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The effect of electrohydrodynamic boundary layers on heat transfer from flame gases
COMBUSTION A N D F L A M E 29, 33-41 (1977)
33
The Effect of Electrohydrodynamic Boundary Layers on Heat Transfer from Flame Gases R. M. CLEMENTS Department of Physics. University of Victoria, Victoria. B.C., V819 2 Y2, Canada
and P. R. SMY Department of Electrical I:)tgineering, University QtAlberta. Edmonton. Alta., T6G 2El, Canada
The effects of dc electric fields on the heat transfer to a cylindrical calorimeter totally immersed in a seeded propane-air flame are investigated. Because of the enhanced ionization, the calorimeter, when biased negatively, is shown to behave as a cylindrical Langmuir probe in a plasma, with large electric fields only in the immediate vicinity of the calorimeter. The increase in heat transfer is strictly a boundary layer phenomenon; it reaches a maximum of about 10% at the highest attainable density (~1018/m 3) and shows good agreement with a simple theory based on sheath-modified flow.
I. INTRODUCTION It is well known that the application of a static electric field to a flame can produce substantial changes in the flame. Not only can the flame shape be altered but also, because gas circulation can be changed, one can exercise a reasonably large degree of control (typically a factor of two) over the heat transfer from the hot flame gases to a solid surface. This was originally discussed in detail by Payne and Weinberg [1] and a recent review of this subject is contained in the book by Lawton and Weinberg [2]. The mechanism involved is the effect of the electric field on the flame ions in changing the gas flow and hence the heat transfer. For the force to exist there must of course be a net surplus of charge in the flame. This readily arises from the unequal mobilities in the positively and negatively charged species.
In a very recent paper Sandhu and Weinberg [3] discuss the effect of the inevitable entrainment of cold gas in an electrostatically accelerated flame. It is concluded that such entrainment will result in a decrease in heat transfer unless the flame is of large cross section. All the effects reported and discussed in the paper relate to the bulk motion of the flame. The authors state that changes in the heat transfer caused by electric fields in the immediate vicinity (that is essentially in the boundary layer) of the surface have a negligible effect. The work described in the present paper is in some ways similar to that of Sandhu and Weinberg but approaches the subject from a quite different viewpoint. In the past few years electrostatic probe theory has been developed to the state where models exist which take account of the fact that most high-pressure plasmas possess significant
Copyright © 1977 by The Combustion Institute Published by Elsevier North-Holland, Inc.
34
_
_
_
R.M. CLEMENTS and P. R. SMY
_
"AL ., -///\,
.,"..
-. _
Fig. l. Simple physical model for positive ion collection when convection dominates the ion flux to the sheath edge. In the area outside the sheath the ion and electron densities are equal, whereas inside the sheath the electron density is zero and the ion density is less than its free stream value.
flow velocities. It is found that experimental measurements which show as much as order-of-magnitude discrepancies with the conventional highpressure stationary plasma theory are found to give good agreement with simple theories which take account of plasma flow in the vicinity of the probe. The basic model which has been developed is shown in Fig. 1 and is described in detail elsewhere [4-6]. For completeness, the physical situation is briefly discussed here. The plasma is considered to consist of positive ions and electrons, the sheath which surrounds the negative probe consists of ions whose flow to the probe is governed by the usual space charge equations. The supply of ions to the sheath edge is due to their convection by the plasma. The thicknesses of any hydrodynamic or thermal boundary layers are considered to be less than the sheath thickness. Within the sheath, there will be large electric fields which, in combination with the ions within the sheath will, by ion drag, alter the flow and heat transfer at the probe. This situation differs from that considered by Sandhu and Weinberg in several ways. First, in their type of flame, it is evident that negative ions exist in substantial quantities and therefore in strict numerical terms the relations for current as a function of density, etc., may require some alteration. Secondly, it is evi-
dent that at their voltages and densities they assume correctly that sheath effects can be neglected (from their figures of 1 /~A/cm 2 and 8000 V the sheath thickness would be ~6 cm). The experimental situation we report in this paper is the one depicted in Fig. 1, i.e., a probe surrounded by a sheath where the sheath is small compared with the flame size. In our case, the previous measurements have shown that the propane-air flame which we use can be considered as consisting of positive ions and electrons only. The electric force within the sheath (where there are no electrons) is naturally related to the product of the ion density and electric field. In order to obtain significant effects while remaining at low sheath radii and probe potentials we have increased the ionization density of the flame by seeding with potassium atoms. By varying the quantity of seed material it is thus possible to vary the sheath radius while keeping the probe potential constant. As an illustration of how thin a sheath one can obtain in this way it is instructive to note that in one case, gas breakdown across the sheath was obtained at voltages as low as 500 V (detailed sheath breakdown studies have been published elsewhere [7]). This value contrasts with potentials of up to 16 kV used by Sandhu and Weinberg. The difference in the two approaches can probably best be summed up by stating that our approach is a "plasma" approach. As such, virtually all the applied voltage appears across a relatively thin sheath with essentially none across the plasma. By ensuring (through seeding) that the sheath remains small with respect to the flame, it is possible to locate the electric field induced distortion of flow to the region away from the flame boundary. In this way the situation approaches a true boundary layer situation. Heat transfer is altered only in the sheath (or boundary layer) region and because there is essentially no field anywhere else, changes in the bulk properties of the flame, in gas circulation, or in cold gas entrainment do not occur. The preceding discussion has only considered the positive ion sheath surrounding an electrode and the implication of the associated strong electric fields on heat transfer. It is not unreason-
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
35
able that a somewhat analogous sheath of electrons should form around a positively biased electrode. However, because of the large difference between ion and electron mobilities the properties of the two sheaths will differ markedly. Compared to the ions, the much higher mobility of the electrons generates a much smaller drag force on the gas for a given current. Thus the direct effect of the electron sheath upon heat transfer will be relatively small. However, if ions are generated in the electron sheath by either thermal or strong electric field processes then they will be repelled from the probe and will transfer their associated momentum to the gas molecules. For such a situation one would expect a decrease in heat transfer.
