Electron vibrational energy transfer under MHD combustion generator conditions

Energy Conversion. Vol. 15, pp. 15-19. Pergamon Press, 1975. Printed in Great Britain

Electron Vibrational Energy Transfer Under MHD Combustion Generator Conditions C, D, HAWLEYtand M, MITCHNERt (First received 26 February 1974; revised 27 January 1975)

Abstract--An analysis of the energy transfer rate between an electron gas and the vibrational modes of diatomic gases has been made. The electron vibrational energy transfer rate may be characterized by G, the electron vibrational energy loss factor. For MI-iD combustion generator conditions, the electron vibrational energy transfer rate dominates and G is approximately equal to 8, the total electron energy loss factor. Detailed numerical predictions have been obtained for Ns and CO. These results indicate that the energy transfer rate is strongly dependent upon the electron velocity distribution function. In particular, for a Maxwellian electron distribution function (which prevails in MHD combustion generators) G has been found to exceed published experimental values of ~ by a factor as large as 50 for N~. Consideration of the conditions under which the experimental values of ~ were obtained shows that the electron distribution function was non-Maxwellian. An analytical re-determination of G and 3, using appropriate non-Maxwellian distribution functions, resolves the discrepancy between the calculated values of G and the measured values of & Energy loss factor Vibrational energy loss factor equilibrium Non-Maxwellian electron distribution

MHD generators Electron molecule collisions Electron nonElectron transport coefficients Nitrogen Carbon monoxide

1. Introduction The total electron energy loss factor 6 has been used to estimate electron non-equilibrium in M H D combustion generators. Crompton and Sutton have experimentally determined 6 for N2, and Demetriades has also experimentally found 6 for many gases, including N2 and CO. Calculations of the electron vibrational energy loss factor 6v have been given by Mnatsakanyan and by Demetriades. For M H D combustion generator conditions, the vibrational energy transfer rate dominates and 6v is approximately equal to & Mnatsakanyan assumed all electron vibrational rate coeffÉcients are given in terms of the rate coefficient for an electron to excite the molecule from the ground vibrational level to the first excited vibrational level. Although the values of 3v derived by Mnatsakanyan agree roughly with those presented in the present work, experimental evidence does not support Mnatsakanyan's assumption about the rate coefficients. Further his theory is valid only for low (300K) vibrational temperatures. Demetriades assumed transitions occur only between adjacent vibrational levels. Experimental cross sections for transitions between non-adjacent vibrational levels are of the same order of magnitude as cross sections for transitions between adjacent vibrational levels. Demetriades' values of G are significantly less for most cases than the values derived in the present work. In this study, the energy transfer rate from an electron gas to the vibrational modes of gaseous diatomic molecules, characterized by 3v, is calculated from electron vibrational excitation cross sections averaged over all electron energies using appropriate electron distribution functions. The energy-dependent electron-molecule

m o m e n t u m transfer cross section is also needed to determine G. The vibrational energy levels of the diatomic molecules are assumed to be equally spaced and in equilibrium with each other at a vibrational temperature Tv.

2. Theory The model utilized to calculate the energy transfer rate from the translational motion of the electrons to the vibrational modes of a diatomic gas is the following. The electron gas is characterized by an electron energy distribution function f(~). The vibrational energy levels of the diatomic molecules are assumed to be equally spaced and in equilibrium with each other at a vibrational temperature Iv. The electron vibrational energy transfer rate may be characterized by 3v, the electron vibrational energy loss factor, defined by the relation

~,,

~ =

n,~e(~

-

~kT~)

6v.

(1)

Here /~e, v is the net energy per unit volume per unit time transferred from the electron gas to the vibrational modes of the molecules. The quantity ne is the electron number density, ve is the energy-averaged electronmolecule momentum transfer collision frequency, ~ is the average electron energy and k is Boltzmann's constant. Similarly the total electron energy transfer rate due to all processes (elastic, rotational, vibrational, electronic, etc.) may be characterized by 3, the total electron energy loss factor, defined by the relation

~,

=

n,~,(~

-

-~kTg) 6.

