sparks maintained
in atmospheric-pressure air by C02 laser radiation.
The variation of the thermal potential with temper
ature, obtained from the data of Reference 11, is shown
in Figure 3. A value of .15 kW/cm was used for the value
of the thermal potential at ignition. This value corres
ponds to the ignition temperature T^ = 12,000 K suggested
by Raizer.10
The absorption coefficient for C02 laser radiation
in air increases rapidly for temperatures greater than
12,000 K, due to the rapidly increasing degree of
ionization.10 It reaches a maximum value of approximately
.85 cm1 at T = 17,000 K and decreases to a minimum of
.38 cm1 at T = 24,000 K. This behavior can be understood
by considering the formula for ^ given in Equation (2.4)
Beyond 17,000 K, first ionization is essentially complete.
The electron density approaches a plateau while the
temperature Tg continues to increase. Therefore,
decreases. For temperatures greater than 24,000 K,
the electron density again increases rapidly due to the
29


12
of for atmospheric-pressure air show a rapid increase
near T = 12,000 K.
The absorption coefficient of air for Nd-glass
laser radiation, which has a wavelength of 1.06 microns,
consists of contributions from both free-free and bound-
free electron transitions. It also exhibits a rapid
increase at temperatures near 12,000 0K.x(^
Consequently, the solution domain of the energy
equation is divided into two regions separated by the
plane x = 0. For x<0 the temperature is sufficiently
low that there is negligible absorption of laser radiation,
i.e. Xy = 0. In the region x > 0, the absorption
coefficient is assumed to be a constant so that
S, = So HlR.-r) n e'*"x
where is the incident laser intensity and H(RT r)
is the Heaviside step function defined to be unity for
0 r Rl and zero for r > R^. The term "spark front"
will be used to denote the plane x = 0.
A complete treatment of the emission and re-absorption
of thermal radiation would require the solution of a
complicated integro-differential equation. It is
included in this analysis through a simple phenomenological
approximation. In the region x < 0, where the temperature
is comparatively low and decreases rapidly with increasing


50
Figure 4. Incident Laser Intensity Required to Maintain
a Spark as a Function of Channel Radius.
Solid curves are for RL = .15 cm and broken
curves are for R^ = .50 cm. Curves 1, 2 and 3
correspond to velocities of 0, 10 and
20 cm/sec, respectively.


APPENDIX I
CONVERGENCE OF THE SERIES SOLUTION
The two-dimensional energy equation describing
subsonic spark propagation was solved in Chapter III
in terms of infinite series. For the solution to be
valid, the series must be shown to converge uniformly
for every point (x,r) in the solution domain 0O £ x co ,
0 £ r £ R .
The following properties of Bessel functions and
their zeroes will be used in proving the convergence of
21
the series:
1 J0(Z) 1 £ l
(Al.l)
I J^z) | 1
(A1.2)
Jl(xom> 0
(Al.3)
lim xon = (n i) 7f
n -* ao
(A1.4)
lim J^iz) .798 z^ cos(z 3 if A)
Z oo
(A1.5)
lim xon Jl2(xon} = *637
ft -+ OP
(A1.6)
The uniform convergence of a series of function can
22
be proved by the Weierstrass M-Test. It states that the
series f (x,r) converges uniformly in an interval of
f\ I
the variables x and r if |fn(x,r) | £ Mn in that interval
and 2 M converges.
61


40
and that these temperatures do exceed 17,000 K. These
results support the temperature variation with velocity
indicated in Table I and in Figures 5-8.
Hall et al.11 compared the results for a two-step
variation of with a one-step model and concluded
that the one-step model was a good approximation. Their
solutions also indicate an increase in temperature
associated with an increase in propagation velocity.
Therefore, the step-function formulation of used
in this analysis should not introduce appreciable error
into the calculations of SQ, the required laser intensity,
or Tffl, the maximum of the radially averaged temperatures.
However, an analysis including the actual temperature
variation of would cause some suppression of temperatures
greater than 17,000 K. For example, there would be a
slight decrease in the area within the 20,000 K isotherm
shown in Figure 6.
For values of x > d the temperature falls below
12,000 K and decreases rapidly. Since the model
retains the value of .74 cm"1, the temperatures predicted
by the model in that region exceed those which actually
occur. The error introduced by this approximation is not
severe sl^nce the spark attenuates the laser intensity to
approximately half its initial value.
The analysis treats the transport of thermal radiation
in an approximate manner. The assumption that the radiation