ing Eq. (2) the electric and hydrodynamic Reynolds numbers are considered equal, which is a realistic approximation for the types of flames considered here. The electric fields within the sheath are of order E ~ V/y s and exert a compressive pressure ½eoEz upon the sheath ions which in turn is transferred via ion drag to the gas well inside the sheath. In practical flow situations the sheath is at its thinnest at the leading edge of the body and becomes progressively thicker towards the rear. In consequence, gas convected into the sheath experiences a s u d d e n j u m p in pressure ~½eo Ez and then a gradual decay to a much lower figure. If we assume that viscous effects can be neglected then Bernoulli's equation holds and flow entering at uf will leave at ur' where
2. THEORY
_P
A solid body immersed in a flowing gas is surrounded by hydrodynamic and thermal boundary layers. The thickness (6) of these layers is usually very similar (because the Prandtl number for combustion gases is approximately one) and is given by 5 ~rp/R 1/2
(1)
where R is the hydrodynamic Reynolds number and rp is the electrode radius. If the gas is ionized and the probe is biased to a negative voltage V then it is also surrounded by a sheath of positive ions whose thickness Ys can be calculated for the case when convection is the mechanism supplying ions to the sheath. By combining Eq. (1) and Eq. (13) of Ref. [4] we obtain for this situation: Ys ~ 5 [REO~2X2] 1/4
(12)
where RE, the electric Reynolds number is, Vfrp/p(kT/e), a, the dimensionless length, is XD/rp and X, the dimensionless potential, is eV/kT. Here of is the flow velocity in the far field, /l is the ion mobility, k is Boltzmann's constant, T is the electron temperature, e is the electronic charge and XD is the Debye length (XD = (eokT/ nee2) 112, where e 0 is the permittivity of free space and ne is the ionization density). In obtain-
(vf '2
vf 2) -
2
eo E2
(3)
2
We assume here that the gas density p remains constant, i.e., that the velocity is small compared with the speed of sound. It should be noted that it is only reasonable to neglect viscous effects if the quantity 5 ~ Ys, i.e., REa2X 2 >> 1. With heat transfer to a tube varying as (flow velocity) 1/2 (see for example [8] ), we see that Eq. (3) implies an increase AQ (~ Q) in the heat transfer to the tube of AQ
1 eoE2
Q
4 pvf 2
(4)
In the absence of recombination effects, the sheath electric fields can be calculated (from measured current and voltage) using the conventional planar space-change collision-dominated expression [9]
j -
9 pe o V2
(5)
8 ys a where j is the ion current density. This is the basis of the various sheath-convection probe expressions which have been shown to work at densities up to 102°/m 3. However, the conditions of the experiment described later are somewhat different from
36
R.M. CLEMENTS and P. R. SMY
those investigated elsewhere. Thus, although the density (~101S/m 3) has been used earlier, the combination of low flow velocity, large probe and probe bias has not been used before. With a probe diameter ~ ½ cm, a flow velocity ~ 3 m/sec we see that the residence time of gas within the sheath region surrounding the probe is of order 2 × 10- a sec. With an ionization density = 2 × 1018/m 3 and a recombination coefficient a ~ 2 × 10 - 1 4 m3/sec the recombination and regeneration time r ~ 2 × 10- 5 sec. In consequence, we can expect the sheath structure to differ from that assumed in the derivation of Eq. (5). However, differentiation of Eq. (5) for the sheath-convection case and a similar analysis for the case when regeneration effects are important (see Ref. [10] for a discussion of the regeneration model and the relevant equations) the electric field at the electrode surface will be
E=A
F-/-V l a / 3
(6)
LeoUJ
where the coefficient A has the value of 1.44 and 1.08, respectively, for the sheath-convection and the regeneration case. The value o f ] here will, of course, depend on whether sheath-convection or regeneration effects dominate. The point is that in either case the appropriate value o f / i n Eq. (6) will be the measured value. It should also be emphasized that the planar analysis can be considered to be reasonably accurate if the sheath is not much larger than the probe. Combining Eqs. (4) and (6) we have
Q ~ 4 \pvf 2] \eo---p/
"
(7)
As noted previously the possibility of ions being created within the sheath is highly significant if we consider the situation of a positively biased electrode. In this case, if ions are created close to the electrode, the action of the radial electric field will be to generate an outward pressure which may reduce the heat transfer to the electrode.
