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Transport of uranium free particles in a d.c. arc plasma burning in air at atmospheric pressure
spectrochimka Acta, Vol.28B,pp.241to262. Pergamon Prea~ 1972.Printed inNorthern Ireland
~~0~
of umniuxn free particles in a kc. arc plasm be in air at atmospheric pressure* R. AVNI and Z. GOLDBART Atomic Energy Commission, Nuclear Research Centre, Negev, P.G. B. 9001, Beer-Sheba, Israel
Abstract-The transport parameter y was calculatedfor the free particlesof umnium in a d.c. arc plasma burning in air. Using the “wire method” the volatilization rate (&,), the total concentration of free particles (n,), the axial velocity (w,) of uranium particles and the plasma cross section (S) were measured. The transport parameter was calculated for cylindricalsymmetry of the arc. The total particle cou~ntr~tion calculated by the wire method was compased to v&es obtained by absolute intensity measurements of ion and atom spectral lines of uranium. This led to an estimate of the moleoukr concentrationof uranium in the d-c. arc plasma. 1. INTROD~JCTION
TEE TRANSPORT parameter y, indicating the flow of the free particles of a given element j in the plasma is [l-4] : Yj = Q&t = vjS(cm3 s-i), (I) where the second equality holds only if cylindrical symmetry and a homogeneous flux of particles parallel to the arc axis is assumed. In equation (1) Qj is the volatilization rate of element j, nt the total particle concentration of this in the plasma (atoms + ions + molecules), V$the axial velocity of the free particle and 8 the cross section of the plasma, perpendicular to the am azis. Convection caused by temperature gradients, diffusion as a result of concentration gradients, and the electric field, affect the value of 7yi. Theoretical models [l-4] for calculating pj are:
(i) the “velocity model” based on the axial migration of particles, and (ii) the “dn,/dt model” based on the simultaneous effects of radial and axial migration of particles of an element in the arc gap. The latter model explains the spatial variation in the particle concentrations as a result of convection, diffusion and the electric field. Experimental methods for determining ySor oj are based on spectral line intensity measu~men~ namely : (i)
Measurement of the transit time [a] of the particles in the plasma; the particle velocity V, in the arc gap is derived from the time of transit over a given distance.
* Partially presented at the XVth C.S.I., M&rid (1969)-Proceedings II. p. 293. [l] [Z] [3] [4]
L. DE GA.LKN,Thesis, University of Amsterdibm (1966). L. DE GALAW, J. Quart. Spew. Radiativs Transfer 5, 735 (1966). P. W. M. J. BOUMANSand L. DE GALAN,Awl. Chem. 89, 674 (1966). P. W. M. J. BOWS, Th+?opyof Spectrodmnical Excitation, Hilger & W&m, London (1966). [6] V. D. MALYKH and M. A. SERD, Opt. Spectpy (U.S.S.R.) I.&303 (1964). 241
242
R. AVNI and Z. GOLDBART
(ii) Calculation of particle concentration from absolute values of line intensities [l-2]. The concentrations thus found represented only atoms and ions (nj). ly*was calculated from nj using an average value of the volatilization rate Q,, determined in the same experiment. Both methods are adequate only when the test element does not form stable molecules, i.e. only the presence of atoms and ions is assumed. The presence of molecules of element j (for example, aluminium oxide or carbide when Al,O, is introduced from the graphite anode crater) in the plasma is neglected for convenience, because the calculation of total particle concentration is complicated even if the molecular spectrum of j is known. The empirical correlation of DE GALAN [l-2] illustrates this : log y, = log &, - log 5Sj = 2.5 + a, (2) where aj is the ionization degree of the atom, and ~5~the average concentration of element j. Thus y, depends on the degree of ionization and not on the degree of dissociation of molecular species. We developed a method which measures also the molecular concentration of the element j, together with the concentration of its atoms and ions. The variation of the total particle concentration was measured radially and axially using wires that passed through the plasma [6]. t
Distikce from5arcaxis,
9
mm
Fig. 1. Radial distribution of uranium (UsOe), nt(z, r)v$, with z parameter, aa derived from 2.0~mm wire portions. Gap = 6-Omm; i = 10 A; carriage velocity = 110 cm 8-l. Distance above the anode: z = O-2 mm - * -; z = 2eomm.. .; z = 3*6mm----; 2 = 5+3mm n w n . [S] R. A~NI and Z. GOLDBART, I. @ect*oc%m. Acta; 28B (to be published) (1973).
