Effect of Charge Mobility on Electric Conduction Driven Dielectric Liquid Flow

Transcription

1 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION Effect of Charge Mobility o Electric Coductio Drive Dielectric Liquid Flow Miad Yazdai ad Jamal Seyed-Yagoobi, Seior Member, IEEE Abstract Electrohydrodyamic (EHD) coductio umig is associated with the heterocharge layers of fiite thickess i the viciity of the electrodes, geerated by the rocess of dissociatio of the eutral electrolytic secies ad recombiatio of the geerated ios. EHD coductio geerated flow relies rimarily uo the asymmetry of the electrodes where the flow is always directed toward the secific directio regardless of the electrodes olarity. However, the differece i the mobilities of ositive ad egative charges could lay a imortat role whe studyig the umig erformace. This aer studies the effects of charge mobility ad its differece betwee the ositive ad egative charges o the heterocharge layer structure ad geerated flow. The umerical simulatio is coducted for a - D rectagular chael with the electrodes embedded agaist the chael wall. The et flow geerated for the symmetric electrodes is hysically illustrated while the imact of mobility differece o the geerated flow for the case of asymmetric electrodes is studied as well. Idex Terms EHD Coductio, dissociatio ad recombiatio, charge mobility. b +/ D d C e E E F (ω) I (z) k B k d k r P Re ehd T sat u NOMENCLATURE ositive/egative charge mobility coefficiet mobility ratio, = b + /b charge diffusio costat chael half height EHD dimesioless umber electro charge electric field vector electric field vector magitude I (ω)/ω modified Bessel fuctio of first tye ad order oe Boltzma uiversal costat dissociatio rate costat recombiatio rate costat egative charge desity uit ormal vector ositive charge desity ressure EHD Reyolds umber saturatio temerature velocity vector Mauscrit received Aril 7, 9. This work was fiacially suorted by the NASA Microgravity Fluid Physics Program. M. Yazdai is curretly PhD studet with the Two Phase Flow ad Heat Trasfer Ehacemet Laboratory i Mechaical, Materials ad Aerosace Egieerig Deartmet, Illiois Istitute of Techology, Chicago, IL, 63 ( myazda@iit.edu). J. Seyed-Yagoobi is IEEE seior member ad rofessor ad chair of the Mechaical, Materials ad Aerosace Egieerig Deartmet, Illiois Istitute of Techology, Chicago, IL, 63 ( yagoobi@iit.edu). EHD referece velocity u velocity vector magitude V alied electric otetial Y w chael wall thickess α dimesioless charge diffusio costat ɛ absolute electric ermittivity η electrode ga µ dyamic viscosity of fluid ν kiematic viscosity of fluid ξ electrode width groud to HV electrode width, = ξ g /ξ HV ω dissociatio rate coefficiet, ω = [ e 3 E ρ mass desity σ electric coductivity of fluid φ otetial field u ehd eq g HV SUBSCRIPTS equilibrium groud electrode high voltage electrode SUPERSCRIPT dimesioless variable I. INTRODUCTION 4πɛk B T sat ] / The EHD umig heomea ivolve the iteractio of electric fields ad flow fields i a dielectric fluid medium. This iteractio betwee electric ad flow fields iduces the fluid motio through the resece of electric body force. EHD coductio umig is associated with the heterocharge layers of fiite thickess i the viciity of the electrodes that are based o the rocess of dissociatio of the eutral electrolytic secies ad recombiatio of the geerated ios [], []. The coductio term here reresets a mechaism for electric curret flow i which charged carriers are roduced ot by ijectio from the electrodes, but by dissociatio of molecules withi the fluid. Feg ad Seyed-Yagoobi [3] coducted a asymtotic aalysis to study the effects of flow covectio o the EHD coductio umig ad the associated eergy trasort. The EHD coductio drive flow rely rimarily uo the asymmetry of the electrodes. The electrodes asymmetry results i the domiace of electric body force ad flow geeratio i oe directio. Yazdai ad Seyed-Yagoobi [4] showed that the flow geeratio is always from the arrower electrode toward the wider electrode regardless of their olarity. They also showed that symmetric charge ad body force distributios