3. EXPERIMENTAL The experimental situation was in many ways similar to that used by Sandhu and Weinberg. The burner used was a commercially manufactured Meker burner. The fuel, propane, was burned with air into which a mist of potassium hydroxide could be introduced. This seeding technique allowed us to attain ionization density levels from lO15/m 3 (unseeded flame) to 1018/m 3. The ionization was measured with a small rotating ion probe using the method and theory previously described [4]. The air/fuel ratio was always approximately stoichiometric-specifically there was no evidence of any charged carbon particles. In fuel rich flames, it has been shown [1] that these particles contribute strongly to any body force on the flame. Numerous measurements using both Langmuir probes and microwave techniques have indicated that, unlike the situation investigated by Sandhu and Weinberg, in our situation the negative charge carriers are essentially all electrons and few negative ions exist. The constant flow calorimeter was a thinwalled stainless-steel tube which was centered in the flame. The low thermal conductivity of stainless steel minimized the heat losses by conduction to those parts of the calorimeter not immersed in the flame. The calorimeter was cooled by dry high-pressure (up to 7 kg/cm 2) air with a flow rate between 0.3 and 0.6 liter/sec (the value depending primarily on the bore of the calorimeter tube). Problems of inadequate mixing which concerned previous reseachers [1,3] who used water as a coolant, were avoided by using air as the coolant. Also, by not using water as a coolant, there were no problems in obtaining good electrical isolation of the calorimeter. The air flow was measured with a standard commercial rotameter while the temperature increase of the air was monitored with iron-constantan thermocouples at the inlet and exhaust of the calorimeter. The output of the thermocouples, which was recorded on a chart recorder, corresponded to air temperature increases of between 60 ° and 100°C degrees. The calorimeter was biased by an unregulated power supply charging a small (0.01 pF) capacitor
ELECTROHYDRODYNAMIC BOUNDARY LAYERS which was directly connected between the calorimeter and the reference electrode. With such an arrangement the onset of breakdown was unmistakable. The potential of the calorimeter and the current between it and the reference electrode were monitored. Besides using the burner top as a reference electrode a cooled copper tube, a cooled stainless-steel grid, and a hot tungsten wire were also used as reference electrodes. These electrodes were normally positioned downstream of the calorimeter so as not to perturb the hydrodynamic flow of the flame gases. 4. RESULTS
As was discussed in the Section 1, the difference between the present study and that of previous workers [1, 3] is that in our experiment the working fluid is a plasma. Thus, a potential applied between two electrodes in the plasma results in strong electric fields (i.e., sheath formation) which are located close to one or both of the electrodes; the field in the bulk plasma is weak and hence it is only in the sheath region where body forces exist which are large enough to materially alter the heat transfer. In order to verify that this was indeed the case and that the observed changes in heat transfer were not being caused by bulk motion of the flame, cold gas entrainment or alteration of the combustion process the following preliminary tests were performed. By using a small movable probe the potential distribution between the calorimeter and the reference electrode was recorded. For the case when the calorimeter was biased negatively such a measurement has already been reported [7] and as expected, essentially all the potential drop appears across the relatively thin sheath surrounding the negative electrode. The field in the bulk plasnra is extremely weak, in fact with the relatively crude technique which we used it was not possible to measure any potential change for different probe positions in the bulk plasma. The calculated electric field in the bulk plasma for our typical conditions is of the order of 5 V/m and as such it seems reasonable that we are unable to measure it using our experimental technique.
37 In the opposite case, when the calorimeter is biased positively, the majority of the potential drop appears as expected, in the vicinity of the reference electrode. In this case the field in the bulk plasma is appreciable (200-300 V/cm). However, at the calorimeter there is a very thin region of high electric field whose thickness is ~1 nun. Clearly, at least for the case when the calorimeter is biased negatively, there are large fields only in the region close to the calorimeter. The variation of measured sheath thickness at the leading edge of a negative probe as a function of ionization density is shown for two values of bias in Fig. 2. Also shown are the calculated values of the sheath thickness based on a model in which regeneration of ions in the sheath is the dominant process [10]. As is evident, the measured sheath thicknesses are in reasonable agreement with those predicted by the model. As another check that bulk (as opposed to sheath) hydrodynamic effects are not responsible for our results, the reference electrode, when the calorimeter was biased negatively, was changed from the burner top to an electrode above the calorimeter. Heat transfer was found to be independent of this change and also independent of the separation between calorinreter and the reference electrode. Presumably a very different result would have been obtained in the situation investigated by Sandhu and Weinberg. Finally, an electrode whose dimensions were the same as the calorimeter was placed close to the calorimeter (but not so close that its sheath would overlap the calorimeter). No change in heat transfer to the calorimeter was observed when this electrode was biased negatively. This again reinforces the view that the application of an electric field did not perturb the bulk plasma properties. Measurements of the increase in heat transfer ~Q with negative bias for a relatively large diameter calorimeter is shown in Fig. 3. Similarly in Fig. 4 we show the increase AQ as a function of ionization density. Also shown in these figures are the values of AQ predicted by Eq. (7). Measurements for a smaller-diameter calorimeter showed a similar trend, however absolute agree-
38
R.M. CLEMENTS and P. R. SMY I
I
!
I
I
I
I
I
I
l
I
i I. 2
I
I
I
I
I
I
I
v
] -r w "1-
0.5
I 0.4
I
I 0.6
J
I 0.8
IONIZATION
i
I 1.0
DENSITY
I
1.4 ne
1.6
1.8
( x 1018/m 3 )
Fig. 2. Comparison of measured sheath thicknesses and those calculated from Ref. [10] for two values of probe bias. The curves are calculated while the crosses are measured values for a negative bias of 500 V and the dots for a negative bias of 300 V. The c o m m o n parameters are: rp = 2.54 × 10 _ 2 m, electrode length = 3.2 × 10 - 2 m, a = 1.5 X 10 - 1 4 m3/sec and/a = 2 × 10 - 3 m2/V.sec.
20
I
I
I
I 400 POTENTIAL
I 500
A
,~ 10
v
0 0 <3
5
2 200
I 300 BIAS
V
(v01ts)
Fig. 3. Variation of heat transfer as a function of negative electrode bias. The curve is from Eq. (7) using the measured electrode currents and p = 0.24 kg/m 3 (scaled for temperature increase from a STP value of 1.2 kg/m3), of = 3 m/sec, and ~ = 2 × 10 - 3 m2/V.sec. The points are experimental for rp = 2.54 × 10 - 3 m and n e = 1.9 × 10--18/ m 3 .
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
39
20
I0 A
o
<~
2
I
0.4
I
I
I
I
I
1
I
0.6
0.8
1.0
1.2
1.4
1.6
1.8
IONIZATION
DENSITY
ne
2.0
(xlOIS/m 3 )
Fig. 4. Variation of heat transfer as a function of ionization density for a negative bias of 525 V. The curve is from Eq. (7) using the measured electrode currents and p = 0.24 kg/m 3 (scaled for temperature increase from a STP value of 1.2 kg/m3), of = 3 m/sec, and/1 = 2 x 10 - 3 m2/V-sec. The points are experimental for rp = 2.54 X 10 - 3 m.
