- Email: [email protected]
Comment on “laser-induced fluorescence measurement of sodium in flames”
COMBUSTION A N D FLAME 34:215-217 (1979)
215
COMMENT Comment on "Laser-lnduced Fluorescence Measurement of Sodium in Flames" R. P. L U C H T and N. M. L A U R E N D E A U
Combustion Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 4 7907
In their observations of fluorescence from the sodium 32p1/2, 3/2 doublet to the ground state 32S1/2 under saturated conditions, Daily and Chan [1] found that the ratio of fluorescence from each of the excited levels was dependent on which level was directly excited by the laser. This difference in fluorescence ratios for the two excitation cases, illustrated in Fig. 9 of Daily and Chan's paper [1], can be explained in detail by an appropriate examination of the rate equations and collisional cross sections for Na atoms in a fuel-air flame. Although Daily and Chan mention that this effect has been explained by Van Vitert et al. [2], we feel a more thorough discussion would be useful. When the 32PI/2 level is directly excited, the rate equations become [3] dN1 dt
can write Q21 _ gl e _ ( E l _ E 2 ) / k Q12 g2
T
(4)
Q31 = gl e_(E l _ E 3 )/k T Q13 g3
(5)
At a temperature of 1000°K, these ratios are 4 X 10 l ° and 2 X 10 l ° , respectively. Since at saturated conditions N1, N2, and N 3 will all be of the same order of magnitude, all terms in (1)-(3) involving collisional excitation from the ground level to the upper levels will be negligible. If we assume steady state, dN 1 dt
N1(B12121 + QI2 + Q13) + N2(B21121
-
dN2 dt
dN3
= ~
= O,
(6)
dt
we can solve (3) for N2/N3: +A21 +Q21)+N3(A31 +Q31) d.N2 -
-
=
dt
(1)
--N2(B21121 +A21 + 021 + Q2s) + Nl(B12/21 + Q12) "1"N3Q32
(2)
dNa_ -
-
-
dt
N2
Q31 +A31 + 032
N3
Q23
(7)
Based on the principle of detailed balancing, we can write
-'N3(Q31 +A31 + Q32) +N2Q23) +N1Q13,
(3)
where levels 1, 2, and 3 denote the 32Sl/2, 32P1/2, and 32P3/2 levels, respectively. From the principle of detailed balancing, we Copyright © 1979 by The Combustion Institute Published by Elsevier North Holland, Inc.
Q23 = g3 e_(E3_E2 )/k T, Q32 g2
(8)
where ga = 4, gz = 2, a n d E a - E 2 = 17 cm - 1 . At a temperature of 1000°K this ratio is equal to 1.95. For flame conditions we can approximate
0010-2180/79/020215+3501.75
216
R.P. LUCHT and N. M. LAURENDEAU
the ratio as Q2 3 g3 - -
-
-
in the flame. The rate constant Q is related to the collisional cross section a (cm 2) by 2.
(9)
Q32 g2
Qij = Nq . ] - oij, 'N 7rla
Substituting (9) into (7) and rearranging, we obtain N2 -- ½ + Qal +A31 lva Q23
(10)
Because levels 2 and 3 have nearly equal energies, the rates of quenching to the ground level should be approximately equal: Q31 = Q21.
(11)
Also, the Einstein coefficients for spontaneous emission are found to be nearly equal for each of the excited levels [4] : Aal =A21 = 6 × 107 sec- 1 .
(12)
Because the fluorescence intensity from each level will be directly proportional to the steadystate population of each level, we can write, for the case where level 2 is directly excited, /31 - - = (½ + 0/)- 1 ,
I21
(13)
where =
Q21 +A21
(14)
Q23 For direct excitation of level 3 we obtain /31 --
121
= 2 + 2c~
(15)
following a similar analysis. We can estimate the electronic quenching rate constants, Q21 and Qaa, and the inelastic collision rate constants, Qza and Qa2, by assuming that the sodium atoms collide only with nitrogen molecules
(16)
where Nq (molecules/cm a) is the number density of the quenching species (N2 in this case), # (g) is the reduced mass of the colliding species, and (8kT/Trla)l/2 (cm/sec) is the average relative velocity of the colliding species. Jenkins [5] reported an electronic quenching cross section (O3t)N z = (O21)N2 for Na-N2 collisions of 21.5 A2 for flame temperatures of 1400-1800°K. Stupavsky and Krause [6] reported inelastic collision cross sections (a23)N 2 and (O32)N 2 of 144 A2 and 76 A 2, respectively, at a temperature of 398°K. It is uncertain how these inelastic collision cross sections will vary with temperature. If we assume that: (1) the only quenching species is N2, (2) the flame temperature is approximately 1500°K at a pressure of one atmosphere, and (3) the cross sections (0"23)N2 and (0"32)N2 are approximately the same at 1500°K as at 398°K, we obtain (0"21 )N 2 0/-- - - - (o23)N 2
21.5 ---0.15, 144
since at these conditions the quenching rate constants are much greater than the rate constant for spontaneous emission. Thus we see that the order of magnitude of c~ should be about 0.1. The presence of other strongly quenching species such as CO, CO2, and O2 will have a strong effect on the value of 0/. It is possible that much of the scatter in Fig. 9 of Daily and Chan's paper [1] is due to the varying composition of the flame gases and the consequent variation in c~. Table 1 lists the results when ~ = 0.05, 0.1, 0.15, and 0.2 is substituted into equations (13) and (15). Cases 1 and 2 refer to direct excitation of levels 2 and 3, respectively. If we try to fit horizontal straight lines to the experimental data in Daily and Chan's Fig. 9, we obtain values of Ial/I21 of 1.6-1.7 and 2.1-2.2 for cases 1 and 2
COMMENT
217 TABLE 1
Therefore, lal/121 should be close to two for both cases.