2 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION over idetical electrodes result i zero et flow geeratio whe the charges have idetical roerties. However, recet exerimetal study by Haaoka et. al [5] revealed that there exists a et geerated flow i the case of symmetric electrodes cofiguratio. Assumig that the diffusio has egligible effect o the charge distributio i macro-scales, the charge distributio symmetry over the idetical electrodes could be broke oly if the charges have differet mobilities. There exist a limited umber of ublicatios focusig o the mobility measuremets i dielectric liquids. Wikur et. al [6] coducted a exerimet to measure the io mobility i -hexae ad a.m solutio of biheyl i -hexae usig a time-of-flight techique. This techique was used i several measuremets of the mobility of ositive ios i hydrocarbos (see for examle [7] ad refereces therei) with the discreacies maily raised by the methods of measuremets. Huag ad Freema [8] measured the mobility of ositive ios i liquid hydrocarbos ad ivestigated the effects of desity ad temerature. They claimed that the ositive ios mobility does ot obey Walde s rule (b + µ. ) but rather has a oliear relatioshi to the viscosity, b + µ.±. as foud earlier for ios i ethers. The time of flight method is later used by Sakamoto ad Yamada [9] to study the io mobilities ad breakdow rocesses i dielectric liquids with razor-blade emitter grid collector used for the electrodes. They reorted the streamig velocities for ositive ad egative agets i differet liquids. Similar techique was used by Casaovas et. al [] to measure the mobility of ositive ad egative ios i olydimethylsiloxae silicoe oil. The curret study exteds the fudametal study by Yazdai ad Seyed-Yagoobi [4] to illustrate the role of charge mobility o the erformace ad characteristics of EHD coductio drive liquid flow. I articular, the imact of uequal mobilities for ositive ad egative charges o the geerated flow for the case of symmetric electrodes is ivestigated ad hysically aalyzed. This study is also exteded to iclude the case of asymmetric electrodes where et flow is exected due to the electrodes asymmetry eve i the case of equal mobilities for ositive ad egative carriers. II. THEORETICAL MODEL The theoretical model reseted here is adoted from the model develoed by Atte ad Seyed-Yagoobi []. Their theoretical model was develoed ad solved aalytically i the absece of ay fluid motio assumig that the ositive ad egative charges ossess idetical mobilities. Jeog et al. [] alied Atte ad Seyed- Yagoobi s model to a comlex electrode geometry ad rovided umerical solutios i the absece of fluid motio as well. The model reseted i this sectio is based o the assumtio that the dielectric liquid is icomressible ad sigle hase with o cotributio of charge ijectio from the electrodes surfaces. The ositive ad egative charge mobilities are differet which is the major dearture of this aer from revious studies. It is assumed that the charge diffusio for ositive ad egative ios is costat ad idetical. The schematic of the umerical domai for a symmetric electrode cofiguratio is reseted i Fig.. Table I resets i/out y x symmetry ξg η V ξhv liquid Fig.. Schematic of umerical domai (ot to scale). Detail of the electrodes desig is rovided i Table I. TABLE I DETAILS OF THE ELECTRODES DESIGNS CHARACTERISTICS. ALL DIMENSIONS ARE NORMALIZED WITH THE CHANNEL HALF HEIGHT, d = mm Electrodes Cofiguratio Symmetric Asymmetric, =.35 Asymmetric, =.86 Chael wall thickess (mm) Y w = 4 (Y w = ) Groud electrode (mm) ξ g = (ξ g = ) ξ g =.7 (ξ g =.35) ξ g = (ξ g = ) solid HV electrode (mm) ξ HV = (ξ HV = ) ξ HV = (ξ HV = ) ξ HV =.7 (ξ HV =.35) i/out Electrode ga (mm) η = (η =.5) the details of the electrode desig for symmetric ad asymmetric electrodes cofiguratios. I steady-state, the system described here is govered by the followig equatios. u () u u = ρ P + ν u + (ρ + ρ )E () The last term o the left had side of equatio. () is the electric body force ad should be obtaied by solvig the Maxwell s relatio ad charge trasort equatios: [( b+ b E = ( ) ; E = φ (3) ɛ ) ( E + ) ] ( u D ) = k r ( eq F (ω) ) (4) Here, the assumtio of thermodyamic equilibrium for free charges is utilized to relate the dissociatio rate to recombiatio rate, k d N = k r eq where eq stads for the charge desity at equilibrium. Fially, with d, u ehd = b + V/d, µ /ρd, V/d, ad eq, resectively as scalig arameters for legth, velocity, ressure, electric field, ad free charges, oe ca obtai the odimesioal equatios goverig the system as follows: u = (5) (u ) u = ( + Re ehd ) P Re ehd u + M C ( ) E (6) E = ( )C ( ), E = φ (7) + d Yw