ment between these measurements and Eq. (7) was somewhat poorer than that indicated by Figs. 3 and 4. This is not unreasonable because with the smaller calorimeter the sheath thickness was considerably greater than the probe radius and hence the simple planar analysis presented in this paper could not be expected to apply. Also, as is implied by Fig. (4), for a flame of low ionization density (n e < 1 0 - 1 S / m ) w e were unable to detect any change in heat transfer; a result in complete agreement with the findings of previous researchers [1-3] who used unseeded, low ionization density flames. We also biased the calorimeter positively with respect to a reference electrode above it. (The burner could not be used as a reference here because the large fields in the region of the burner top strongly altered the appearance of the reaction zone.) A decrease in heat transfer, whose magni-
tude was of the same order as the observed increase in heat transfer, was measured. The effect depended only on the applied voltage and not on the ionization density (as long as this density was low enough so that a large voltage-i.e., greater than a few kilovolts, could be applied and breakdown not take place). The effect was also independent of the calorimeter diameter for the sizes investigated (1-5 ram). These results tend to suggest that it is not a sheath effect which is responsible for the decrease in heat transfer. To test this hypothesis we repeated one of the tests described at the beginning of this section. Here the calorimeter was not biased but was close to an electrode which, for the present case, was positively biased. Unlike the results obtained when the calorimeter was biased negatively, there was a change in heat transfer, i.e., it decreased. Hence the evidence strongly indicates that cold gas en-
40 trainment, as described by Sandhu and Weinberg, is the dominant mechanism responsible for our observations. Because the cooling which we observed does not appear to be a plasma/sheath phenomena our subsequent discussion is specific to the increase in heat transfer.
5. DISCUSSION The measurements in Fig. 2 confirm the plasma nature of the working fluid in our measurements. The observed sheath thicknesses are found to be small compared to the electrode separation as they are calculated to be. Calculations of the sheath thicknesses for previous measurements [3] however confirm the assumptions made in those measurements, i.e., that the sheaths were large compared with the probe separation. Agreement between our simple planar theory and experiment in Figs. 3 and 4 is as good as can be expected. It is clear from Fig. 2 that the sheath geometry is in fact not planar since the sheath radius is about twice the probe radius. Figures 3 and 4 do establish that substantial perturbation of heat transfer due to sheath effects is expected and does occur under our experimental conditions. The effect of hydrodynamic boundary layer flow upon the ion current to a probe is now fairly well understood [11- 15]. In this paper we have investigated the inverse effect, i.e., the effect of probe generated electric forces upon the hydrodynamic boundary layer. As Sandhu and Weinberg point out, such an interdependent system is a very complex one to analyse in an accurate manner and here we are content to obtain a very approximate estimate of what might be expected in a sheath/boundary layer situation. That such a situation can exist does however seem well established by our measurements of sheath thickness. It is clear that in these experiments, unlike those of Sandhu and Weinberg, the electrostatic forces upon the plasma are located to the vicinity of the probe. For the situation which we have investigated to be of practical interest it is clear that one must have ionization densities well in excess of those found in the usual unseeded combustion reaction. Such densities do occur in devices which
R.M. CLEMENTS and P. R. SMY are of practical interest such as plasma jets, welding arcs and also in combustion MHD plasmas. For the latter, ionization densities over an order of magnitude greater than the maximum values attained in this experiment are typical. However in combustion MHD, flow velocities are also much higher (typically two orders of magnitude). The net result then is that it is somewhat questionable whether the effects considered in the present paper will materially alter the heat transfer to the electrodes. Nevertheless it is possible that in some situations of interest heat transfer could be altered by the effects considered here. On the other hand, welding arcs possess flow velocities which are low ( ~1 m/sec or less), hence in this case heat transfer might be augmented by the electric fields. 6. CONCLUSIONS The work described here is a logical extension of previous measurements by Weinberg and coworkers [ 1 - 3 ] . The work covers a situation which will eventuahy be reached with increasing gas temperature, i.e., a situation where the ionization density becomes so large that plasma/sheath effects intrude. Our approach here is an approximate one, at even higher densities a more accurate planar analysis which takes full account of the coupling between the ion and fluid flows might well be warranted. From our measurements we can state with some assurance that the heat transfer perturbation is in good agreement with calculations based upon ion drag forces in the sheath. While cooling could, in principle, be caused by a somewhat similar sheath effect, in which positive ions created in the electron sheath would result in a net outwardly directed force, in practice our measured cooling seems to be readily accounted for by the well established ideas relating to cooling by cold gas entrainment. However, there may well be some electrode/plasma situations in which this type of cooling is important. The authors wouM like to acknowledge the assistance o f J. R. Ramsay and the financial assistance o f the National Research Council o f Canada and the Universities o f Victoria and Alberta.
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
REFERENCES 1. Payne, K. G., and Weinberg, F. J., Proc. Roy. Soe. (London) A250, 316 (1959). 2. Lawton, J., and Weinberg, F. J., Electrical Aspects of Combustion, Clarendon Press, Oxford, 1969. 3. Sandhu, S. S., and Weinberg, F. J., Combust. Flame 25,321 (1975). 4. Ciements, R. M., and Smy, P. R., J. AppL Phys. 41, 3745 (1970). 5. Clements, R. M., and Smy, P. R., Proc. lEE 117, 1421 ~1970). 6. Clements, R. M., and Smy, P. R., Can. J. Phys. 49, 2540 (1971). 7. Clements, R. M., and Stay, P. R., J. Phys. D: Appl. Phys. 6, 1253 (1973). 8. Schlichting, H., Boundary-Layer Theory, McGrawHill, New York, 1968.