NOMENCLATURE
coefficient of the series solution
radial heat conduction parameter
coefficient of the series solution for 0, defined
by Equation (3.8)
coefficient of the series solution
coefficient of the series solution for 9Z defined
by Equation (3.9)
coefficient of the series solution
specific heat capacity, joule/gm K
coefficient of the series solution for Oz defined
by Equation (3.10)
spark length, cm
electron
ionization potential, ev
axial function for the solution of 0 ,
axial function for the solution of @K
Gaunt factor
radial function for the solution of 9,
radial function for the solution of 0K
Planck's constant, ev-sec
Heaviside step function
zero order Bessel function of the first kind
first order Bessel function of the first kind
viii


11
In that case the free electrons in the presence of
scattering centers are accelerated by the electric field
of the laser radiation. The electron and heavy-particle
temperatures are then equilibrated by elastic collisions.
Although both neutral particles and ions can serve
as scattering centers, absorption in the field of neutral
13
particles is important only in a very weakly ionized gas.
At typical spark temperatures the ionization is appreciable
so that only absorption due to electron-ion collisions is
considered.
The inverse bremsstrahlung absorption coefficient
for a high-temperature, ionized gas is given by the
13
modified Kramers formula ^
8
3. 4x 10
Zz9
cm1 (2.4)
where N+ is the number density of positive ions with
charge Z, Ng is the number density of electrons, Tg is
the electron temperature, \) is the radiation frequency
and g is the Gaunt factor. In the single ionization
temperature range, the absorption coefficient is approx-
imately proportional to N. /T s. Therefore, under equilibrium
conditions, will exhibit a rapid increase at temper
atures where the ionization becomes appreciable. For
atmospheric-pressure air this temperature lies in the range
10,000 K to 15,000 K.^ In fact, equilibrium calculations-^-0


54
1
R
L
i
0 5 1.0 1.5 2.0
X, cm
Figure 8. Isotherms for a Spark with RT = .50 cm
J-j *
Rc = 1*2^ L and u = 20 cm/sec. (1) T = 0,
(2) T = 10,000 K, (3) T = 14,000 K,
(4) T = 17,000 K.


where the thermal flux potential
e = j a (t)
o
has been used to replace T as the dependent variable.
Equation (2.3) can be linearized if suitable
approximations are made for the temperature dependence
of ?*, and 0^. The first approximation is to
assume c^/ A equal to a constant which reduces to
a constant. Figure 2 shows the variation of this ratio
for atmospheric-pressure air plotted from the data of
Reference 11. The definite temperature dependance,
apparent in the figure, indicates that the assumption is
a shortcoming in the theory. However, the assumption is
a reasonable first approximation in the temperature range
10,000 25,000 K which is of greatest interest for
subsonic laser sparks. In fact, it will be shown in
Chapter VI that this approximation is valid for the
velocities considered in this analysis. Although the
authors of Reference 11 used analytical approximations
for the temperature variations of enthalpy and thermal
conductivity, their approximations are equivalent to
assuming a constant c / A
Un-ionized air is transparent to C02 laser radiation
which has a wavelength of 10.6 microns. However,
absorption can occur by inverse bremsstrahlung (free-free
transitions of the electrons) when air becomes ionized.


60
Table I
Physical Properties of Laser Sparks
of .15 cm and .50 cm Radius
Bc
cm
u
cm/sec
kW/cm
Tm
103 K
d
cm
Vpo
BL -
.15 cm
.15
0.
.24
15.
.83
.46
.15
10.
.28
16.
1.02
.53
.15
20.
.32
17.
1.24
.60
.30
0.
.21
14.
.66
.39
.30
10.
.25
15.
.94
.50
.30
20.
.31
17.
1.26
.61
RL
.50 cm
.50
0.
.19
13.5
.50
.31
.50
10.
.23
14.5
.84
.46
.50
20.
.30
16.5
1.26
.61
1.00
0.
.17
13.
.34
.22
1.00
10.
.22
14.
.74
.42
1.00
20.
.29
16.
1.21
.59