3 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION 3 TABLE II SUMMARY OF ELECTROSTATIC AND FLOW BOUNDARY CONDITIONS TABLE III NUMERICAL VALUES OF DIMENSIONLESS NUMBERS FOR THE BASIC CASE HV electrode Groud electrode = u φ = φ = =, = =, = Liquid/solid iterface φ s φ = (, ) = u u i Ilet/outlet i = u = u out out φ i = φ out, φ i = φ out (, ) i = (, ) (, ) i = (, ) out, out, Chael outer wall - φ = - Dimesioless umber Numeric value Re ehd 38 C. α M 3.58 [( ) ( ) E + ] ( u α ( C ) = C ) (F (ω) ) (8) where Lagevie s aroximatio for dielectric fluids [3] is used to relate the recombiatio costat to the liquid roerties; that is: k r = b ++b ɛ. The resultig dimesioless coefficiets i the above equatios are defied as follows: Re ehd = b +V ν, M = ( ɛ ), ρb + + C = σd b + ɛv, α = D b + V = k BT ev, = b + b with the charge cocetratio at equilibrium, eq, defied as eq = σ b + +b. The liquid is i cotact with the chael wall which is assumed to be solid a isulator. Therefore, with o volumetric electric charges withi the solid zoe, the Lalace equatio govers the otetial field: φ s = (9) The boudary coditios are reseted i Table II. Note that the cotiuity of otetial field is alied across the solid/liquid iterface where the liquid is i cotact with the wall while the chael outer surface is grouded. No boudary coditio is required for velocity ad electric charges iside the solid wall sice the corresodig equatios are ot solved withi the solid zoe. I additio, the zero flux boudary coditio for electric charges across the solid/liquid iterface is based o the egligibly small double layer thickess which imlies that o charges cross the solid/liquid iterface as the volumetric electric charge iside the solid zoe is assumed to be zero. I additio, boudary coditios o the electrodes surfaces imly that charges with the same olarity as the electrodes do ot accumulate o the surface of the electrodes while o diffusio of the charges with oosite sig exists right at the surface of the electrode. However, there exists charge diffusio i the uer layers withi the heterocharge layer. III. RESULTS AND DISCUSSIONS The results reseted i this sectio illustrate the role of charge mobility o the erformace of EHD coductio um. The referece values selected to calculate the dimesioless arameters ad br = corresod to the roerties of refrigerat R-3 as the workig fluid []. The values of the dimesioless umbers for the base case are reseted i Table III ad alied voltage of kv. Note that the values reseted i this aer for has bee arbitrarily selected due to the lack of measured mobility ratios esecially for refrigerat R-3. These results are reseted to solely illustrate the effects of charges mobility o the geerated flow. First we examie the imact of charge mobility for symmetric electrodes cofiguratio where o flow is exected whe the mobility of ositive ad egative charges is idetical. Later i this sectio, the role of charge mobility for asymmetric electrodes is ivestigated where et flow is geerated due to the electrodes asymmetry i the case of idetical mobilities. I additio, the variatio of et geerated flow over the rage of mobility ratios for all electrodes cofiguratios is reseted ad aalyzed. Note that this rage A. Symmetric Electrodes Cofiguratio The results of electric field, et charge desity ad electric body force for symmetric electrodes cofiguratio ad idetical mobility for ositive ad egative charges are reseted i Fig.. The electrodes symmetry results i symmetric distributio of electric field ad thus, et charge desity. I additio, the iter-electrode regio is characterized by high itesity electric field (see Fig. a) which yields i higher charge desity i this regio (Fig. b). Therefore, the resultat electric body force distributio is idetical but i oosite directios above the electrodes surfaces. The flow streamlies reseted i Fig. 3 cofirm zero et geerated flow which, of course, is associated with local flow circulatios due to the local body force distributio. The symmetric distributio of charges over the surfaces of idetical electrodes will be broke oce differet mobilities are assiged for ositive ad egative charges. This mechaism alog with the distributio of resultat electric body force is illustrated i Fig. 4 for =. Note that this value of mobility ratio has ot bee secifically measured for R-3 but has bee arbitrarily chose by cosiderig the measuremet by Casaovas et. al [] for urified silico oil cotaiig C 7 F 4. Here, egative charges have smaller mobility coefficiet tha the ositive charges. Therefore, there exists more tedecy for ositive charges to migrate toward the electrodes due to the exteral electric field as govered by Eq. (8). The cosequece is higher ositive charge desity o the surface of the groud electrode as observed i Fig. 4a which results i the domiace

