|Table of Contents|
Table of Contents
List of Tables
List of Figures
Chapter 1. Introduction
Chapter 2. Theoretical development
Chapter 3. Solution of the equations
Chapter 4. Simplified one-dimensional analysis
Chapter 5. Example calculations
Chapter 6. Discussion of results
Chapter 7. Conclusions
Appendix 1. Convergence of the series solution
Appendix 2. Effect of radiation on subsonic sparks
TWO-DIMENSIONAL MODEL POE THE SUBSONIC
PROPAGATION OF LASER SPARKS
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,
to my parents
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.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . iii
LIST OF TABLES . . . . . . . . v
LIST OF FIGURES . . . . . . . vi
NOMENCLATURE . . . . . . . . viii
ABSTRACT . . . . . . . . . xii
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
I CONVERGENCE OF THE SERIES
SOLUTION . . . . . . . 61
II EFFECT OF RADIATION ON
SUBSONIC SPARKS . . . . . . 64
REFERENCES . . . . . . . . . ?4
BIOGRAPHICAL SKETCH . . . . . . 77
LIST OF TABLES
I Physical Properties of Laser Sparks
of .15 cm and .50 cm Radius . . . . 60
LIST OF FIGURES
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
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
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 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
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
vx axial velocity component, cm/sec
x axial co-ordinate, cm
xM axial location of the maximum temperature, cm
xon nth zero of J
Y zero order Bessel function of the second kind
Z ionic charge
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,
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
absorption coefficient for radiation of frequency
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
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
Jad Hanna Batteh
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.
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.
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
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.
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.
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.
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
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
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,
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
%/X C P + P Vr C P
dT) + + 5 0 (2.1)
X )X r
(YVX) + 0 (2.2)
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
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
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
where the thermal flux potential
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
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
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
= 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)
\ 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
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)
This equation results in a relationship between the propagation velocity and the required laser intensity.
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.
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
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
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"
The constants an are determined by the joining conditions.
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)
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
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 ) -( &
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
~~ozrW RSo (XT 4 ()X(O* Rml
The solution of the homogeneous equation for eh is assumed to have the form
GV (x,r = F (x)
Substitution into Equation (3.2) results in the set of ordinary differential equations
--AF -( + rm) F= 0
/ 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
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)
+ 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
E) z- e0_. B, -T- ( xo,. %
Rc rt (3.7)
where the coefficients are given by
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
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,
PO I e (3.12)
where P A is the power absorbed by the spark and Po is the incident power.
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)
In Equations (4.1) and (4.2), the radial heat conduction 10
term has been simplified, as suggested by Raizer, by
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
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
/ ( 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
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
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
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.
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.
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
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.
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
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
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
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.
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
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.
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
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
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
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
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
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?,
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.
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.
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.
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.
Figure 1. Co-ordinate System for the Steady State Propagation of a Laser
Spark in a Channel.
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
( -%) (C:OK)
0 10 20 30
T, 10' K
Figure 3. Variation of Thermal Conductivity and Thermal Potential e
with Temperature for Atmospheric-pressure Air.
1.0 1.5 2.0
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.
I I I I I I I
0 .3 .6 .9
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.
I I I I I I I
0 .3 .6 .9
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.
I I t I I I I I
-.5 0 .5 1.5 2.0
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.
SI I II I 1 III
0 .5 1.0 1.5 2.0
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.
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.
.52 Energy Loss 4
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
-. 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.
0 1 2
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
Loss .4 '
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.
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
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)
lim J1(z) = .798 z" cos(z 3 ~' /4) (Al.5)
lim Xon J 12(xon) = .637 (Al.6)
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.
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
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)
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
lir ICnI / Ani =1
and 2 JAn converges, then IBnland ICnI
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
Therefore, the series 1r Xon 2 Tn converges.
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
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
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
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
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)
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
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
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
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.
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.
I I I lII II I IlI
0 .5 1.0 1.5 2.0 2.5 3.0
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.
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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.
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.
Dennis R. Keefer, Chai/x'man Associate Professor of
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
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
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
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 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.
/xVean, Collegig of Engineering
Dean, Graduate School