41 9. Cobine, J. D., Gaseous Conductors, Dover, New York, 1958, p. 129. 10. Ciements, R. M., Oliver, B. M., Noor, A. 1., and Smy, P. R., Electron. Lett., in press. 11. Lam, S.H.,AIAAJ. 2,256 (1964). 12. Clements, R. M., and Smy, P. R., J. Appl. Phys. 44, 3550 (1973). 13. Vasilieva, I. A., and Nefedov, A. P., International Conference on Gas Discharges, London, England, Sept. 15-18, 1970, IEE, London, 1970, p. 162. 14. Ashin, M. 1., Vasilieva, I. A., and Nefedov, A. P., High Temp. 7, 574 (1969). 15. Chung, P. M., Talbot, L., and Touryan, K. J., Electric Probes in Stationary and Flowing Plasmas: Theory and Application, Springer-Verlag, New York, 1975. Received 10 June 19 76; revised 7 October 19 76
33
The Effect of Electrohydrodynamic Boundary Layers on Heat Transfer from Flame Gases R. M. CLEMENTS Department of Physics. University of Victoria, Victoria. B.C., V819 2 Y2, Canada
and P. R. SMY Department of Electrical I:)tgineering, University QtAlberta. Edmonton. Alta., T6G 2El, Canada
The effects of dc electric fields on the heat transfer to a cylindrical calorimeter totally immersed in a seeded propane-air flame are investigated. Because of the enhanced ionization, the calorimeter, when biased negatively, is shown to behave as a cylindrical Langmuir probe in a plasma, with large electric fields only in the immediate vicinity of the calorimeter. The increase in heat transfer is strictly a boundary layer phenomenon; it reaches a maximum of about 10% at the highest attainable density (~1018/m 3) and shows good agreement with a simple theory based on sheath-modified flow.
I. INTRODUCTION It is well known that the application of a static electric field to a flame can produce substantial changes in the flame. Not only can the flame shape be altered but also, because gas circulation can be changed, one can exercise a reasonably large degree of control (typically a factor of two) over the heat transfer from the hot flame gases to a solid surface. This was originally discussed in detail by Payne and Weinberg [1] and a recent review of this subject is contained in the book by Lawton and Weinberg [2]. The mechanism involved is the effect of the electric field on the flame ions in changing the gas flow and hence the heat transfer. For the force to exist there must of course be a net surplus of charge in the flame. This readily arises from the unequal mobilities in the positively and negatively charged species.
In a very recent paper Sandhu and Weinberg [3] discuss the effect of the inevitable entrainment of cold gas in an electrostatically accelerated flame. It is concluded that such entrainment will result in a decrease in heat transfer unless the flame is of large cross section. All the effects reported and discussed in the paper relate to the bulk motion of the flame. The authors state that changes in the heat transfer caused by electric fields in the immediate vicinity (that is essentially in the boundary layer) of the surface have a negligible effect. The work described in the present paper is in some ways similar to that of Sandhu and Weinberg but approaches the subject from a quite different viewpoint. In the past few years electrostatic probe theory has been developed to the state where models exist which take account of the fact that most high-pressure plasmas possess significant
Copyright © 1977 by The Combustion Institute Published by Elsevier North-Holland, Inc.
34
_
_
_
R.M. CLEMENTS and P. R. SMY
_
"AL ., -///\,
.,"..
-. _
Fig. l. Simple physical model for positive ion collection when convection dominates the ion flux to the sheath edge. In the area outside the sheath the ion and electron densities are equal, whereas inside the sheath the electron density is zero and the ion density is less than its free stream value.
flow velocities. It is found that experimental measurements which show as much as order-of-magnitude discrepancies with the conventional highpressure stationary plasma theory are found to give good agreement with simple theories which take account of plasma flow in the vicinity of the probe. The basic model which has been developed is shown in Fig. 1 and is described in detail elsewhere [4-6]. For completeness, the physical situation is briefly discussed here. The plasma is considered to consist of positive ions and electrons, the sheath which surrounds the negative probe consists of ions whose flow to the probe is governed by the usual space charge equations. The supply of ions to the sheath edge is due to their convection by the plasma. The thicknesses of any hydrodynamic or thermal boundary layers are considered to be less than the sheath thickness. Within the sheath, there will be large electric fields which, in combination with the ions within the sheath will, by ion drag, alter the flow and heat transfer at the probe. This situation differs from that considered by Sandhu and Weinberg in several ways. First, in their type of flame, it is evident that negative ions exist in substantial quantities and therefore in strict numerical terms the relations for current as a function of density, etc., may require some alteration. Secondly, it is evi-
dent that at their voltages and densities they assume correctly that sheath effects can be neglected (from their figures of 1 /~A/cm 2 and 8000 V the sheath thickness would be ~6 cm). The experimental situation we report in this paper is the one depicted in Fig. 1, i.e., a probe surrounded by a sheath where the sheath is small compared with the flame size. In our case, the previous measurements have shown that the propane-air flame which we use can be considered as consisting of positive ions and electrons only. The electric force within the sheath (where there are no electrons) is naturally related to the product of the ion density and electric field. In order to obtain significant effects while remaining at low sheath radii and probe potentials we have increased the ionization density of the flame by seeding with potassium atoms. By varying the quantity of seed material it is thus possible to vary the sheath radius while keeping the probe potential constant. As an illustration of how thin a sheath one can obtain in this way it is instructive to note that in one case, gas breakdown across the sheath was obtained at voltages as low as 500 V (detailed sheath breakdown studies have been published elsewhere [7]). This value contrasts with potentials of up to 16 kV used by Sandhu and Weinberg. The difference in the two approaches can probably best be summed up by stating that our approach is a "plasma" approach. As such, virtually all the applied voltage appears across a relatively thin sheath with essentially none across the plasma. By ensuring (through seeding) that the sheath remains small with respect to the flame, it is possible to locate the electric field induced distortion of flow to the region away from the flame boundary. In this way the situation approaches a true boundary layer situation. Heat transfer is altered only in the sheath (or boundary layer) region and because there is essentially no field anywhere else, changes in the bulk properties of the flame, in gas circulation, or in cold gas entrainment do not occur. The preceding discussion has only considered the positive ion sheath surrounding an electrode and the implication of the associated strong electric fields on heat transfer. It is not unreason-
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
35
able that a somewhat analogous sheath of electrons should form around a positively biased electrode. However, because of the large difference between ion and electron mobilities the properties of the two sheaths will differ markedly. Compared to the ions, the much higher mobility of the electrons generates a much smaller drag force on the gas for a given current. Thus the direct effect of the electron sheath upon heat transfer will be relatively small. However, if ions are generated in the electron sheath by either thermal or strong electric field processes then they will be repelled from the probe and will transfer their associated momentum to the gas molecules. For such a situation one would expect a decrease in heat transfer.