72
Since the 6,000 K isotherm crosses near x = 0, the width
of the zone in which the temperature falls from 12,000 K
to 6,000 K is .10 cm. This distance is less them the
absorption length of .25 cm for the vuv radiation emitted
from the core. Therefore, the model, though simple, is
at least consistent in that the radiation can penetrate to
regions where T < 6,000 K.
The intention of the discussion presented above was
to describe, in a qualitative way, the effect re-absorption
of thermal radiation would have on spark propagation. A
future investigation keying on the processes occurring in
the spark front would be useful in understanding the propa
gation mechanism. In particular, a one-dimensional model
of a nonequilibrium gas under the simultaneous influence
of two radiation fields would be a logical extension to
this analysis.


39
of small radius sparks. It is worthwhile to consider the
simplifying assumptions and the limitations they impose
on the use of the model.
The temperature dependence of cp/ A was neglected
in the analysis. This approximation has no effect on
threshold calculations (u = 0) since the ratio appears
only in the convective loss term. For the propagation
velocities considered in this analysis, convection is
significant only near the spark front, as discussed in
Chapter V. There the temperatures are not much different
from the ignition temperature Ti# Therefore, c / A an
be considered constant at a value typical of temperatures
near 12,000 K.
The absorption coefficient ^ was assumed to be
zero for x < 0 and a constant equal to .7^ cm-1 for
x > 0. The two regions are separated by the plane x = 0
where the temperature averaged over the laser radius
equals the ignition temperature T^. Actually the absorp
tion coefficient has a maximum value of approximately
.85 cm"1 at T = 17,000 K and decreases to .60 cm"1 at
20,000 K. Therefore, temperatures in excess of 17,000 K
are more difficult to achieve than the model supposes.
Raizer10 has calculated maximum temperatures for the
one-dimensional case taking into account the variation
of with temperature. His analysis shows that the
maximum temperature increases with propagation velocity


19
Since the J0(x0nr//Rc^ form a comPlete> orthogonal
set in the range 0 r H they can be used to form
c
a unique series expansion of any function of r in that
interval. In particular, the function SQ H(B^ r)
can be expanded in terms of an infinite series. Therefore,
for Equation (3.^) to be valid, the tn must be the
coefficients of such a series expansion. The coefficients
can be found by using the orthogonality property1^
0
where cinm. is the Kronecker delta and J-^ is the first
order Bessel function of the first kind. Multiplying
Equation (3.4) by rJ0^xomr/^Rc^ integrating from 0 to Bc
and applying the orthogonality condition gives
0
The value of the integral, obtained with the aid of
Equation (11.3.20) of Beference 15, is R RT Jn (*,BT/B )/x.
c l i oxi c on
The resulting equation for the constants bn of the
particular solution is


APPENDIX II
EFFECT OF RADIATION ON SUBSONIC SPARKS
It was determined in Chapter VI that spark propagation
at velocities of several meters per second by thermal con
duction results in temperatures which exceed the experimentally
observed temperatures. Therefore, it has been postulated
that re-absorption of thermal radiation plays a dominant
role in the propagation of laser sparks at high velocities.
12
Jackson and Nielsen obtained propagation velocities
of several meters per second by including radiative trans
fer in a one-dimensional model. Their solution was based
on calculations of the radiative emission and absorption
20
for equilibrium air. The recent results of Keefer et al.
suggest, however, that highly nonequilibrium processes are
occurring in the cooler plasma regions.
Except for two limiting cases, the solution of the
energy equation including radiative transfer poses a problem
of considerable difficulty. In the optically thick limit,
where the mean free path of the thermal radiation Lv is
much less than some characteristic dimension of the spark L,
the radiative energy flux can be written as Xr V T, where
X r (T) is the radiative analogue of the thermal conductivity.
In that case, the effect of radiative transport is merely an
64


^3
could lead instead, through photoionization and inverse
bremsstrahlung absorption, to a nonequilibrium region
ahead of the spark. A discussion of this phenomenon
and its effects on spark propagation is given in Appendix II.



UJ3/oas-lu o V//d3
48
Figure 2. Variation of c / A with Temperature
for Atmospheric-pressure Air.