(2)

Here/~e is the net energy per unit volume per unit time transferred from the electron gas to all modes of the molecules. The quantity Tg is the gas (translational)

t Department of Mechanical Engineering, High Temperature Gas-dynamics Laboratory, Stanford University, Stanford, California 94305, U.S.A. 15

16

C . D . HAWLEY and M. M I T C H N E R

temperature. From Equations (1) and (2) if Tv is equal to To then 8 is greater than or equal to By. (For purely elastic collisions 8 equals 2me/raM, where me and mM are the masses of the electron and the molecule respectively.) Later discussion indicates 3,, is approximately equal to a for conditions of interest. From a detailed consideration of energy transfer due to electrons inducing transitions from the jth to the lth vibrational levels, it may be shown that l~e, v = nenM~v 1 -- exp ~

j = l l>j

threshold (0"3 eV < e < 1"5 eV) behavior of the cross section Q~2,v(~) was taken from Burrow and Schulz, and the electron-Nz momentum-transfer collision cross section as given by Frost and Phelps was employed. The electron vibrational excitation cross sections Qell v(E) where l = 2-9 for CO were again taken from Schulz. No theoretical cross sections of the form Q~Z,v(~) where j :~ 1 and l :A 1 were available, so these cross sections were (implicitly) set equal to zero. The electron-CO momentum-transfer collision cross section was taken from Hake and Phelps. 4. Results for a Maxwellian Electron Distribution

(3) Here nM is the molecular number density and ev is the energy spacing between adjacent vibrational levels The rate coefficient S~l, v is given by the relation v:

(2 f -\me/

f ( , ) ¢l/ZQJet, v(') d,.

If the electron distribution function is Maxwellian then 8v may be calculated as a function of Te and Tv using the cross-section data discussed above. Figure 1 displays 8v as a function of Te on curves of constant Tv for N2. Similarly Fig. 2 gives 8v as a function of Te on curves of constant Tv for CO.

(4)

I

i

I

f

r

5162i

o

The electron energy distribution f(~) is normalized by the relation

2.16 2

gO

f f ( , ) d e = I.

(5)

o

i(5 2

The quantity Q~t, v(~) is the cross section for an electron of energy e to excite the molecule from the jth to the lth vibrational level (or to de-excite, if l < j). Excitation and de-excitation cross sections are related (due to detailed balancing) by the equation (for j < l)

5.s6 3 3,,,. 2.1(5 3

• QJet, v(,) = [, -- (l -- y) ¢vlO['¼ v[, - (t - j ) ,v].

(6)

The above equations simplify if the electron distribution is Maxwellian and can therefore be characterized by an electron temperature Te. The electron Maxwellian distribution is given by fM(e) = ~

(kTe) -3/2 ,1/z exp ~ e

"

id 3

5.16 4

(7) 2.1(~ 4

Thus the average electron energy ~ becomes ~kT. and (for j < l) S' t , v = e x p ( -(l-j)kTe ~v) l~Se,v.

(8)

L~

Tv : 3 0 0 K

__1 2000

~ _ _ _

I 6000

I

I tO000

r

T~, i,:; Fig. l. The electron vibrational energy loss factor for N~ calculated assuming a Maxwellian electron distribution function (2 m e / m M = 3"89 × 10-5).

To determine /~e, v, the electron vibrational cross For both N2 and CO, 8v rises very rapidly with sections ~.,, O jt A%), the electron energy distribution func- increasing electron temperature until Te ~ 5000K. This tion f(~) must be known. It is necessary to also use Oe behavior may be related to the structures of the electron to calculate By. vibration excitation cross sections which exhibit a similar rapid rise with increasing electron energy. For 3. Cross-Section Data typical MHD combustion temperatures (3000 K) calculaThe electron vibrational excitation cross sections tions indicate that electron energy transfer to rotational Q~t, v(E) for Nz where l = 2-9 have been measured by and electronic modes is unimportant (less than 5 per Schulz. These cross sections, with their magnitudes cent) compared to electron vibrational energy transfer. scaled up by a factor of 1"86 as suggested by Englehardt, Therefore calculated values of av are approximately Phelps and Risk, were utilized in the calculations equal to 8. No known direct measurements of a have v(E) and a~, v(,) where j :A 1 and l :A 1, been reported for N2 and CO under conditions where below. For Qe,1:10 Chen's theoretical cross sections were used. The the electron distribution function was clearly Maxwellian.