Transport of uranium free particles in a d.c. am plasma
243
The radial distribution of the particle flux, +, z)~,, of uranium (Fig. 1) obtained by neutron activation analysis of the deposits on the wires IS] were fitted to either an exponential or a Bessel function. These are in fact theoretical solutions of the dlz,/dt model [3-41. The ‘wire method”, as will be shown further on, allows an experimental check of the theoretical model. 2. T~EORE~CA~ CO~S~ERATIO~~ The dB,fdt model, as described by BOUMANS[3-4-J considers the flow of particles through an intitesimal element of volume, and is written in cylindrical coordinates (T, z) for a steady state:
(3) where D and V$are the diffusion coefficient and the axial particle velocity. Solution of this partial differential equation for a 10 mm arc gap with a core radius (R) of about 5 mm were given by GINSEL [7] and BA~INCE[S] for constant values of vj and D. GINSEL[7] considered the supply of particles to the plasma from a point source; at approp~ate boundary eo~ditio~ [3] his solution has an exponential form:
n,(z, r) =
Qr
27rDm
exp
- 2
(W
- z)]
BAYINCK[8] considered a disc shaped source for particle supply to the plasma; at app~p~ate bounda~ condition [3, 4, 81 his solution is a first-order Bessel function:
n,(z/R, r/R) = s
5 %=IB’UJo2(&)
J,(L,-$=p
+ J~2(&z)l
(-;B)
where B = 1/V
+ A,2 -
w
B’ = dWa + An2+ W and
A, is an infinite series of positive numbers satisfying the boundary conditions, J,, J1 are the zeroth and first order Bessel functions, R is the radius of the arc core and a the radius of the anode crater. When R is known, function n,(z, r)/Q, can be calculated from Equation (5) using appropriate values of a, D and vs. The results given in Fig. 1 [radial ~stribution of particle flux A&, r)v,] can be fitted to a first order BesseI functions for different z. This agreement between Equation (5) and Fig. 1 was tested in the following way: assume that the plot in [7] L. A. GWSEL, Thesis, State University of Utrecht (1933). [S] H. BAVINCK,Rep. No. TW-98, I&thematicrtl Centre, Amsterdam (19&t. 2
R. AVNI end Z. GOLDBART
244
Fig. 1 follows Equation (4). In this case the ratio w,/2D is obtained directly from the slope (see appendix) of the exponential plots in Figs. 4 and 5. The experimental values of ~,/2 D led to the next three steps :
(i) The values of w,/2D were introduced into Equation (5) and the ratio nt(z, T)/ &, was calculated from the numerical solution of the Bessel function. These values were compared with the values of nt(z, r)/&, obtained experimentally as follows. (ii) From the values of v,/2D the axial velocity (v,) of the uranium particle for different z was calculated with known values of the diffusion coefficient, D [l-4].
(iii) With the w,values thus obtained, the total particle concentration of uranium n,(z, r) was calculated using the radial distribution of the particle flux shown in Fig. 1, at different z. The volatilization rate Q,, was measured as described previously [6]. In this way values of the ratio nt(z, r)/&, were experimentally obtained. The agreement between n,(z, r)/&, from steps (i) and (iii) demonstrates that the assumptions were appropriate and that the plots in Fig. 1 behave similarly to the Bessel function given by Equation (6). The diffusion coefficient, D, was obtained from the relationship used by DE GALAN [l]namely : D = const.
mu*76
1 AG@Y&
(6)
in which M* is the reduced mass, d, the diameter of the diffusing particles and T is the absolute temperature in the given region of the d.c. arc. With the experimental values of nt(z, r) and the volatilization rate [6] (Q, and &(r)) the radial and axial distribution of the transport parameter (y,) was calculated using Equation (1). Furthermore, y, was also calculated from the known values of P)~and the cross section (AS’)of the plasma. The total particle concentration nt(z, r) obtained by the ‘wire method” represents the total concentration of atoms, ions and molecules of uranium in the plasma. These values were compared with those of the particle concentration n,(z, r) obtained by measuring the absolute intensity of the uranium spectral lines, i.e. atom and ion lines. From this comparison the molecular concentration of uranium in the plasma could be calculated. 3. EXPERIMENTAL The apparatus, the operating conditions and the neutron activation analysis method were the same as described in a previous paper [6]. The radial distribution of the particle flux nt(r)v, was measured by cutting the wire into 2.0 mm portions. The highest y-activity, i.e. the highest particle flux was attributed to the arc core. The measurement of the absolute intensity of uranium line spectra for the calculation of ng was described elsewhere [9]. Figure 2 shows the schematic wire displacement for measuring the uranium particle flux at various heights (z) in the plasma. [g] R. Am,Spectrochim.Acta2SB, 619 (1968).
Transport of uranium free particles in a d.c. arc plasma
245
Fig. 2. Schematic representationof wire displacement in the plasma. Wires 1, 2, 3, 4 at Merent 2. I
Fig. 3a. Schematic radial voltage distribution for measuring the plasma and core radii.
The plasma diameter at different heights was also measured by the %ire method” using the oscillogram given in Fig. 3. The z axis is indicated by each pea,k (cf. Fig. 3a). The width (2r,) of the arc plasma W&S measured for a given wire velocity [ 10, 111. This width is derived from the distance between the lower inflection points [lo] R. AVNI, Spe&ochim. Acta 23B, 697 (1968). [ll] R. A~NI, Thesis, Weizmaxm Institute of Science, Rehovoth (1970).
R. AVNI and Z. GOLDBART
246
Fig. 3b. Oscilloscopic picture of the wires for S measurement. Ordinate 10 V cm-l; Abscissa 5 ma cm-l. Left, the anode region. Right, the cathode region. (Fig. 3a), i.e. the entrance and the exit of each wire into and from the plasma. The arc core diameter (2rJ was obtained by measuring the width on the peak between the two higher inflection points (Fig. 3a) ; this change is assumed to be caused by the entrance of the wire into the arc core [lo, 111. In cylindrical symmetry the width thus obtained represents the diameter of the core. 4. RESULTS AND DISCTJSSION 4.1 Particle velocity The slopes of the lines in Figs. 4 and 5 give the values of v,/2D by Equation (4). From this parameter, the axial particle velocity v, is obtained after calculation of the diffusion coefficient D. Using Equation (6) and the measured temperature [ll] of the different regions in the plasma (1 < r Q 3 mm and z up to 8-O mm) D was calculated. Figure 6 is a plot of the experimental values of the axial velocity of the uranium particle. Within the experimental error, v, is constant over the arc gap except for the anode region. This indicates that the assumption vj is constant for the two solutions [Equations (4) and (5)] of Equation (3) is adequate for a real d.c. arc plasma except in the vicinity of the anode. By addition of 4 % UF, to the U,O, in the anode crater, the volatilization rate, Qj, of uranium particles is increased [12]. Figure 6 shows that this did not affect the axial velocity of the uranium particles. 4.2 Total particle concerdration in the plasma The radial distribution of the particle flux nt(T, z)vj measured on the activated aluminium wires is given in Figs. 4 and 5. In a previous paper [6] the proportionality factor between the particles collected by the wire and the particle flux in the plasma, [12]