a
131/121, Case 1
131//21, Case 2
0.05 0.10 0.15 0.20
1.82 1.67 1.54 1.43
2.10 2.20 2.30 2.40
respectively. For the flame conditions of Daily and Chan's experiment, it appears that a was approximately equal to 0.1. If the flame diluent gas were argon, the intensity ratio would be nearly equal for each case of direct excitation. For collisions of sodium atoms with argon [5, 7] (a21) Ar < 0.3 and (a2a) Ar = 110 A 2, SO a should be on the order of 0.001.
REFERENCE 1. 2.
3. 4.
5. 6. 7.
Daffy, J. W. and Chan, C., Combust. Flame, 33:4753 (1978). Van Vitert, B., Van Calcar, R. A., Hollander, T. J. and T. J. Alkemade, C. Th. J., private communication to J. W. Daffy, Utrecht, 1977. Daily, J. W., Applied Opt. 16, 568 (1977). Penner, S. S., Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, Mass., 1959, p. 24. Jenkins, D. R., Proc. Roy. Soc. (London) A293, 493 (1966). Stupavsky, M. and Krause, L., Can. J. Phys. 46, 2127 (1968). Pitre, J. and Krause, L., Can. J. Phys. 51, 334 (1971).
Received 12 May 1978; revised 19 July 1978
215
COMMENT Comment on "Laser-lnduced Fluorescence Measurement of Sodium in Flames" R. P. L U C H T and N. M. L A U R E N D E A U
Combustion Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 4 7907
In their observations of fluorescence from the sodium 32p1/2, 3/2 doublet to the ground state 32S1/2 under saturated conditions, Daily and Chan [1] found that the ratio of fluorescence from each of the excited levels was dependent on which level was directly excited by the laser. This difference in fluorescence ratios for the two excitation cases, illustrated in Fig. 9 of Daily and Chan's paper [1], can be explained in detail by an appropriate examination of the rate equations and collisional cross sections for Na atoms in a fuel-air flame. Although Daily and Chan mention that this effect has been explained by Van Vitert et al. [2], we feel a more thorough discussion would be useful. When the 32PI/2 level is directly excited, the rate equations become [3] dN1 dt
can write Q21 _ gl e _ ( E l _ E 2 ) / k Q12 g2
T
(4)
Q31 = gl e_(E l _ E 3 )/k T Q13 g3
(5)
At a temperature of 1000°K, these ratios are 4 X 10 l ° and 2 X 10 l ° , respectively. Since at saturated conditions N1, N2, and N 3 will all be of the same order of magnitude, all terms in (1)-(3) involving collisional excitation from the ground level to the upper levels will be negligible. If we assume steady state, dN 1 dt
N1(B12121 + QI2 + Q13) + N2(B21121
-
dN2 dt
dN3
= ~
= O,
(6)
dt
we can solve (3) for N2/N3: +A21 +Q21)+N3(A31 +Q31) d.N2 -
-
=
dt
(1)
--N2(B21121 +A21 + 021 + Q2s) + Nl(B12/21 + Q12) "1"N3Q32
(2)
dNa_ -
-
-
dt
N2
Q31 +A31 + 032
N3
Q23
(7)
Based on the principle of detailed balancing, we can write
-'N3(Q31 +A31 + Q32) +N2Q23) +N1Q13,
(3)
where levels 1, 2, and 3 denote the 32Sl/2, 32P1/2, and 32P3/2 levels, respectively. From the principle of detailed balancing, we Copyright © 1979 by The Combustion Institute Published by Elsevier North Holland, Inc.