8
since most of the absorbed radiation appears as a temper
ature increase as opposed to directed fluid motion. Viscous
dissipation is also neglected since the loss due to
dissipation is less than the losses due to thermal radiation
and heat conduction in the region of laser-plasma interaction.
For this model the energy and continuity equations referred
to a co-ordinate system fixed to the spark are
CP
£T
X
+ PVr CP
2 T
} r
(2.1)
+ 5
(2.2)
In the above equations p is the density, T is the
temperature, vx and vr are the axial and radial velocity
components, respectively, cp(T) is the specific heat
capacity, A(T) is the thermal conductivity, S is the
intensity of the laser beam, (T) is the absorption
coefficient at frequency )) and $ is the net energy lost
by radiation per cnr per sec. The density is assumed to
be related to the temperature and pressure by the perfect
gas law.
In order to solve for both the velocity field and the
temperature field, the two momentum equations must be
included in the analysis. Clearly, the solution of these
four, nonlinear, coupled partial differential equations is


58
Figure 12. Axial Variation of the Volumetric Energy-
Loss Terms for R^ = .50 cm, Rc = 1.25 RL
and u = 0. (1) Thermal radiation,
(2) Radial heat conduction, (3) Axial
heat conduction.


28
A more accurate determination of this parameter
can be obtained from the two-dimensional solution
presented in Chapter III. By using Equation (3.6) to
calculate & and the radial derivative, substituting
into Equation (4.3) and evaluating at x = 0 the following
equation for A is obtained:
where Tq = (An/xon)Ji(xonBL/Rc) The convergence of
the infinite series is demonstrated in Appendix I.


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Dennis R. Keefer, Chairman
Associate Professor of
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Professor of Engineering
Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
I/.
Ulrich H. Kurzwegp
Associate Professor of
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Associate Professor of
Mathematics


1
T
rl
1
Figure A-
0 .5 1.0 1.5 2.0 2.5 3.0
x, c m
. Isotherms for a Spark with RL = .50 cm, Rc = 2 R^, u = 1 m/sec
and T^ = 6,000 K. (1) T = 8,000 K, (2) T = 12,000 K,
(3) T = 16,000 K, (4) T = 17,000 K.


LIST OP TABLES
Table Page
I Physical Properties of Laser Sparks
of .15 cm and .50 cm Radius 60
v


3
Ignition can occur in a gas which is usually transparent to
laser radiation if the laser intensity creates sufficient
ionization in a region to initiate absorption of the laser
energy.2^ Due to the high intensities required, ignition
has been observed only with the focused output of pulsed
lasers. The initial breakdown region quickly develops
into a laser spark. Shook waves, driven by the absorption
of laser radiation, propagate outward from the hot spark
and heat the surrounding areas. In particular, the cold
gas ahead of the spark and in the path of the laser beam is
ionized and begins to absorb the incident radiation. Thus,
a new absorbing layer is created and the spark front
propagates in a direction opposite the laser flux due to a
mechanism similar to that occuring in the detonation of
explosives. Typical spark temperatures are 10^ to 10^ K.
As the spark propagates, it is observed to expand radially
to fill the cone formed by the focused laser beam.
In 1969 Bunkin et al.^ demonstrated for the first
time that a laser intensity lower than that required for
breakdown could maintain an externally initiated discharge.
They found that Nd-glass laser radiation at an intensity
¡L 2
of 10 kW/cm could maintain an air spark which had been
initially ignited by an electrical discharge. This
intensity was two orders of magnitude less than that
required to initiate breakdown. In this case the incident
laser intensity is insufficient for generating a shock wave.


14
- 4- -1
z r
^ 2
rn 0
(2.6)
- 50 H(Rl-OX, e-
>(v x
The boundary conditions at x = + and at the channel
radius R are
c
0, (- oO, r) = ez { oO} r) = 0, Cx.Rc) = i (x,Rt)= o (2.7)
Since the temperature and the heat flux must be continuous
at x = 0, the Joining conditions are
0, (o,r)= 02,(0, r)
(2.8)
Equations (2.7) (2.9),together with the restriction
that 0 be finite, completely determine the solution.
There is, however, one further condition which must be
met. Since x = 0 has been defined as the plane where
appreciable absorption begins, the temperature at x = 0
as determined from the differential equations must