R. AVNI and A. BOTJKOBZA,Spectvdhn. Acta 24B, 515 (1969).
Transport of uranium free particles in a d.c. arc plasma
Distance
from arc axis,
Fig. 4. Radial distribution of uranium from 2.0~mm wire portions. Gap = 110 cm&. Distance above anode: z = 2 = 6.9 A; 2
247
mm
(U,O,), n$(z, r)v,, with z parameter as 8.0 mm; i = 10 A; carriage velocity 0.2 mm +; z = 2.2 mm 0; z = 4.0 W; = 7.8 mm 0.
was shown to be near unity within experimental error. In other words the wires measure the absolute values of the particle flux in the plasma. Using the values of w,the experimental total particle concentrations were obtained from the particle flux. Table 1 shows the axial distribution of the total particle concentration (n,) in an 8-Omm arc gap as measured from the wires together with the particle concentration (nj) calculated from the absolute intensity of the uranium lines 4289 and 4310& According to this table the molecular concentration of uranium in the arc core of the plasma (R = 2-O mm) is about 20 times that of the atom and ion concentration. The “wire method” does not specify the kind of uranium molecules found in the plasma, but only their sum. Carbides, oxides and nitrides of uranium may be in the plasma [13-141. The high concentration of molecules and the relatively low concentration of uranium atoms and ions explain the relatively high temperatures of the plasma in spite of the low first ionization potential of uranium 6.22eV [15]. The agreement between Equation (5)and the radial plots in Figs. 1, 4 and 5 is [13] H. NICKEL, Spectrochim. Acta 21,363 (1965). [14] R. RAUTSCHKE, Spectrochim. Acta 23B, 65 (1968). [la] J. B. MANN,J. Chma. Phys. 40,1632 (1964).
248
R. Aarr
tend Z. GOLDBART
Distance from arc axis,
mm
Fig. 5. Radial distribution of uranium (UaOr, + 4% UP,), n&z, r)~,, with z pammeter aa derived from 2.O-mm wire portions. Gap = 8-Omm; i = 10 A; carriage velocity 110 cm.s+. Distance above the anode: z = 0.2 mm +; z = 2.2 mm 0; z = 4.0 mm n ; z = 6.9 mm A; z = 7*8mm 0.
._ i \ \ 500
T \,
4,;
,,,,,, 4
6
Distance.2.
mm
2 Anode
6 Cathode
Fig. 6. Axial distribution of uranium particle 4%,‘,uF,.
velocity
(21,) A
UsOa; H rr,O, +
Transport of uranium free particles in a d.c. arc plasma
249
Table 1. Axial distribution of total particle concentration. Arc gap = 8-Omm; P = 2.0 mm; i = 10 A; 35 s exposure Line intensity (b) method
Wire method (a) 2,
mm
nt
x lo-r1 crnw3
n.3 X lo-r1 crnm3
Matrix Anode 0.0
nt -nj Iti
-
270.0 110.0 90.0 110*0 60.0 -
nt x lo-r1 crnm3 Matrix U,O, + 4% UF,
u303
-
0.2 1.8 3.5 6.0 7.8 Cathode 8.0
Wire method (a)
-
11.0 6.0 5.0 4.0 3.0 -
-
23.5 17.3 17.0 26.5 19.0 -
300.0 150.0 100.0 lOO*O 80.0 -
(a) Mean value over 4 carriages of wires; relative standard deviation 25 ‘A. (b) Mean values over 4 spectra (d.c. arc focused on slit); relative standard deviation 20%.
shown in Table 2. For a closer comparison the 2*0-mm wire portions with the highest y-activity, were cut into 05-mm sections. As described, the ratio of n,(z, r)/Q, calculated by Equation (5) with the experimental values of v,/2D is compared with the same ratio measured by the wire method. The results in Table 2 show good agreement between the values of n,/Q, obtained by the two methods. Table 3 shows the radial distribution of nt of uranium in the anode, central and cathode regions of the plasma as described previously [9, lo]. The radial diffusion of uranium particles, which is strong in the anode region, decreases towards the cathode. Along the axis a lower concentration was found in the cathode region compared to the nt values in the anode region. The decrease of the particle concentration towards the cathode is caused by radial Table 2. Ratio nt(z, r)/Q, calculated with the Bessel solution (Equation 6) and measured by the wire method at z = 1.5 mm above the anode in an 8.0 mm gap. Crater radius (a) = 2.0 mm Equation (5)* Radius
mm 0.00 0.25 0.50 0.75 I.00 1.25 I.60
Temperature K 6700 6600 6300 5900 5500 6000 4600
Wire method
4Qr x 1O-88.cm+ 7.54 7.62 7.90 8.16 8.15 8.07 7.45
Wire portion@ mm
Q, x 10-14 s-1
% x 10-l'
cm-3$
M&r x
IO8 s.om-*
05
2.8
20.0
7.15
I.0
6.5
60.0
7.70
1.5 2.0
11.0 13.0
85.0 100.0
7.70 7.69
* Computer calculations using experimental values of vJ2D. t The wires were cut into 4 portions. Each wire portion was measured separately for yradiation. $ Relative standard deviation from 4 wires 25 %.