ing Eq. (2) the electric and hydrodynamic Reynolds numbers are considered equal, which is a realistic approximation for the types of flames considered here. The electric fields within the sheath are of order E ~ V/y s and exert a compressive pressure ½eoEz upon the sheath ions which in turn is transferred via ion drag to the gas well inside the sheath. In practical flow situations the sheath is at its thinnest at the leading edge of the body and becomes progressively thicker towards the rear. In consequence, gas convected into the sheath experiences a s u d d e n j u m p in pressure ~½eo Ez and then a gradual decay to a much lower figure. If we assume that viscous effects can be neglected then Bernoulli's equation holds and flow entering at uf will leave at ur' where
2. THEORY
_P
A solid body immersed in a flowing gas is surrounded by hydrodynamic and thermal boundary layers. The thickness (6) of these layers is usually very similar (because the Prandtl number for combustion gases is approximately one) and is given by 5 ~rp/R 1/2
(1)
where R is the hydrodynamic Reynolds number and rp is the electrode radius. If the gas is ionized and the probe is biased to a negative voltage V then it is also surrounded by a sheath of positive ions whose thickness Ys can be calculated for the case when convection is the mechanism supplying ions to the sheath. By combining Eq. (1) and Eq. (13) of Ref. [4] we obtain for this situation: Ys ~ 5 [REO~2X2] 1/4
(12)
where RE, the electric Reynolds number is, Vfrp/p(kT/e), a, the dimensionless length, is XD/rp and X, the dimensionless potential, is eV/kT. Here of is the flow velocity in the far field, /l is the ion mobility, k is Boltzmann's constant, T is the electron temperature, e is the electronic charge and XD is the Debye length (XD = (eokT/ nee2) 112, where e 0 is the permittivity of free space and ne is the ionization density). In obtain-
(vf '2
vf 2) -
2
eo E2
(3)
2
We assume here that the gas density p remains constant, i.e., that the velocity is small compared with the speed of sound. It should be noted that it is only reasonable to neglect viscous effects if the quantity 5 ~ Ys, i.e., REa2X 2 >> 1. With heat transfer to a tube varying as (flow velocity) 1/2 (see for example [8] ), we see that Eq. (3) implies an increase AQ (~ Q) in the heat transfer to the tube of AQ
1 eoE2
Q
4 pvf 2
(4)
In the absence of recombination effects, the sheath electric fields can be calculated (from measured current and voltage) using the conventional planar space-change collision-dominated expression [9]
j -
9 pe o V2
(5)
8 ys a where j is the ion current density. This is the basis of the various sheath-convection probe expressions which have been shown to work at densities up to 102°/m 3. However, the conditions of the experiment described later are somewhat different from
36
R.M. CLEMENTS and P. R. SMY
those investigated elsewhere. Thus, although the density (~101S/m 3) has been used earlier, the combination of low flow velocity, large probe and probe bias has not been used before. With a probe diameter ~ ½ cm, a flow velocity ~ 3 m/sec we see that the residence time of gas within the sheath region surrounding the probe is of order 2 × 10- a sec. With an ionization density = 2 × 1018/m 3 and a recombination coefficient a ~ 2 × 10 - 1 4 m3/sec the recombination and regeneration time r ~ 2 × 10- 5 sec. In consequence, we can expect the sheath structure to differ from that assumed in the derivation of Eq. (5). However, differentiation of Eq. (5) for the sheath-convection case and a similar analysis for the case when regeneration effects are important (see Ref. [10] for a discussion of the regeneration model and the relevant equations) the electric field at the electrode surface will be
E=A
F-/-V l a / 3
(6)
LeoUJ
where the coefficient A has the value of 1.44 and 1.08, respectively, for the sheath-convection and the regeneration case. The value o f ] here will, of course, depend on whether sheath-convection or regeneration effects dominate. The point is that in either case the appropriate value o f / i n Eq. (6) will be the measured value. It should also be emphasized that the planar analysis can be considered to be reasonably accurate if the sheath is not much larger than the probe. Combining Eqs. (4) and (6) we have
Q ~ 4 \pvf 2] \eo---p/
"
(7)
As noted previously the possibility of ions being created within the sheath is highly significant if we consider the situation of a positively biased electrode. In this case, if ions are created close to the electrode, the action of the radial electric field will be to generate an outward pressure which may reduce the heat transfer to the electrode.
3. EXPERIMENTAL The experimental situation was in many ways similar to that used by Sandhu and Weinberg. The burner used was a commercially manufactured Meker burner. The fuel, propane, was burned with air into which a mist of potassium hydroxide could be introduced. This seeding technique allowed us to attain ionization density levels from lO15/m 3 (unseeded flame) to 1018/m 3. The ionization was measured with a small rotating ion probe using the method and theory previously described [4]. The air/fuel ratio was always approximately stoichiometric-specifically there was no evidence of any charged carbon particles. In fuel rich flames, it has been shown [1] that these particles contribute strongly to any body force on the flame. Numerous measurements using both Langmuir probes and microwave techniques have indicated that, unlike the situation investigated by Sandhu and Weinberg, in our situation the negative charge carriers are essentially all electrons and few negative ions exist. The constant flow calorimeter was a thinwalled stainless-steel tube which was centered in the flame. The low thermal conductivity of stainless steel minimized the heat losses by conduction to those parts of the calorimeter not immersed in the flame. The calorimeter was cooled by dry high-pressure (up to 7 kg/cm 2) air with a flow rate between 0.3 and 0.6 liter/sec (the value depending primarily on the bore of the calorimeter tube). Problems of inadequate mixing which concerned previous reseachers [1,3] who used water as a coolant, were avoided by using air as the coolant. Also, by not using water as a coolant, there were no problems in obtaining good electrical isolation of the calorimeter. The air flow was measured with a standard commercial rotameter while the temperature increase of the air was monitored with iron-constantan thermocouples at the inlet and exhaust of the calorimeter. The output of the thermocouples, which was recorded on a chart recorder, corresponded to air temperature increases of between 60 ° and 100°C degrees. The calorimeter was biased by an unregulated power supply charging a small (0.01 pF) capacitor
ELECTROHYDRODYNAMIC BOUNDARY LAYERS which was directly connected between the calorimeter and the reference electrode. With such an arrangement the onset of breakdown was unmistakable. The potential of the calorimeter and the current between it and the reference electrode were monitored. Besides using the burner top as a reference electrode a cooled copper tube, a cooled stainless-steel grid, and a hot tungsten wire were also used as reference electrodes. These electrodes were normally positioned downstream of the calorimeter so as not to perturb the hydrodynamic flow of the flame gases. 4. RESULTS
As was discussed in the Section 1, the difference between the present study and that of previous workers [1, 3] is that in our experiment the working fluid is a plasma. Thus, a potential applied between two electrodes in the plasma results in strong electric fields (i.e., sheath formation) which are located close to one or both of the electrodes; the field in the bulk plasma is weak and hence it is only in the sheath region where body forces exist which are large enough to materially alter the heat transfer. In order to verify that this was indeed the case and that the observed changes in heat transfer were not being caused by bulk motion of the flame, cold gas entrainment or alteration of the combustion process the following preliminary tests were performed. By using a small movable probe the potential distribution between the calorimeter and the reference electrode was recorded. For the case when the calorimeter was biased negatively such a measurement has already been reported [7] and as expected, essentially all the potential drop appears across the relatively thin sheath surrounding the negative electrode. The field in the bulk plasnra is extremely weak, in fact with the relatively crude technique which we used it was not possible to measure any potential change for different probe positions in the bulk plasma. The calculated electric field in the bulk plasma for our typical conditions is of the order of 5 V/m and as such it seems reasonable that we are unable to measure it using our experimental technique.