27
Equation (4.10) in his one-dimensional analysis. For
large propagation velocities, K and both approach
unity, if the variation of A with u is neglected. Then
Haizer's result that the laser intensity is proportional
p
to u at large velocities is obtained.
For values of x greater than the point of maximum
temperature for C02 laser radiation, 2 can be approx
imated by
0;
(x d)
so that d is the length of the spark.
By setting the derivative of 02 in Equation (4.5)
equal to zero, the location of the maximum temperature is
found to be
(<'-0 (l- e'*'*)
2
Raizerlu uses a highly empirical method for calculating
the thermal conduction parameter. If the axial variations
are neglected within a cylinder of radius R^ with strongly
cooled walls, the thermal potential will vary approximately
as JQ(x01r/RL). When substituted into Equation (4.3), this
2
distribution results in the value A = xQ1 =5.8. To
take into account the fact that the temperature is still
quite high at r = R^, Raizer reduces this value by and
uses an effective A = 2.9.
Xm *U'-D-2K ^


1 u mean free path of radiation of frequency cm
L characteristic spark dimension, cm
_2
m thermal radiation parameter, cm
Mn terms of the series used for comparison in the
Weierstrass M-Test.
n summation index
N electron number density, cm

N+ positive ion number density, cm
p
A power absorbed by the spark, kW
PQ incident laser power, kW
r radial co-ordinate, cm
R channel radius, cm
Rl laser beam radius, cm
p
S laser intensity, kW/cm
2
SQ incident laser intensity, kW/cm
S' incident laser intensity for a reduced ignition
2
temperature, kW/cm
t coefficient of the series solution
n
T temperature, K
Te electron temperature, K
heavy-particle temperature, K
ignition temperature, K
Tm maximum value of the radially averaged temperatures, K
(A^/x.^JJ, (x_Rt/R)
n n on 1 on L c
u spark propagation velocity, cm/sec
vr radial velocity component, cm/sec
ix


Figure 3. Variation of Thermal Conductivity X and Thermal Potential 0
with Temperature for Atmospheric-pressure Air.
-p-
VO


31
With these values of the parameters, Equations (3.6) -
(3.11) were used to study the propagation velocities,
threshold intensities and temperature profiles for air
sparks of radius .15 and .50 cm. Calculations were
carried out for propagation velocities from 0 to 20 cm/sec.
The channel radius was varied from Rc = R^ to RQ = 2R^.
A computer program was written to evaluate the series.
It was found that twenty terms were sufficient to insure
convergence for the range of parameters studied.
The variation of laser intensity with channel radius
for several propagation velocities is shown in Figure 4.
For a spark of .15 cm radius, a decrease in channel radius
results in a significant increase in the laser intensity
required to maintain the spark. The presence of the
channel boundary constricts the radial temperature profile.
Therefore, the radial heat conduction loss from the spark
is increased as R decreases, and a higher laser intensity
is required to overcome the increased losses.
The energy loss per unit volume due to thermal
_o
conduction in the radial direction is proportional to R^ ,
whereas the loss per unit volume due to optically thin
radiation is independent of radius. Therefore, thermal
radiation becomes the dominant loss mechanism for large
radius sparks. In that case the presence of a boundary
should have little effect on spark propagation. This is
demonstrated in Figure 4, which shows that the required


36
maximum has teen used to nondimensionalize the axial
distance. For Figures 12 and 13, xm equals .129 cm
and .231 cm, respectively.
Figures 12 and 13 illustrate the dominant role of
thermal radiation in the dissipation of energy for sparks
of large radius. Radial heat conduction accounts for
only 10% of the energy loss. Again, axial heat conduction
and convection, for the velocities considered here, are
important only near the spark front.
It is apparent from Figures 10 13 that the laser
intensity for a propagating spark must exceed the threshold
intensity so that the additional loss due to convection
at the spark front can be overcome. However, near the
temperature maximum convective losses are negligible so
that the increased laser intensity results in a higher
maximum temperature for a propagating spark than for a
stationary one. Beyond xm, convection actually is a
heating term. The spark length increases with propagation
velocity due to this additional source of energy and due
to the increase of laser intensity with propagation
velocity. These results are illustrated in Figures 3-8
and in Table I.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
NOMENCLATURE viii
ABSTRACT x*i
CHAPTER
I INTRODUCTION 1
II THEORETICAL DEVELOPMENT 7
III SOLUTION OF THE EQUATIONS 16
IV SIMPLIFIED ONE-DIMENSIONAL
ANALYSIS 24
V EXAMPLE CALCULATIONS 29
VI DISCUSSION OF RESULTS 37
VII CONCLUSIONS 44
APPENDIX
I CONVERGENCE OF THE SERIES
SOLUTION 61
II EFFECT OF RADIATION ON
SUBSONIC SPARKS 64
REFERENCES 74
BIOGRAPHICAL SKETCH 77
iv