250
R. AVNI and Z. GOLDBART Table 3. Radial distribution of nt. Arcgap=S*Omm; i=lOA Plasma region Anode[l”l
Centre[lol
Cathode[lO]
rnun
n, x lo-l1 cmW3
o-2 2-4 4-6 o-2 24 4-6 o-2 2-4 4-6
270.0 35.0 5.0 100.0 45.0 9.0 60.0 30.0 7.0
Relative standard deviation % (a) 25 25 25 25 35 35 25 35 35
(a) The relative standard deviation is obtained from the passage of 4 carriages. (16 wires in the anode region; 8 wires in the centre region; 4 wires in the cathode region.) Table 3a. Normalized values of the particle concentration,nt/nt (anode). Cf. Table 3. Arc gap = 8.0 mm; i = 10 A nt/nt (anode) Plasma region (r) (2)
O-2 mm
2-4 mm
4-6 mm
Anode Centre Cathode
1.0 0.37 0.22
1.0 1.30 0.85
1.0 1.80 1.40
diffusion of particles. So the particle concentration in the vicinity of the axis (r = O-2 mm) decreases continuously with increasing height above the anode. Table 3a illustrates this effect and shows particle concentrations that have been normalized to the corresponding values in the anode region. For r = 2-4 mm and r = 4-6 mm a slight increase toward the cathode was obtained. Diffusion apparently proceeds so that the particle concentration decreases steeply only in the axial zone and does not vary appreciably in the outer zones. 4.3 Transport parameter of uranium (y) The radial and axial transport parameter of free uranium particles was calculated using Equation (1). Table 4 shows the radial distribution of yj in the anode, central and cathode regions of the plasma. The increase of yu,outside the arc core (r > 2 mm) shows the radial transport in the arc mantle. The discrepancy in the transport parameter calculated by Q,/ n, with v,S indicates that the second way cannot be used for computing the radial distribution of y, because:
(i)
w
vjS was assumed to hold only for a vertical movement of the particles in the cylinder. The 2 mm wire sections used to measure the particle flux do not represent any real radius for the plasma cross section calculation; the wires could not be cut into smaller portions to obtain a detailed picture over the arc cross section.
251
Trsnsport of umnium free particles in e d.c. sre plssms Table 4. Rsdid distribution of y,. Arc gap 8.0 mm; i = 10 A
centre Cathode
130
48.0 370.0 130.0 290.0 220.0 420.0
o-2 2-4 o-2 24 o-2 2-4
Anode
120 280
* Man v&w over 4 esrrkges of wires. Relative stsndsxd de&&ion 25 % in the anode region and 30 % in the centre cbndcathode regions. t Relative standard deviation 40 % from 4 passages of wires (Figs. 3 and 3a).
Figure 7 shows the axial distribution of the transport parameter in the arc core, for a radius of r = 2 mm, when U,O, and U,O, + 4 %UF, were initially placed in the anode cavity. The Merence in y, for U,O, and U30, + 4%UF,, at the same axial velocity (Fig. 6) and at almost constant value of n, (Table I), shows that 8 higher volatilization rate Qj resulted from the addition of UF, to the U,O, matrix
lw
The transport parameter for aluminium evaluated by DE GALAN [1] [Equation (2)] yAl = 2-7 x lo6 em3 s-l for a mixture of Al,O,/C = 8 is very high compared to the values given in Fig. 7. Al0 molecules which were found in a d.c. arc burning in air [16] were not computed in his nj values i.e. the values of cn,are too small. The
I I
I
3
I
1
I
8
7
5
Anode
Cathode Distance,z,
mm
Fig. ‘7. Axial distribution of yj. A U,O, 110 cm s-1 carriage velocity; 0 UaC, 250 cm s-r carriagevelocity; n TJsO, + 4 % UF,, 110 cm s-1 carriage velocity. [ls] R. W. 8. PEARSE snd A. G. GAYDON, The I~~~~~~ t Hall Ltd. London (1965).