Electron Vibrational Energy Transfer Under M H D Combustion Generator Conditions '

I

I

I

17

frequency re(c) is assumed constant, then

1

5.10-2

ve(O =

e

~e----

(13)

,

md~e and

2.1() z

1(~2

De = ~ e --, /ze

(14)

where De is the electron diffusion coefficient. Then a may be expressed in the form

5.16 3

a=

3(D, )"

05)

V 'm"

e-- -- kT# tZe

2.1(53

1(53 5,1() 4

2.1C74

i

I 6000

2000

Te,

i

I I0000

K

Fig. 2. The electron vibrational energy loss factor for CO calculated assuming a Maxwellian electron distn'bufion function (2 mdraM = 3.89 X 10-5).

5. Results for a Non-Maxwellian E l e c t r o n Distribution

The total electron energy loss factor a has been experimentally found for Nz both by Crompton and Sutton and by Demetriades. Consideration of the conditions under which the experimental values of ~ were obtained shows that the electron distribution function was nonMaxwellian. For brevity, only N2 is discussed in detail here. Results for CO will be briefly presented at the end of this section. Crompton and Sutton have determined a for room temperature N2 by equating the total energy transfer rate Be with the Joule dissipation rate according to the relation 1~, - - J ` 8 _

creE ~.

(9)

Crompton and Sutton measured Dellze as a function of the applied electric field, and utilizing Nielsen's measurements of Ue, found 8 as a function of the applied electric field. Further to find a as a function of the electron 'temperature' the average energy ~ was set equal to ~ k T e (valid only if the electron distribution is Maxwellian). The method of data reduction of Crompton and Sutton makes two assumptions. One assumption is that the elastic collision frequency is a constant, when in fact it is a sharply peaked function. The second assumption is that the electron distribution is Maxwellian. It has been shown by Nighan that, due to the relatively low fractional ionization typical of low pressure molecular gas discharges ( n e / n u < 10-~), the electron distribution function is clearly non-Maxwellian. Englehardt, Phelps and Risk have compiled the data of Crompton and Sutton and of Nielsen as well I

t

I

~

I

(from Fig. l)

i6 2

a //-

5.1o-3 /

~,,,B

/ /

oe

Here Je is the electron current density, ere is the electron electrical conductivity and E is the applied electric field. The electrical conductivity can be written in terms of the electron mobility t~e as ere = enet~e,

L

5.10"z

/

2.1(~3

/ / /

l(~ 3

/

/

/

/ l

(10)

5.10-4.

(11)

2,10-4

l I

\

where e is the electron charge and where i~e = Ue/E.

The quantity Ue is the electron drift velocity. Therefore, from equations (2), (9-11) the following formula for a may be deduced a=

eUeE

02)

If the energy-dependent momentum-transfer collision 2

I';,--; .2

,

.6

I

I 1.0

I

2/3 ~ , eV

Fig. 3.

Comparison of the theoretical

electron

vibrational

energy loss factor av (taken from Fig. 1) with the mmlsured total electron energy loss factor 8 for Ns. The theoretical 8v is calculated assuming a Maxwellian electron distribution function. The measured a I~ obtained from experlinents similar to those of Crompton and Sutton with the data reduced assum/ng a constant electron collision frequency (T~ : Tg = 300 K).

18

C.D.