20
XOrx RC 3^UOr0
[ + *) + ^
1
The solution of the homogeneous equation for @K is
assumed to have the form
ew(X,r1 = £ (x^G^lO
n.
Substitution into Equation (3.2) results in the set of
ordinary differential equations
oC (d* + fn) F, = o
J X
where the are separation constants. The solutions
of these equations are obtained in a manner analagous
to the procedure used for Ql Imposing the boundary
conditions Fn( aO ) = 0 and Gn(Rc) = 0 results in
eh= I
a
T0(x on~)
'(i-K)*
+ m
]


25
In Equations (4.1) and (4.2), the radial heat conduction
term has been simplified, as suggested by Raizer,10 by
setting
2 / 3e\ A e
RL \3r
(4.3)
K
<= Rl
The parameter A is assumed to be independent of x and r
but it is a function of spark and channel properties.
For the one-dimensional equations, the boundary
conditions are
Qt (- oo) = 0Z ( oo) = o
The joining conditions are given by
e (o') = 2 (o) = eL
d§,
X / x = o
The solutions of Equations (4.1) and (4.2) subject
to these conditions are
f ( i + O*
e,(x}= e. e
^ ( X )
0L £
. t-O-r')*
+ Q ( I e ) e
(4.4)
(4.5)


1
O .3 .6 .9
x, cm
Figure 5. Isotherms for a Spark with HL = .15 cm, Rc = 1.25 RL and u = 0.
(1) T = 0, (2) T = 10,000 K, (3) T = 14,000 K, (4) T = 16,000 K.


56
Figure 10. Axial Variation of the Volumetric Energy
Loss Terms for R^ = .15 cm, Rc = 1.25 RL
and u = 0. (1) Radial heat conduction,
(2) Thermal radiation, (3) Axial heat
conduction.


67
of N+, 0+ and electrons above their equilibrium values. If
the relaxation time is short enough, the concentrations will
equilibrate to values characteristic of some higher tempera
ture. The net effect of the radiation is then an equilib
rium heating of the cold gas, as assumed by Jackson and
12
Nielsen. One reaction which is essential for equilibration
is the electron-recombination reaction
N+ + e + M * N + M (A2.1)
where M is an atomic or molecular species. At electron
temperatures near 12,000 K, the reaction rate constants
for such reactions are on the order of 10^ cm^/sec.2^
There is, however, the competing reaction
N+ + N + M > N2+ + M (A2.2)
for the excess N+. The constant for this reaction has
not been accurately determined. An estimate for tempera
tures near 12,000 K is 10-^0 ^ cm^/sec.2^ For
equilibrium, atmospheric-pressure air at 12,000 K the
nitrogen atom number density is approximately six times
the electron number density,^ so that reaction (A2.2) may
proceed at a much faster rate than reaction (A2.1). Conse
quently, a nonequilibrium concentration of N2+ may exist
Q Of)
outside the hot plasma core. In fact, investigators *
have reported strong N2+ radiation outside the core region.
The role of reaction (A2.1) in the relaxation scheme increases
with increasing electron number density. The relative