37 In the opposite case, when the calorimeter is biased positively, the majority of the potential drop appears as expected, in the vicinity of the reference electrode. In this case the field in the bulk plasma is appreciable (200-300 V/cm). However, at the calorimeter there is a very thin region of high electric field whose thickness is ~1 nun. Clearly, at least for the case when the calorimeter is biased negatively, there are large fields only in the region close to the calorimeter. The variation of measured sheath thickness at the leading edge of a negative probe as a function of ionization density is shown for two values of bias in Fig. 2. Also shown are the calculated values of the sheath thickness based on a model in which regeneration of ions in the sheath is the dominant process [10]. As is evident, the measured sheath thicknesses are in reasonable agreement with those predicted by the model. As another check that bulk (as opposed to sheath) hydrodynamic effects are not responsible for our results, the reference electrode, when the calorimeter was biased negatively, was changed from the burner top to an electrode above the calorimeter. Heat transfer was found to be independent of this change and also independent of the separation between calorinreter and the reference electrode. Presumably a very different result would have been obtained in the situation investigated by Sandhu and Weinberg. Finally, an electrode whose dimensions were the same as the calorimeter was placed close to the calorimeter (but not so close that its sheath would overlap the calorimeter). No change in heat transfer to the calorimeter was observed when this electrode was biased negatively. This again reinforces the view that the application of an electric field did not perturb the bulk plasma properties. Measurements of the increase in heat transfer ~Q with negative bias for a relatively large diameter calorimeter is shown in Fig. 3. Similarly in Fig. 4 we show the increase AQ as a function of ionization density. Also shown in these figures are the values of AQ predicted by Eq. (7). Measurements for a smaller-diameter calorimeter showed a similar trend, however absolute agree-
38
R.M. CLEMENTS and P. R. SMY I
I
!
I
I
I
I
I
I
l
I
i I. 2
I
I
I
I
I
I
I
v
] -r w "1-
0.5
I 0.4
I
I 0.6
J
I 0.8
IONIZATION
i
I 1.0
DENSITY
I
1.4 ne
1.6
1.8
( x 1018/m 3 )
Fig. 2. Comparison of measured sheath thicknesses and those calculated from Ref. [10] for two values of probe bias. The curves are calculated while the crosses are measured values for a negative bias of 500 V and the dots for a negative bias of 300 V. The c o m m o n parameters are: rp = 2.54 × 10 _ 2 m, electrode length = 3.2 × 10 - 2 m, a = 1.5 X 10 - 1 4 m3/sec and/a = 2 × 10 - 3 m2/V.sec.
20
I
I
I
I 400 POTENTIAL
I 500
A
,~ 10
v
0 0 <3
5
2 200
I 300 BIAS
V
(v01ts)
Fig. 3. Variation of heat transfer as a function of negative electrode bias. The curve is from Eq. (7) using the measured electrode currents and p = 0.24 kg/m 3 (scaled for temperature increase from a STP value of 1.2 kg/m3), of = 3 m/sec, and ~ = 2 × 10 - 3 m2/V.sec. The points are experimental for rp = 2.54 × 10 - 3 m and n e = 1.9 × 10--18/ m 3 .
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
39
20
I0 A
o
<~
2
I
0.4
I
I
I
I
I
1
I
0.6
0.8
1.0
1.2
1.4
1.6
1.8
IONIZATION
DENSITY
ne
2.0
(xlOIS/m 3 )
Fig. 4. Variation of heat transfer as a function of ionization density for a negative bias of 525 V. The curve is from Eq. (7) using the measured electrode currents and p = 0.24 kg/m 3 (scaled for temperature increase from a STP value of 1.2 kg/m3), of = 3 m/sec, and/1 = 2 x 10 - 3 m2/V-sec. The points are experimental for rp = 2.54 X 10 - 3 m.