4 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION 4 E f f Fig. 4. Dimesioless cotours of et charge desity ad electric body force with = ad = Fig.. Dimesioless cotours of electric field, et charge desity, ad (c) electric body force with = ad =. (c) Fig. 5. Flow streamlies with =. ad = Fig. 3. Flow streamlies with = ad =. of electric body force directig toward egative x-directio (see Fig. 4b). The flow streamlies dislayed i Fig. 5 illustrate the et flow motio toward egative x-directio. The variatio of et geerated flow with mobility ratio for differet electrodes sacigs is reseted i Fig. 6. The dimesioless flow rate at the chael outlet is defied as: Q et = u ehd d d u out dy () The geerated flow iitially icreases ad evetually dros with the mobility ratio. This sceario ca be exlaied as follows: The iitial icrease i the geerated flow is due to the more roouced asymmetric charge distributio over the electrodes surfaces. However, larger values of mobility ratio suress the migratio of egative charges toward the HV electrode surface to yield lower charge desity over the electrode surface as illustrated i Fig. 7. I additio, the electric field has a more uiform distributio aroud the groud electrode sice charges ted to have a uiolar distributio o the surface of the groud electrode as the heterocharge layer o the HV electrode is iclied to disaear. Therefore, the two edges of the groud electrode are characterized by high itesity electric body force i oosite directios as reseted i Fig. 7, which adversely imact the geerated liquid flow. Also show i Fig. 6 is the effect of electrode ga o the geerated flow. The flow is exected to icrease as the electrode ga is reduced due to the higher electric filed itesity withi the iter-electrode regio. B. Asymmetric Electrodes Cofiguratio The fudametal alicatio of EHD coductio drive flow is based o the asymmetric electrodes geometry. As illustrated i Fig. 8, the cosequece of electrodes asymmetry Q et x 4 =, Re ehd = 38, C =. η = η = η = Fig. 6. Variatio of et geerated flow with mobility ratio for symmetric electrodes, =.

5 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION 5 - f f Fig. 7. Dimesioless cotours of et charge desity ad electric body force with = 6 ad = Fig. 9. dimesioless electric body force cotours ad flow streamlies =.35 ad =. f x 4 =.35, Re ehd = 38, C = Fig. 8. dimesioless electric body force cotours ad flow streamlies with =.35 ad =. Q et η = η = η = Fig.. Variatio of et geerated flow with mobility ratio for Asymmetric electrodes, =.35. is the domiace of the electric body force directed toward the broke symmetry; from the arrower electrode toward the wider electrode causig the et flow geerated from left to the right of the chael. However, the resece of oosig electric body force results i local reversed flow which remais limited to the regio close to the electrodes. The imact of uequal mobility coefficiets for ositive ad egative charges o the electric body force ad flow field for =.35 is deicted i Fig. 9. As discussed earlier, with >, the ositive charge desity over the surface of the groud electrode surasses the egative charge desity over the HV electrode. For the give cofiguratio, this will ehace the oosig body force o the edge of the groud electrode to icrease the circulatig flow (see Fig. 9b) ad lower the geerated flow. Figure resets the variatio of geerated flow with mobility ratio. The immediate dro of the flow rate for > is due to the icrease i the oosig electric body force o the edge of the arrower (groud) electrode. However, the course is evetually reversed for larger values of mobility ratios er our earlier discussio. Noticeably, the uward tred starts at smaller mobility ratios ad leads to a slightly larger flow rate for large mobility ratios whe the ga betwee the electrodes exads. This is because the electric field ecouters less variatio across the electrodes edges whe the ga is larger yieldig i earlier trasitio to seudo-uiolar charge distributio over the electrode surface. Fially, before cocludig this study, we reset the variatio of et flow with mobility ratio for the case of =.86 where the broke symmetry requires the flow to be geerated from right to the left. As observed i Fig., the flow icreases with the mobility ratio as the direct cosequece of romoted body force o the right edge of the groud electrode. However, the trasitio to a symmetric distributio over the electrode surface occurs at much larger mobility ratios sice the mobile charge carriers are ow migratig toward the wider electrode. The threshold at which the tred shifts is see to occur far beyod = 6 ad therefore is ot reseted here.