Q23 = g3 e_(E3_E2 )/k T, Q32 g2
(8)
where ga = 4, gz = 2, a n d E a - E 2 = 17 cm - 1 . At a temperature of 1000°K this ratio is equal to 1.95. For flame conditions we can approximate
0010-2180/79/020215+3501.75
216
R.P. LUCHT and N. M. LAURENDEAU
the ratio as Q2 3 g3 - -
-
-
in the flame. The rate constant Q is related to the collisional cross section a (cm 2) by 2.
(9)
Q32 g2
Qij = Nq . ] - oij, 'N 7rla
Substituting (9) into (7) and rearranging, we obtain N2 -- ½ + Qal +A31 lva Q23
(10)
Because levels 2 and 3 have nearly equal energies, the rates of quenching to the ground level should be approximately equal: Q31 = Q21.
(11)
Also, the Einstein coefficients for spontaneous emission are found to be nearly equal for each of the excited levels [4] : Aal =A21 = 6 × 107 sec- 1 .
(12)
Because the fluorescence intensity from each level will be directly proportional to the steadystate population of each level, we can write, for the case where level 2 is directly excited, /31 - - = (½ + 0/)- 1 ,
I21
(13)
where =
Q21 +A21
(14)
Q23 For direct excitation of level 3 we obtain /31 --
121
= 2 + 2c~
(15)
following a similar analysis. We can estimate the electronic quenching rate constants, Q21 and Qaa, and the inelastic collision rate constants, Qza and Qa2, by assuming that the sodium atoms collide only with nitrogen molecules
(16)
where Nq (molecules/cm a) is the number density of the quenching species (N2 in this case), # (g) is the reduced mass of the colliding species, and (8kT/Trla)l/2 (cm/sec) is the average relative velocity of the colliding species. Jenkins [5] reported an electronic quenching cross section (O3t)N z = (O21)N2 for Na-N2 collisions of 21.5 A2 for flame temperatures of 1400-1800°K. Stupavsky and Krause [6] reported inelastic collision cross sections (a23)N 2 and (O32)N 2 of 144 A2 and 76 A 2, respectively, at a temperature of 398°K. It is uncertain how these inelastic collision cross sections will vary with temperature. If we assume that: (1) the only quenching species is N2, (2) the flame temperature is approximately 1500°K at a pressure of one atmosphere, and (3) the cross sections (0"23)N2 and (0"32)N2 are approximately the same at 1500°K as at 398°K, we obtain (0"21 )N 2 0/-- - - - (o23)N 2
21.5 ---0.15, 144
since at these conditions the quenching rate constants are much greater than the rate constant for spontaneous emission. Thus we see that the order of magnitude of c~ should be about 0.1. The presence of other strongly quenching species such as CO, CO2, and O2 will have a strong effect on the value of 0/. It is possible that much of the scatter in Fig. 9 of Daily and Chan's paper [1] is due to the varying composition of the flame gases and the consequent variation in c~. Table 1 lists the results when ~ = 0.05, 0.1, 0.15, and 0.2 is substituted into equations (13) and (15). Cases 1 and 2 refer to direct excitation of levels 2 and 3, respectively. If we try to fit horizontal straight lines to the experimental data in Daily and Chan's Fig. 9, we obtain values of Ial/I21 of 1.6-1.7 and 2.1-2.2 for cases 1 and 2
COMMENT
217 TABLE 1
Therefore, lal/121 should be close to two for both cases.
a
131/121, Case 1
131//21, Case 2
0.05 0.10 0.15 0.20
1.82 1.67 1.54 1.43
2.10 2.20 2.30 2.40
respectively. For the flame conditions of Daily and Chan's experiment, it appears that a was approximately equal to 0.1. If the flame diluent gas were argon, the intensity ratio would be nearly equal for each case of direct excitation. For collisions of sodium atoms with argon [5, 7] (a21) Ar < 0.3 and (a2a) Ar = 110 A 2, SO a should be on the order of 0.001.
REFERENCE 1. 2.
3. 4.
5. 6. 7.
Daffy, J. W. and Chan, C., Combust. Flame, 33:4753 (1978). Van Vitert, B., Van Calcar, R. A., Hollander, T. J. and T. J. Alkemade, C. Th. J., private communication to J. W. Daffy, Utrecht, 1977. Daily, J. W., Applied Opt. 16, 568 (1977). Penner, S. S., Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, Mass., 1959, p. 24. Jenkins, D. R., Proc. Roy. Soc. (London) A293, 493 (1966). Stupavsky, M. and Krause, L., Can. J. Phys. 46, 2127 (1968). Pitre, J. and Krause, L., Can. J. Phys. 51, 334 (1971).
Received 12 May 1978; revised 19 July 1978