ment between these measurements and Eq. (7) was somewhat poorer than that indicated by Figs. 3 and 4. This is not unreasonable because with the smaller calorimeter the sheath thickness was considerably greater than the probe radius and hence the simple planar analysis presented in this paper could not be expected to apply. Also, as is implied by Fig. (4), for a flame of low ionization density (n e < 1 0 - 1 S / m ) w e were unable to detect any change in heat transfer; a result in complete agreement with the findings of previous researchers [1-3] who used unseeded, low ionization density flames. We also biased the calorimeter positively with respect to a reference electrode above it. (The burner could not be used as a reference here because the large fields in the region of the burner top strongly altered the appearance of the reaction zone.) A decrease in heat transfer, whose magni-
tude was of the same order as the observed increase in heat transfer, was measured. The effect depended only on the applied voltage and not on the ionization density (as long as this density was low enough so that a large voltage-i.e., greater than a few kilovolts, could be applied and breakdown not take place). The effect was also independent of the calorimeter diameter for the sizes investigated (1-5 ram). These results tend to suggest that it is not a sheath effect which is responsible for the decrease in heat transfer. To test this hypothesis we repeated one of the tests described at the beginning of this section. Here the calorimeter was not biased but was close to an electrode which, for the present case, was positively biased. Unlike the results obtained when the calorimeter was biased negatively, there was a change in heat transfer, i.e., it decreased. Hence the evidence strongly indicates that cold gas en-
40 trainment, as described by Sandhu and Weinberg, is the dominant mechanism responsible for our observations. Because the cooling which we observed does not appear to be a plasma/sheath phenomena our subsequent discussion is specific to the increase in heat transfer.
5. DISCUSSION The measurements in Fig. 2 confirm the plasma nature of the working fluid in our measurements. The observed sheath thicknesses are found to be small compared to the electrode separation as they are calculated to be. Calculations of the sheath thicknesses for previous measurements [3] however confirm the assumptions made in those measurements, i.e., that the sheaths were large compared with the probe separation. Agreement between our simple planar theory and experiment in Figs. 3 and 4 is as good as can be expected. It is clear from Fig. 2 that the sheath geometry is in fact not planar since the sheath radius is about twice the probe radius. Figures 3 and 4 do establish that substantial perturbation of heat transfer due to sheath effects is expected and does occur under our experimental conditions. The effect of hydrodynamic boundary layer flow upon the ion current to a probe is now fairly well understood [11- 15]. In this paper we have investigated the inverse effect, i.e., the effect of probe generated electric forces upon the hydrodynamic boundary layer. As Sandhu and Weinberg point out, such an interdependent system is a very complex one to analyse in an accurate manner and here we are content to obtain a very approximate estimate of what might be expected in a sheath/boundary layer situation. That such a situation can exist does however seem well established by our measurements of sheath thickness. It is clear that in these experiments, unlike those of Sandhu and Weinberg, the electrostatic forces upon the plasma are located to the vicinity of the probe. For the situation which we have investigated to be of practical interest it is clear that one must have ionization densities well in excess of those found in the usual unseeded combustion reaction. Such densities do occur in devices which
R.M. CLEMENTS and P. R. SMY are of practical interest such as plasma jets, welding arcs and also in combustion MHD plasmas. For the latter, ionization densities over an order of magnitude greater than the maximum values attained in this experiment are typical. However in combustion MHD, flow velocities are also much higher (typically two orders of magnitude). The net result then is that it is somewhat questionable whether the effects considered in the present paper will materially alter the heat transfer to the electrodes. Nevertheless it is possible that in some situations of interest heat transfer could be altered by the effects considered here. On the other hand, welding arcs possess flow velocities which are low ( ~1 m/sec or less), hence in this case heat transfer might be augmented by the electric fields. 6. CONCLUSIONS The work described here is a logical extension of previous measurements by Weinberg and coworkers [ 1 - 3 ] . The work covers a situation which will eventuahy be reached with increasing gas temperature, i.e., a situation where the ionization density becomes so large that plasma/sheath effects intrude. Our approach here is an approximate one, at even higher densities a more accurate planar analysis which takes full account of the coupling between the ion and fluid flows might well be warranted. From our measurements we can state with some assurance that the heat transfer perturbation is in good agreement with calculations based upon ion drag forces in the sheath. While cooling could, in principle, be caused by a somewhat similar sheath effect, in which positive ions created in the electron sheath would result in a net outwardly directed force, in practice our measured cooling seems to be readily accounted for by the well established ideas relating to cooling by cold gas entrainment. However, there may well be some electrode/plasma situations in which this type of cooling is important. The authors wouM like to acknowledge the assistance o f J. R. Ramsay and the financial assistance o f the National Research Council o f Canada and the Universities o f Victoria and Alberta.
ELECTROHYDRODYNAMIC BOUNDARY LAYERS
REFERENCES 1. Payne, K. G., and Weinberg, F. J., Proc. Roy. Soe. (London) A250, 316 (1959). 2. Lawton, J., and Weinberg, F. J., Electrical Aspects of Combustion, Clarendon Press, Oxford, 1969. 3. Sandhu, S. S., and Weinberg, F. J., Combust. Flame 25,321 (1975). 4. Ciements, R. M., and Smy, P. R., J. AppL Phys. 41, 3745 (1970). 5. Clements, R. M., and Smy, P. R., Proc. lEE 117, 1421 ~1970). 6. Clements, R. M., and Smy, P. R., Can. J. Phys. 49, 2540 (1971). 7. Clements, R. M., and Stay, P. R., J. Phys. D: Appl. Phys. 6, 1253 (1973). 8. Schlichting, H., Boundary-Layer Theory, McGrawHill, New York, 1968.
41 9. Cobine, J. D., Gaseous Conductors, Dover, New York, 1958, p. 129. 10. Ciements, R. M., Oliver, B. M., Noor, A. 1., and Smy, P. R., Electron. Lett., in press. 11. Lam, S.H.,AIAAJ. 2,256 (1964). 12. Clements, R. M., and Smy, P. R., J. Appl. Phys. 44, 3550 (1973). 13. Vasilieva, I. A., and Nefedov, A. P., International Conference on Gas Discharges, London, England, Sept. 15-18, 1970, IEE, London, 1970, p. 162. 14. Ashin, M. 1., Vasilieva, I. A., and Nefedov, A. P., High Temp. 7, 574 (1969). 15. Chung, P. M., Talbot, L., and Touryan, K. J., Electric Probes in Stationary and Flowing Plasmas: Theory and Application, Springer-Verlag, New York, 1975. Received 10 June 19 76; revised 7 October 19 76