6 CONF. PRESENTATION SESSION : PARTICLES IN FLOWS & FLOW ELECTRIFICATION 6 Q et.5 x =.86, Re ehd = 38, C =. η =.6 η = η = [9] S. Sakamoto ad H. Yamada, Otical study of coductio ad breakdow i dielectric liquids, Electrical Isulatio, IEEE Trasactios o, vol. EI-5, o. 3,. 7 8, Jue 98. [] J. Casaovas, R. Grob, A. Chemi, J. P. Guelfucci, ad J.-P. Crie, Io mobility measuremets i a 5 cst viscosity olydimethylsiloxae silicoe oil, Electrical Isulatio, IEEE Trasactios o, vol. EI-, o., , Aril 985. [] P. Atte ad J. Seyed-Yagoobi, Electrohydrodyamically iduced dielectric liquid flow through ure coductio i oit/lae geometry, IEEE Tras. Dielectr. Electr. Isul, vol.,. 3, 3. [] S. I. Jeog, J. Seyed-Yagoobi, ad P. Atte, Theoretical/umerical study of electrohydrodyamic umig through coductio heomeo, IEEE Tras. Dielectr. Electr. Isul, vol. 39,. 355, 3. [3] P. Lagevi, Recombmaiso et mobilites des ios das les gag, A. Chim. Phys, vol. 8,. 433, 93. Fig.. Variatio of et geerated flow with mobility ratio for asymmetric electrodes, =.86. IV. CONCLUSIONS The effect of uequal mobility ratios of ositive ad egative charge carriers ( = b + /b > ) o the erformace of EHD coductio umig was ivestigated. Net flow was observed for the case of symmetric electrodes ad > which is attributed to the o-symmetric charge distributio over the electrodes. The et flow decreased for large mobility ratios as the migratio of egative charges is suressed to result i relatively uiform charge ad body force distributio over the surface of groud electrode. The imact of charge mobility for asymmetric electrodes was aalyzed as well. With >, the flow was adversely imacted comared to the case of idetical mobilities whe it was geerated from the groud electrode toward the HV electrode, but it was ositively affected by the uequal mobility coefficiets whe it was geerated from the HV electrode toward the groud electrode. ACKNOWLEDGMENT This work was fiacially suorted by the NASA Microgravity Fluid Physics Program. REFERENCES [] J. Seyed-Yagoobi, Electrohydrodyamic umig of dielectric liquids, J. Electrostatics, vol. 63, , 5. [] M. Yazdai ad J. Seyed-Yagoobi, Numerical ivestigatio of electrohydrodyamic-coductio umig of liquid film i the resece of evaoratio, ASME J. Heat Trasfer, vol. 3, 8. [3] Y. Feg ad J. Seyed-Yagoobi, Electrical charge trasort ad eergy coversio with fluid flow durig electrohydrodyamic coductio umig, Physics of Fluids, vol. 9, o. 5,. 57, 7. [4] M. Yazdai ad J. Seyed-Yagoobi, Electrically iduced dielectric liquid film flow based o electric coductio heomeo, IEEE. Tras. o Dielectric liquids, i ress, 9. [5] R. Haaoka, I. Takahashi, S. Takata, T. Fukami, ad Y. Kamamaru, Proerties of ehd um with combiatio of rod-to-rod ad meshy arallel lates electrode assemblies, IEEE Tras. Dielectrics ad Electrical Isulatios, vol. 6, o., , Aril 9. [6] P. S. Wiokur, M. L. Roush, ad J. Silverma, Io mobility measuremets i dielectric liquids, The Joural of Chemical Physics, vol. 63, o. 8, , 975. [7] S. Ishii, T. Aoki, M. Nagao, ad M. Kosaki, The mobility measuremet of ositive charge carrier i -hexae, Oct 994, [8] S. S.-S. Huag ad G. R. Freema, Positive io mobilities i gaseous, critical, ad liquid hydrocarbos: Desity ad temerature effects, The Joural of Chemical Physics, vol. 7, o. 3, , 979.