of ~0~~
Sp&rs, Chapman
R. AVNI and Z. GOLDBART
252
volatilization rate of Al,O, from the anode is estimated as being of the same order of magnitude as that for UaO,. The val values of DE GALAN[l] are too high due to the summation of only atoms and ions for particle concentration. The decrease of nt towards the cathode was used for analysing impurities in uranium [lo, 121 exposing on the photographic plate only the cathode region instead of the whole arc gap. In the cathode region the uranium lines spectra obtained is less dense leaving the impurity spectral lines undisturbed. APPENDIX
If we assume that Equation (4) describes the spatial distribution of n, and that vj varies only with z, the flux equation can be expressed as
Q(x, r) = n,(z, r) . v&) =
Q&)
47rDdi373
exp
The slope of the curve ln @ vs. r is obtained by differentiating In Q with respect to r: am
-=-
ar
&+
-
1
$4 [ l/z2 + p-2 + 2
1*
For the anode region therefore
v,/2D was evaluated by point to point calculation over all the experimental points with subtraction of the mean value of l/r in each point. For T > 0.4 cm, v,/2D can be directly derived from Figs. 4 and 5, within an error of 15 %. The solution for the anode region is valid for z = 0.2 cm with the same error as for z = 0. For higher z values the simplification used is:
for dm > O-6 cm, within the same error of 15 %. The computation of v,/2D was done after dividing the value of the slope between every two points by the mean value of
~~0~
of umniuxn free particles in a kc. arc plasm be in air at atmospheric pressure* R. AVNI and Z. GOLDBART Atomic Energy Commission, Nuclear Research Centre, Negev, P.G. B. 9001, Beer-Sheba, Israel
Abstract-The transport parameter y was calculatedfor the free particlesof umnium in a d.c. arc plasma burning in air. Using the “wire method” the volatilization rate (&,), the total concentration of free particles (n,), the axial velocity (w,) of uranium particles and the plasma cross section (S) were measured. The transport parameter was calculated for cylindricalsymmetry of the arc. The total particle cou~ntr~tion calculated by the wire method was compased to v&es obtained by absolute intensity measurements of ion and atom spectral lines of uranium. This led to an estimate of the moleoukr concentrationof uranium in the d-c. arc plasma. 1. INTROD~JCTION
TEE TRANSPORT parameter y, indicating the flow of the free particles of a given element j in the plasma is [l-4] : Yj = Q&t = vjS(cm3 s-i), (I) where the second equality holds only if cylindrical symmetry and a homogeneous flux of particles parallel to the arc axis is assumed. In equation (1) Qj is the volatilization rate of element j, nt the total particle concentration of this in the plasma (atoms + ions + molecules), V$the axial velocity of the free particle and 8 the cross section of the plasma, perpendicular to the am azis. Convection caused by temperature gradients, diffusion as a result of concentration gradients, and the electric field, affect the value of 7yi. Theoretical models [l-4] for calculating pj are:
(i) the “velocity model” based on the axial migration of particles, and (ii) the “dn,/dt model” based on the simultaneous effects of radial and axial migration of particles of an element in the arc gap. The latter model explains the spatial variation in the particle concentrations as a result of convection, diffusion and the electric field. Experimental methods for determining ySor oj are based on spectral line intensity measu~men~ namely : (i)
Measurement of the transit time [a] of the particles in the plasma; the particle velocity V, in the arc gap is derived from the time of transit over a given distance.
* Partially presented at the XVth C.S.I., M&rid (1969)-Proceedings II. p. 293. [l] [Z] [3] [4]
L. DE GA.LKN,Thesis, University of Amsterdibm (1966). L. DE GALAW, J. Quart. Spew. Radiativs Transfer 5, 735 (1966). P. W. M. J. BOUMANSand L. DE GALAN,Awl. Chem. 89, 674 (1966). P. W. M. J. BOWS, Th+?opyof Spectrodmnical Excitation, Hilger & W&m, London (1966). [6] V. D. MALYKH and M. A. SERD, Opt. Spectpy (U.S.S.R.) I.&303 (1964). 241
242
R. AVNI and Z. GOLDBART
(ii) Calculation of particle concentration from absolute values of line intensities [l-2]. The concentrations thus found represented only atoms and ions (nj). ly*was calculated from nj using an average value of the volatilization rate Q,, determined in the same experiment. Both methods are adequate only when the test element does not form stable molecules, i.e. only the presence of atoms and ions is assumed. The presence of molecules of element j (for example, aluminium oxide or carbide when Al,O, is introduced from the graphite anode crater) in the plasma is neglected for convenience, because the calculation of total particle concentration is complicated even if the molecular spectrum of j is known. The empirical correlation of DE GALAN [l-2] illustrates this : log y, = log &, - log 5Sj = 2.5 + a, (2) where aj is the ionization degree of the atom, and ~5~the average concentration of element j. Thus y, depends on the degree of ionization and not on the degree of dissociation of molecular species. We developed a method which measures also the molecular concentration of the element j, together with the concentration of its atoms and ions. The variation of the total particle concentration was measured radially and axially using wires that passed through the plasma [6]. t
Distikce from5arcaxis,
9
mm
Fig. 1. Radial distribution of uranium (UsOe), nt(z, r)v$, with z parameter, aa derived from 2.0~mm wire portions. Gap = 6-Omm; i = 10 A; carriage velocity = 110 cm 8-l. Distance above the anode: z = O-2 mm - * -; z = 2eomm.. .; z = 3*6mm----; 2 = 5+3mm n w n . [S] R. A~NI and Z. GOLDBART, I. @ect*oc%m. Acta; 28B (to be published) (1973).