HAWLEY

and M. MITCHNER

as more recent measurements of De/tze and Ue. They have curve-fit these data over a wide range of applied electric fields. Figure 3 displays ~v taken from Fig. I (calculated assuming a Maxwellian distribution) contrasted with 8 which we have calculated from the electron transport coefficient data of Englehardt, Phelps and Risk using the method of data reduction of Crompton and Sutton [Equation (15)]. Two assumptions were made above in comparing 8 with By. We may examine first Crompton and Sutton's assumption that the electron distribution function was Maxwellian. From Equations (2) and (12), we can write t~e nen M

--

I

J

Io"~

I

5.10~'~

2" I0a~" lOa-'

measured values of E e / n e n ~

--

5.1d 361 o

calculated volues of Ee,v/ne n M

2.1C~3~

1636

I

i

2

E -nM

I

2.1o-2 +

Io -2

+/

/

5.10 -3 8,

Bv

/

2.10-3

/°

+8

5.10- 4

o

8v

- - - curve fit for By

2.jd4

I .2

z

I .6

l

I 1.0

z/3 g, eV Fig. 5. Comparison of the total electron energy loss factor a with the electron vibrational energy loss factor 8v for N~. The quantity a is determined from experimental drift velocity data and non-Maxwellian electron distributions, and ao is calculated using these same non-lVlaxwellian electron distributions (Tv = Tq = 300 K).

1

2.1633

%

I

(16)

rIM

~

r

i 0 -3

efeE

Measurements of Ue may therefore be viewed as providing a direct measure of the parameter l~e/nenM, independent of any assumptions concerning ~e. The corresponding parameter Lee,v/nenM may be calculated from Equation (3), employing the non-Maxwellian distribution functions of Nighan and of Englehardt, Phelps and Risk calculated for the experimental conditions under which Ue was measured. This comparison is shown in Fig. 4. The agreement is good and I

I

i

I

6 -zo mz ,10 v o l t s -

I0

F i g u r e 4. C o m p a r i s o n of the measured total electron energy transfer rate for N~ with the theoretical electron vibrational energy transfer rate calculated using non-Maxwellian electron distribution functions (Tv = Ta = 300 K).

supports the validity of (a) the expression provided by Equation (3), (b) the electron vibrational excitation cross sections used and (c) the calculated non-Maxwellian electron distributions used. We may also conclude that values of ~ determined under experimental conditions similar to those of Crompton and Sutton cannot be assumed to apply to other situations where the electrons may have a Maxwellian distribution. Figure 5 shows By re-computed using the above

mentioned non-Maxwellian distribution functions compared with a calculated dropping the constant re(E) assumption. To determine ~e and ~ the non-Maxwellian distributions were utilized. Thus a is now partially experimentally (Ue) and partially theoretically (~e and g) determined. Figure 5 shows good agreement between theory and experiment and supports the validity of the method of calculating ~v as provided by Equations (3) and (1). Demetriades has determined 8 experimentally for various electron and vibrational 'temperatures' using a microwave technique. Electron temperatures ranged between 3200 K and 7600 K, while Tv ranged between 1700 K and 6100 K. A typical value of ne/nM given by Demetriades is 10-G. Nighan has calculated electron distribution functions in an e - - N 2 plasma with ne/nM < 10-5. Nighan's work shows that for values of "Te' and Tv typical of Demetriades' experiments the electron distribution function is non-Maxwellian. Since Demetriades' experimental conditions satisfy the criterion ne/nM < 10-5, Nighan's results would indicate that Demetriades' measurements of 3 were also made under conditions where the electron distribution function was non-Maxwellian. This conclusion is further supported by a quantitative criterion for the existence of a Maxwellian distribution given by Demetriades when this criterion is evaluated for Demetriades' experimental conditions. Figure 6 displays ~ as measured by Demetriades versus 2/3g, plotted without regard for variation in Tv. Also shown for comparison are curves for ,~v calculated assuming Maxwellian and non-Maxwellian electron distributions. The total energy loss factor ~ as derived

Electron Vibrational Energy Transfer Under M H D Combustion Generator Conditions l

