Part of 2010_Walmsley Electrostatics

Part of 2010_Walmsley Electrostatics - H.L. Walmsley /...

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Unformatted text preview: H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 2) 3) 4) 5) 6) 7) hydrocarbon/air mixtures [1], the charge-transfer chargecollection efficiency associated with a typical test setup (disc with same area as a 150 mm square sheet) is 0.81, 0.75 or 0.69 with 10 mm, 12.5 mm and 15 mm radius electrodes respectively. Thus the typical degree of underestimation due to induced-charge errors is modest but not negligible. The lowest charge-collection efficiency occurs with the largest sphere and the smallest possible disc. The largest sphere we considered was 15 mm radius and for this, with our best estimate of the post-discharge charge distribution on the disc surface, the minimum charge-collection efficiency we found for a 60 nC charge-transfer was 0.56. With worst-case assumptions about the breakdown voltage and the distribution of charge on the disc surface after the discharge, this reduced to an overall worst-case charge-collection efficiency of 0.40. However the combination of conditions associated with this low level of efficiency is thought unlikely to occur in practice. It is possible to allow for a reduced charge-transfer chargecollection efficiency by applying a correction factor to the measured charge-transfers. The results in Conclusion 2) indicate that, for results relating to a 60 nC charge-transfer threshold and with the expected post-discharge charge distribution, a correction factor of Â2 would ensure that hazardous equipment was not wrongly classified as safe. A correction factor of Â2.5 would deal with our worst-case charge distribution. These corrections would, of course, only be needed for fast, unshielded probe measurements and would not be required for measurements made with slow and/or shielded probes as these have a high charge-transfer charge-collection efficiency. Lower charge-transfer thresholds have been suggested [1] for more sensitive materials (30 nC for Gas Group IIb and 10 nC for Gas Group IIc). There is an inverse relationship between charge-transfer and charge-collection efficiency so efficiencies are lower for these smaller charge-transfers. In addition, for smaller charge-transfers, the efficiencies vary more steeply with the geometric parameters. Because of the low chargecollection efficiency and increased sensitivity to parameter changes it seems best to avoid the use of fast, unshielded probes for measuring charge-transfers around the 30 nC and 10 nC thresholds. Our results suggest an overall lower limit to the charge-transfer charge-collection efficiency of around 0.2 although this conclusion depends on the relationship we have used between discharge patch size and disc-sphere separation, which is unverified. If this relationship, or an equivalent, is ultimately confirmed, it may be acceptable to interpret data for more sensitive materials with the lower-limit efficiency and hence get results that will err on the safe side. With such low levels of charge-collection efficiency, the circuit charge-transfers corresponding to the 10 nC and 30 nC thresholds would only be 2 nC and 6 nC respectively and it is debatable whether there would be any benefit in measuring and interpreting such small circuit charge-transfers rather than simply requiring that no discharges should be detected by a measurement circuit of adequate sensitivity. Although our results were obtained with charged discs, they are expected to apply approximately to flat surfaces of other shapes and equivalent area. The calculations in this paper cover discharges from isolated charged discs to earthed spheres. We expect measurement efficiencies to be higher for the following cases: a. Discharges from discs located near earthed planes (because the presence of the earth plane will reduce the charge coupling between disc and sphere). 17 b. Discharges from convex charged objects (because the convex shape will reduce coupling to the sphere). c. Discharges with higher charge-transfers. If charge-transfer results for cases like these are interpreted using the measurement efficiencies estimated in this paper, the resulting hazard assessments should err on the safe side. Thus hazardous items should not be wrongly classified as safe although some (marginally) safe items may be wrongly classified as hazardous. 