76
25. Wilson, K. H. and Nicolet, W. E., "Spectral Absorption
Coefficients of Carbon, Nitrogen and Oxygen Atoms,"
J. Quant. Spectrosc. Radiat. Transfer, 12, 1257 (1972).
26. Samson, J. A. R. and Cairns, R. B., "Absorption and
Photoionization Cross Sections of 02 and N2 at Intense
Solar Lines," J. Geophys. Res.. 69. 4583 (1964).
27. Bortner, M. H., Kummler R. H. and Baurer, T., "Summary
of Suggested Rate Constants," Reaction Rate Handbook
(2nd Edition), edited by M. H. Bortner and T. Baurer,
Santa Barbara, Calif.: DASIAC, DoD Nuclear Information
and Analysis Center, 1972.
28. Bardsley, J. N. and Biondi, M. A., "Dissociative
Recombination," Advances in Atomic and Molecular Physics
Volume 6. edited by D. R. Bates and I. Esterman,
New York: Academic Press, 1970.
29. Phelps, A. V., Electron Attachment and Detachment
Processes," Reaction Rate Handbook (2nd Edition),
edited by M. H. Bortner and T. Baurer, Santa Barbara,
Calif.: DASIAC, DoD Nuclear Information and Analysis
Center, 1972.
30. Ferguson, E., "Ion-Neutral Reactions A. Thermal Processes,"
Reaction Rate Handbook (2nd Edition)edited by M. H.
Bortner and T. Baurer, Santa Barbara, Calif.: DASIAC,
DoD Nuclear Information and Analysis Center, 1972.
31. Ferrari, C. and Clarke, J. H., "On Photoionization
Ahead of a Strong Shock Wave," Supersonic Flow.
Chemical Processes and Radiative Transfer, edited by
D. B. Olfe and V. Zakkay, New York: Pergamon
Press, 1964.


axial velocity component, cm/sec
axial oo-ordinate, cm
axial location of the maximum temperature, cm
nth zero of JQ
zero order Bessel function of the second kind
ionic charge
?0S u/X .cm-1
2 2 2
separation constant, equal to xQn /R cm"
[l -t- 4 A / J
r 1 ,/z
Li t (4/o1 r!) R + a)J
r i '/a
Li t(2x.^/dRa') J
I I + [(x.^/Rc^t nr]]
exponential factors in the series solution, cm"1
thermal potential, kW/cm
ignition thermal potential, kW/cm
reduced ignition thermal potential, kW/cm
thermal potential in the region x < 0, kW/cm
thermal potential in the region x > 0, kW/cm
solution of the homogeneous equation for Qz kW/cm
particular solution of the equation for > kW/cm
thermal potential averaged over the laser radius,
kW/cm


69
7 T 2ft
Its rate constant is on the order of 10' cnr/sec.
Three-body recombination
X+ + e + M X + M (A2.5)
where X is N, 0 or N2, also plays a significant role.
The reaction rate constant for (A2.5) at electron
temperatures near 6,000 K is typically 5 x 102^ cm^/sec
when M is an atomic or molecular species and
5 x 10 cm /sec when M is an electron. Electron
attatchment to oxygen atoms and molecules may also be
important since the negative ions can react with other
species to form stable compounds.2^>3
If the recombination reactions are fast enough, the
electron number density will relax to its equilibrium value
before the electrons can be appreciably heated by the laser
radiation. The assumption of rapid relaxation is made
12
implicitly in the analysis of Jackson and Nielsen. If
chemical relaxation occurs too slowly, a nonequilibrium
layer will form which will enhance the absorption of laser
radiation.
The development of the nonequilibrium layer occurs
due to the combined absorption of thermal radiation and
laser radiation. The absorption of thermal radiation
induces appreciable ionization ahead of the spark analagous
to the effect of precursor radiation in shock waves.The
free electrons are then selectively heated by inverse


*


2
the laser beam thereby eliminating the need for special
equipment. It is possible, therefore, to obtain pure
plasmas burning in free space. These characteristics plus
the high temperatures attainable in laser sparks extend
the possible applications beyond those of lower frequency
discharges. For example, a recent study1 explored the
feasibility of using laser sparks to increase the enthalpy
of the flowing gas in a hypersonic wind tunnel. These
laser-sustained discharges also offer the researcher
interested in plasma physics a unique opportunity to study
an unconfined plasma in the laboratory.
However, the occurrence of laser sparks is often
undesirable. A laser beam of sufficiently high intensity
can initiate a spark in a gas or by interaction with solid
materials. The intensity of the laser radiation is
attenuated in passing through the spark due to the conversion
of radiation energy to thermal energy which takes place in
the spark. Therefore, the intensity of laser radiation
which can be transported to a target is limited by the
threshold intensity for the creation of a spark. Exceeding
this intensity results in the creation of a laser spark
which absorbs a portion of the incident radiation ahead of
the target.
Laser sparks generally fall into two categories
depending on whether the laser beam ignites the discharge
or merely maintains an externally initiated discharge.



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