5.10-2

i

I

5.16z

'

I

19

I

-

t

1

~vl(Maxwellian f - from F i g . 2 ) - ~

4-

~v (Moxwellicln f - from Fig.l)

2d0-2

/

2.1() 2 +

i02

/ +/~i /,

[

5.10-3

2.1(ff3

/

/

/

// //

// /

5.10-3

/,

(reduced from e x p e r assuming Z/e = consf.]

i/

2.16 3 . ×

/

xX X X X , ,,~/

5.1(94

j.ij

/

/

IG 3 ×x

/

////

i0-2

v (non-Moxwelfian ffrom Fig.5) /

//

Sv(n~o-Mo*~ellio~fl

+

o

/ ,o

x6/~

163

~"

+

' ~ " ' ~ ( r e d u c e d from exper, assuming , Ue =const.- from Fig.3)

x

x

3¢

5.1G 4

x

/

x x

2.1(:54

.2

.6

1.0

z/3 ~ , eV

Fig. 6. Comparison for N~ of the total electron energy loss factor ~ as measured by Demetriades ( x , plotted without regard for variation in T~) with other values of 8 and the electron vibrational energy loss factor ~v (Tv = T~ = 300 K except for Demetriades' values of 8). The symbol O represents ~v (taken from Fig. 5) calculated assuming non-Maxwellian electron distributions, and 4- represents 8 (taken from Fig. 5) reduced from experimental drift velocity data and using non-Maxwellian electron distributions.

from electron transport data at Tv = 300K either assuming ~e is constant, or taking into account the energy dependence, is also shown. Demetriades' values of 3 are much less than 3v calculated assuming a Maxwellian distribution. Although a strict comparison is not possible (because no electron distribution functions for high (Tv > 1000K) vibrational temperatures are available), there is a rough correspondence between Demetriades' values and the non-Maxwellian 3v curve (--------). The differences between Demetriades' values and the non-Maxwellian 3v curve likely result from two effects. First, Demetriades' measured values of 3 correspond to values of Tv in the range between 1700 K and 6100 K, whereas the 3v curve was calculated for Tv = 300 K. Since 3v increases with increasing Tv (see Fig. 1) Demetriades' values of ~ would be expected to be larger than the 3v curve as shown. Secondly includes rotational energy transfer while 3v does not, which again would explain larger values of 8, particularly at lower average electron energies. Results for CO, corresponding to those shown in Fig. 6 for Nz, are displayed in Fig. 7 compared with Demetriades' values of ~ for CO.

2.1G4

.2

I

I .6

L

l 1.0

z/3 ~ , eV Fig. 7. Comparison for CO of the total electron energy loss factor 8 as measured by Demetriades ( × , plotted without regard for variation in Tv) with other values of 8 and the electron vibrational energy loss factor ~v (Tv = Tg = 300 K except for Demetriades' values of 8). The symbol O represents 8v calculated assuming non-Maxwellian electron distributions, and 4- represents 8 reduced from experimental drift velocity data and using non-Maxwelllan electron distributions.

6. Conclusions

The electron vibrational energy loss factor 8v (approximately equal to the total energy loss factor 8) has been determined for N2 and CO. These results indicate that 8v is strongly dependent upon the electron velocity distribution function. In particular, for a Maxwellian electron distribution function (which prevails in M H D combustion generators) ~v has been found to exceed published experimental values of 8 for Nz by a factor as large as 50. Consideration of the conditions under which the experimental values of 8 were obtained shows that the electron distribution function was nonMaxwellian. An analytical re-determination of 8v and ~, using appropriate non-Maxwellian distribution functions, resolves the discrepancy between the calculated values of ~v and the measured values of 8. For calculations for Nz and CO involving ~ under M H D combustion generator conditions it is recommended that the results presented in Figs. 1 and 2 be employed. Acknowledgements--This work was supported by the Air Force Office of ScientificResearch under Contract No. AFOSR 1:4462073-C-0045.