Transport of uranium free particles in a d.c. am plasma
243
The radial distribution of the particle flux, +, z)~,, of uranium (Fig. 1) obtained by neutron activation analysis of the deposits on the wires IS] were fitted to either an exponential or a Bessel function. These are in fact theoretical solutions of the dlz,/dt model [3-41. The ‘wire method”, as will be shown further on, allows an experimental check of the theoretical model. 2. T~EORE~CA~ CO~S~ERATIO~~ The dB,fdt model, as described by BOUMANS[3-4-J considers the flow of particles through an intitesimal element of volume, and is written in cylindrical coordinates (T, z) for a steady state:
(3) where D and V$are the diffusion coefficient and the axial particle velocity. Solution of this partial differential equation for a 10 mm arc gap with a core radius (R) of about 5 mm were given by GINSEL [7] and BA~INCE[S] for constant values of vj and D. GINSEL[7] considered the supply of particles to the plasma from a point source; at approp~ate boundary eo~ditio~ [3] his solution has an exponential form:
n,(z, r) =
Qr
27rDm
exp
- 2
(W
- z)]
BAYINCK[8] considered a disc shaped source for particle supply to the plasma; at app~p~ate bounda~ condition [3, 4, 81 his solution is a first-order Bessel function:
n,(z/R, r/R) = s
5 %=IB’UJo2(&)
J,(L,-$=p
+ J~2(&z)l
(-;B)
where B = 1/V
+ A,2 -
w
B’ = dWa + An2+ W and
A, is an infinite series of positive numbers satisfying the boundary conditions, J,, J1 are the zeroth and first order Bessel functions, R is the radius of the arc core and a the radius of the anode crater. When R is known, function n,(z, r)/Q, can be calculated from Equation (5) using appropriate values of a, D and vs. The results given in Fig. 1 [radial ~stribution of particle flux A&, r)v,] can be fitted to a first order BesseI functions for different z. This agreement between Equation (5) and Fig. 1 was tested in the following way: assume that the plot in [7] L. A. GWSEL, Thesis, State University of Utrecht (1933). [S] H. BAVINCK,Rep. No. TW-98, I&thematicrtl Centre, Amsterdam (19&t. 2
R. AVNI end Z. GOLDBART
244
Fig. 1 follows Equation (4). In this case the ratio w,/2D is obtained directly from the slope (see appendix) of the exponential plots in Figs. 4 and 5. The experimental values of ~,/2 D led to the next three steps :
(i) The values of w,/2D were introduced into Equation (5) and the ratio nt(z, T)/ &, was calculated from the numerical solution of the Bessel function. These values were compared with the values of nt(z, r)/&, obtained experimentally as follows. (ii) From the values of v,/2D the axial velocity (v,) of the uranium particle for different z was calculated with known values of the diffusion coefficient, D [l-4].
(iii) With the w,values thus obtained, the total particle concentration of uranium n,(z, r) was calculated using the radial distribution of the particle flux shown in Fig. 1, at different z. The volatilization rate Q,, was measured as described previously [6]. In this way values of the ratio nt(z, r)/&, were experimentally obtained. The agreement between n,(z, r)/&, from steps (i) and (iii) demonstrates that the assumptions were appropriate and that the plots in Fig. 1 behave similarly to the Bessel function given by Equation (6). The diffusion coefficient, D, was obtained from the relationship used by DE GALAN [l]namely : D = const.
mu*76
1 AG@Y&
(6)
in which M* is the reduced mass, d, the diameter of the diffusing particles and T is the absolute temperature in the given region of the d.c. arc. With the experimental values of nt(z, r) and the volatilization rate [6] (Q, and &(r)) the radial and axial distribution of the transport parameter (y,) was calculated using Equation (1). Furthermore, y, was also calculated from the known values of P)~and the cross section (AS’)of the plasma. The total particle concentration nt(z, r) obtained by the ‘wire method” represents the total concentration of atoms, ions and molecules of uranium in the plasma. These values were compared with those of the particle concentration n,(z, r) obtained by measuring the absolute intensity of the uranium spectral lines, i.e. atom and ion lines. From this comparison the molecular concentration of uranium in the plasma could be calculated. 3. EXPERIMENTAL The apparatus, the operating conditions and the neutron activation analysis method were the same as described in a previous paper [6]. The radial distribution of the particle flux nt(r)v, was measured by cutting the wire into 2.0 mm portions. The highest y-activity, i.e. the highest particle flux was attributed to the arc core. The measurement of the absolute intensity of uranium line spectra for the calculation of ng was described elsewhere [9]. Figure 2 shows the schematic wire displacement for measuring the uranium particle flux at various heights (z) in the plasma. [g] R. Am,Spectrochim.Acta2SB, 619 (1968).
Transport of uranium free particles in a d.c. arc plasma
245
Fig. 2. Schematic representationof wire displacement in the plasma. Wires 1, 2, 3, 4 at Merent 2. I
Fig. 3a. Schematic radial voltage distribution for measuring the plasma and core radii.
The plasma diameter at different heights was also measured by the %ire method” using the oscillogram given in Fig. 3. The z axis is indicated by each pea,k (cf. Fig. 3a). The width (2r,) of the arc plasma W&S measured for a given wire velocity [ 10, 111. This width is derived from the distance between the lower inflection points [lo] R. AVNI, Spe&ochim. Acta 23B, 697 (1968). [ll] R. A~NI, Thesis, Weizmaxm Institute of Science, Rehovoth (1970).
R. AVNI and Z. GOLDBART
246
Fig. 3b. Oscilloscopic picture of the wires for S measurement. Ordinate 10 V cm-l; Abscissa 5 ma cm-l. Left, the anode region. Right, the cathode region. (Fig. 3a), i.e. the entrance and the exit of each wire into and from the plasma. The arc core diameter (2rJ was obtained by measuring the width on the peak between the two higher inflection points (Fig. 3a) ; this change is assumed to be caused by the entrance of the wire into the arc core [lo, 111. In cylindrical symmetry the width thus obtained represents the diameter of the core. 4. RESULTS AND DISCTJSSION 4.1 Particle velocity The slopes of the lines in Figs. 4 and 5 give the values of v,/2D by Equation (4). From this parameter, the axial particle velocity v, is obtained after calculation of the diffusion coefficient D. Using Equation (6) and the measured temperature [ll] of the different regions in the plasma (1 < r Q 3 mm and z up to 8-O mm) D was calculated. Figure 6 is a plot of the experimental values of the axial velocity of the uranium particle. Within the experimental error, v, is constant over the arc gap except for the anode region. This indicates that the assumption vj is constant for the two solutions [Equations (4) and (5)] of Equation (3) is adequate for a real d.c. arc plasma except in the vicinity of the anode. By addition of 4 % UF, to the U,O, in the anode crater, the volatilization rate, Qj, of uranium particles is increased [12]. Figure 6 shows that this did not affect the axial velocity of the uranium particles. 4.2 Total particle concerdration in the plasma The radial distribution of the particle flux nt(T, z)vj measured on the activated aluminium wires is given in Figs. 4 and 5. In a previous paper [6] the proportionality factor between the particles collected by the wire and the particle flux in the plasma, [12]