8) Charges on a concave plastic surface that, to some extent, wraps around a test sphere would have more coupling to the sphere than the charges on a plane disc and hence we would expect a reduced charge-collection efficiency. We therefore do not recommend the use of fast, unshielded charge-transfer probes for measuring discharges from concave surfaces without further investigation. Acknowledgements The author would like to thank J. Chubb, J. Smallwood, U. von Pidoll and P. Holdstock for valuable discussions during the preparation of this work. Appendix A. Analytical equations for analysing breakdown and charge-transfer of an earthed-sphere/charged-insulatingdisc gap A.1. Basic equations Equations for the voltages and on-axis axial electric field components in the disc-sphere system have been given by, for example, Heidelberg [14]. In order to highlight key parameters we present them in dimensionless form using the characteristic quantities L*, E*, F*, s* and Q* which are defined in the Glossary. We found it useful to link the characteristic electric field to the breakdown criterion in order to explicitly include the role of breakdown phenomena. With the above characteristic quantities, the dimensionless forms of Heidelberg’s equations for the voltage and axial electric field on axis can be written: ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À À Á Á F0 ðz0 Þ ¼ D0 À z0 2 þR02 À D0 À z0 s0 À E0 ðz0 Þ s0 1 z02 ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á2 À Á2 À Á D0 z0 À 1 þ R0 z0 À D0 z0 À 1 (A.1) À0 Á D À z0 ¼ À1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á2 D0 À z0 þR02 9 8hÀ i > > D0 z0 À 1ÁÀD0 z0 À 2Á þ R02 z02 < Á= À 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À D0 z0 À 2 À 03 Á2 À > > z: ; D0 z0 À 1 þR02 z02 (A.2) and the charge induced on the sphere by the charge on the disc is 0 Qs s0 ¼ À2 i hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D02 þ R02 À D0 (A.3) where the primes indicate dimensionless quantities and the symbols are as defined in the Glossary. 18 H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 charge density of Às ad ¼ Às0 s*ad. The discharge charge-transfer, 0 DQd , is given by: Useful special cases of equations (A.1) and (A.2) are: a) The voltage on axis at the disc surface (z0 ¼ D0 ): ÀÁ F0 D0 s0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 À ! Á2  1 D02 À 1 þ R0 D0 À D02 À 1 ¼ R0 À 02 D 0 0 02 DQd ¼ Àad rd2 s0 ¼ reff s0 (A.4) a) The axial electric field component on axis (in the air) at the disc surface (z0 ¼ D0 ): (A.8a) 0 DQc The circuit charge-transfer is ¼ þ where DQi0 is the change inqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi induced charge on the sphere. From equation (A.3), 0 DQi0 ¼ 2s0 ad ½ D02 þ rd2 À D0 Š and hence: & 0 DQc ¼ s0 ad 2 The 0 DQd DQi0 ! ' qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 02 D02 þ rd À D0 À rd charge-collection efficiency (A.9) can be defined as 0 0 3m ¼ DQc =DQd and hence is given by: E0 À 0Á D s0 8h 9 i   > 02 02 02 02 = >  1 < D À1 D À2 þR D rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ À1 À 03 À D02 À 2  2 > D> : ; D02 À 1 þR02 D02 (A.5) 3m ¼ 1 À 2ad h 02 reff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 02 D02 þ reff =ad À D0 If the discharge patch radius is roughly dependent on the 0 separation of disc and sphere centres we can set reff ¼ KD0 and thus write: 0 DQd ¼ ÀK 2 D02 s0 a) The axial electric field component at the sphere surface (z0 ¼ 1): 3m ¼ 1 À E 0 ð1Þ s0 8hÀ 9 i > D0 À 1ÁÀD0 À 3Á þ R02 > < Á= À0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼À À D À3 À Á2 > > : ; D0 À 1 þR02 (A.6) (A.10a) 2ad K 2 D0 ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ K 2 =ad À 1 (A.8b) (A.10b) The various forms of equations (A.8) and (A.10) give the results we require for the charge-transfer and charge-collection efficiency in terms of the disc charge density, discharge geometry and the characteristics of the discharge patch (mean radius and charge loss fraction). A.2. Breakdown conditions Appendix B. Critical breakdown fields and voltages Because we have chosen the characteristic electric field to be the critical breakdown field at the sphere surface, the breakdown criterion is E0 (1) ¼ 1. From this, equation (A.6) leads directly to the dimensionless disc charge density required for breakdown at the distance, D0 , (we ignore polarity effects): The most accurate breakdown criterion (see, for example, the review in [16]) is generally thought to be that of Pedersen [15], which states that breakdown occurs when: 8hÀ 9À1 i > D0 À 1ÁðD0 À 3Þ þ R02 > < = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s0 ¼ À ðD0 À 3Þ À Á2 > > : ; D0 À 1 þR02 Zg ½MaxfðEðzÞ À pBÞ; 0gŠ2 dz ¼ pC 2 (B.1) 0 (A.7) By definition, the characteristic charge density, s*, is determined