R. AVNI and A. BOTJKOBZA,Spectvdhn. Acta 24B, 515 (1969).
Transport of uranium free particles in a d.c. arc plasma
Distance
from arc axis,
Fig. 4. Radial distribution of uranium from 2.0~mm wire portions. Gap = 110 cm&. Distance above anode: z = 2 = 6.9 A; 2
247
mm
(U,O,), n$(z, r)v,, with z parameter as 8.0 mm; i = 10 A; carriage velocity 0.2 mm +; z = 2.2 mm 0; z = 4.0 W; = 7.8 mm 0.
was shown to be near unity within experimental error. In other words the wires measure the absolute values of the particle flux in the plasma. Using the values of w,the experimental total particle concentrations were obtained from the particle flux. Table 1 shows the axial distribution of the total particle concentration (n,) in an 8-Omm arc gap as measured from the wires together with the particle concentration (nj) calculated from the absolute intensity of the uranium lines 4289 and 4310& According to this table the molecular concentration of uranium in the arc core of the plasma (R = 2-O mm) is about 20 times that of the atom and ion concentration. The “wire method” does not specify the kind of uranium molecules found in the plasma, but only their sum. Carbides, oxides and nitrides of uranium may be in the plasma [13-141. The high concentration of molecules and the relatively low concentration of uranium atoms and ions explain the relatively high temperatures of the plasma in spite of the low first ionization potential of uranium 6.22eV [15]. The agreement between Equation (5)and the radial plots in Figs. 1, 4 and 5 is [13] H. NICKEL, Spectrochim. Acta 21,363 (1965). [14] R. RAUTSCHKE, Spectrochim. Acta 23B, 65 (1968). [la] J. B. MANN,J. Chma. Phys. 40,1632 (1964).
248
R. Aarr
tend Z. GOLDBART
Distance from arc axis,
mm
Fig. 5. Radial distribution of uranium (UaOr, + 4% UP,), n&z, r)~,, with z pammeter aa derived from 2.O-mm wire portions. Gap = 8-Omm; i = 10 A; carriage velocity 110 cm.s+. Distance above the anode: z = 0.2 mm +; z = 2.2 mm 0; z = 4.0 mm n ; z = 6.9 mm A; z = 7*8mm 0.
._ i \ \ 500
T \,
4,;
,,,,,, 4
6
Distance.2.
mm
2 Anode
6 Cathode
Fig. 6. Axial distribution of uranium particle 4%,‘,uF,.
velocity
(21,) A
UsOa; H rr,O, +
Transport of uranium free particles in a d.c. arc plasma
249
Table 1. Axial distribution of total particle concentration. Arc gap = 8-Omm; P = 2.0 mm; i = 10 A; 35 s exposure Line intensity (b) method
Wire method (a) 2,
mm
nt
x lo-r1 crnw3
n.3 X lo-r1 crnm3
Matrix Anode 0.0
nt -nj Iti
-
270.0 110.0 90.0 110*0 60.0 -
nt x lo-r1 crnm3 Matrix U,O, + 4% UF,
u303
-
0.2 1.8 3.5 6.0 7.8 Cathode 8.0
Wire method (a)
-
11.0 6.0 5.0 4.0 3.0 -
-
23.5 17.3 17.0 26.5 19.0 -
300.0 150.0 100.0 lOO*O 80.0 -
(a) Mean value over 4 carriages of wires; relative standard deviation 25 ‘A. (b) Mean values over 4 spectra (d.c. arc focused on slit); relative standard deviation 20%.
shown in Table 2. For a closer comparison the 2*0-mm wire portions with the highest y-activity, were cut into 05-mm sections. As described, the ratio of n,(z, r)/Q, calculated by Equation (5) with the experimental values of v,/2D is compared with the same ratio measured by the wire method. The results in Table 2 show good agreement between the values of n,/Q, obtained by the two methods. Table 3 shows the radial distribution of nt of uranium in the anode, central and cathode regions of the plasma as described previously [9, lo]. The radial diffusion of uranium particles, which is strong in the anode region, decreases towards the cathode. Along the axis a lower concentration was found in the cathode region compared to the nt values in the anode region. The decrease of the particle concentration towards the cathode is caused by radial Table 2. Ratio nt(z, r)/Q, calculated with the Bessel solution (Equation 6) and measured by the wire method at z = 1.5 mm above the anode in an 8.0 mm gap. Crater radius (a) = 2.0 mm Equation (5)* Radius
mm 0.00 0.25 0.50 0.75 I.00 1.25 I.60
Temperature K 6700 6600 6300 5900 5500 6000 4600
Wire method
4Qr x 1O-88.cm+ 7.54 7.62 7.90 8.16 8.15 8.07 7.45
Wire portion@ mm
Q, x 10-14 s-1
% x 10-l'
cm-3$
M&r x
IO8 s.om-*
05
2.8
20.0
7.15
I.0
6.5
60.0
7.70
1.5 2.0
11.0 13.0
85.0 100.0
7.70 7.69
* Computer calculations using experimental values of vJ2D. t The wires were cut into 4 portions. Each wire portion was measured separately for yradiation. $ Relative standard deviation from 4 wires 25 %.
250
R. AVNI and Z. GOLDBART Table 3. Radial distribution of nt. Arcgap=S*Omm; i=lOA Plasma region Anode[l”l
Centre[lol
Cathode[lO]
rnun
n, x lo-l1 cmW3
o-2 2-4 4-6 o-2 24 4-6 o-2 2-4 4-6
270.0 35.0 5.0 100.0 45.0 9.0 60.0 30.0 7.0
Relative standard deviation % (a) 25 25 25 25 35 35 25 35 35
(a) The relative standard deviation is obtained from the passage of 4 carriages. (16 wires in the anode region; 8 wires in the centre region; 4 wires in the cathode region.) Table 3a. Normalized values of the particle concentration,nt/nt (anode). Cf. Table 3. Arc gap = 8.0 mm; i = 10 A nt/nt (anode) Plasma region (r) (2)
O-2 mm
2-4 mm
4-6 mm
Anode Centre Cathode
1.0 0.37 0.22
1.0 1.30 0.85
1.0 1.80 1.40
diffusion of particles. So the particle concentration in the vicinity of the axis (r = O-2 mm) decreases continuously with increasing height above the anode. Table 3a illustrates this effect and shows particle concentrations that have been normalized to the corresponding values in the anode region. For r = 2-4 mm and r = 4-6 mm a slight increase toward the cathode was obtained. Diffusion apparently proceeds so that the particle concentration decreases steeply only in the axial zone and does not vary appreciably in the outer zones. 4.3 Transport parameter of uranium (y) The radial and axial transport parameter of free uranium particles was calculated using Equation (1). Table 4 shows the radial distribution of yj in the anode, central and cathode regions of the plasma. The increase of yu,outside the arc core (r > 2 mm) shows the radial transport in the arc mantle. The discrepancy in the transport parameter calculated by Q,/ n, with v,S indicates that the second way cannot be used for computing the radial distribution of y, because:
(i)
w
vjS was assumed to hold only for a vertical movement of the particles in the cylinder. The 2 mm wire sections used to measure the particle flux do not represent any real radius for the plasma cross section calculation; the wires could not be cut into smaller portions to obtain a detailed picture over the arc cross section.
251
Trsnsport of umnium free particles in e d.c. sre plssms Table 4. Rsdid distribution of y,. Arc gap 8.0 mm; i = 10 A
centre Cathode
130
48.0 370.0 130.0 290.0 220.0 420.0
o-2 2-4 o-2 24 o-2 2-4
Anode
120 280
* Man v&w over 4 esrrkges of wires. Relative stsndsxd de&&ion 25 % in the anode region and 30 % in the centre cbndcathode regions. t Relative standard deviation 40 % from 4 passages of wires (Figs. 3 and 3a).
Figure 7 shows the axial distribution of the transport parameter in the arc core, for a radius of r = 2 mm, when U,O, and U,O, + 4 %UF, were initially placed in the anode cavity. The Merence in y, for U,O, and U30, + 4%UF,, at the same axial velocity (Fig. 6) and at almost constant value of n, (Table I), shows that 8 higher volatilization rate Qj resulted from the addition of UF, to the U,O, matrix
lw
The transport parameter for aluminium evaluated by DE GALAN [1] [Equation (2)] yAl = 2-7 x lo6 em3 s-l for a mixture of Al,O,/C = 8 is very high compared to the values given in Fig. 7. Al0 molecules which were found in a d.c. arc burning in air [16] were not computed in his nj values i.e. the values of cn,are too small. The
I I
I
3
I
1
I
8
7
5
Anode
Cathode Distance,z,
mm
Fig. ‘7. Axial distribution of yj. A U,O, 110 cm s-1 carriage velocity; 0 UaC, 250 cm s-r carriagevelocity; n TJsO, + 4 % UF,, 110 cm s-1 carriage velocity. [ls] R. W. 8. PEARSE snd A. G. GAYDON, The I~~~~~~ t Hall Ltd. London (1965).
of ~0~~
Sp&rs, Chapman
R. AVNI and Z. GOLDBART
252
volatilization rate of Al,O, from the anode is estimated as being of the same order of magnitude as that for UaO,. The val values of DE GALAN[l] are too high due to the summation of only atoms and ions for particle concentration. The decrease of nt towards the cathode was used for analysing impurities in uranium [lo, 121 exposing on the photographic plate only the cathode region instead of the whole arc gap. In the cathode region the uranium lines spectra obtained is less dense leaving the impurity spectral lines undisturbed. APPENDIX
If we assume that Equation (4) describes the spatial distribution of n, and that vj varies only with z, the flux equation can be expressed as
Q(x, r) = n,(z, r) . v&) =
Q&)
47rDdi373
exp
The slope of the curve ln @ vs. r is obtained by differentiating In Q with respect to r: am
-=-
ar
&+
-
1
$4 [ l/z2 + p-2 + 2
1*
For the anode region therefore
v,/2D was evaluated by point to point calculation over all the experimental points with subtraction of the mean value of l/r in each point. For T > 0.4 cm, v,/2D can be directly derived from Figs. 4 and 5, within an error of 15 %. The solution for the anode region is valid for z = 0.2 cm with the same error as for z = 0. For higher z values the simplification used is:
for dm > O-6 cm, within the same error of 15 %. The computation of v,/2D was done after dividing the value of the slope between every two points by the mean value of