by the critical breakdown field, Eb, at the sphere surface. Thus the real disc charge density, s ¼ s0 s*, required to cause breakdown with a given disc-sphere separation (or gap) can be determined by combining equation (A.7) with a calculation of Eb (see Appendix B). Alternatively, if the charge density on the disc is given, equation (A.7) can be solved for D0 to obtain the equivalent breakdown gap (g0 ¼ D0 À 1). A.3. Discharge charge-transfers and charge-transfer collection efficiency where E(z) is the axial electric field in the gap, g is the gap length, z is the distance from the surface of the highly stressed electrode (which is the earthed sphere in the cases of interest to us), p is the pressure and B and C are constants. The electric field can be written as E(z) ¼ E0 (z)E0 where, in the case of a gap between a charged disc and an earthed sphere, E0 is the electric field at the surface of the sphere. The dimensionless field, E0 (z), depends only on the geometry and it is known, equation (B.1) can be solved to find Eb, which is the value of E0 that produces breakdown. Solving equation (B.1) with lengths in mm, p in bar, B ¼ 2.42 kV/mm bar and C ¼ 2.08 kV/ (mm bar)0.5 gives Eb in kV/mm. The breakdown voltage, Fb, is then: Fb ¼ Eb Zg E0 ðzÞdz (B.2) 0 We assume that, when a discharge occurs, it affects a patch of radius rd on the disc and that within this patch the charge density changes by a fixed multiple, Àad of the initial charge density, s. We discuss how to obtain the equivalent top hat radius and mean charge loss factor in Appendix C. With the above simplified representation of post-discharge conditions, the post-discharge voltages, electric fields and charges are found by adding the contributions from two co-aligned charged discs, the original disc and a second of radius rd carrying a uniform This gives voltages in kV when used with the constants above and lengths in mm. The Pedersen method is relatively complex because it requires the solution of equation (B.1). Heidelberg [14] quotes a suggestion by Bouwers [18], that the critical field can be obtained from the solution of: ða=lÞðEb l=2vÞexpð À v=Eb lÞð1 À Eb l=2vÞ ¼ 1 (B.3) H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 where l is the mean free path of an electron (taken as 3.7  10À7 m) and v is the ionization potential of O2 (taken as 12.5 V). Landers [17] uses an expression by Steinbigler [19], which may be written as (B.4) where E00 ¼ 2.28  106 V/m and acrit ¼ 10À2 m. In the present work we solved equation (B.1) for Eb in the discsphere geometry using equations (A.2) and (A.6) to define E0 (z) and compared the results to the values obtained from the simpler expressions in equations (B.3) and (B.4) (see Figs. 2 and 3 of the main text). When testing the incendivity of discharges from plastics without earthed backings, we are usually concerned with electrode radii between 10 mm and 15 mm, and gaps of a few tens of mm. The discussion in the main text shows that equation (B.4) is accurate enough for our purpose under these conditions and its simplicity matches that of the analytical solutions. We have therefore adopted this breakdown criterion for our calculations. It leads to a characteristic charge density of:  s* ¼ 23a 30 Eb ¼ 23a 30 E00 1 þ ðacrit =aÞ1=3   01=3 ¼ 23a 30 E00 1 þ acrit  (B.5) where a0 crit ¼ acrit =a. For sphere radii between 10 mm and 15 mm 0 1 =3 we have 2:0 < ð1 þ acrit Þ < 2:145. Hence, to a fair approximation, we can simplify equation (B.5) to Eb w 2.07E00 ¼ 4.72 MV/m, which leads to an approximate fixed characteristic charge density of s* ¼ 43a30E00 ¼ 84 mC/m2. Once s* is established it leads, via equation (A.7), to the disc charge density required for breakdown. The integral in equation (B.2) is already available as equation (A.4), which can be written: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi &  2 À !' Á2  1 0 0 Fb ¼ Eb as R À 02 D02 À 1 þ R0 D0 À D02 À 1 D (B.6) This expression was used to generate Fig. 4 of the main text. Appendix C. Representing the discharged patch as a charge disturbance with a ‘‘top-hat’’ distribution The distribution of removed charge in a symmetrical discharge patch can be expressed as: a(r) ¼ (s(r) À s)/s , where r is the distance from the centre of the patch, s(r) is the charge density in the patch after discharge and s is the initial (uniform) charge density. Our simple model deals only with uniform charge densities so we need to approximate this distribution of charge by an equivalent top-hat charge distribution. To do this we use the top-hat distribution that has the same mean radius and total charge deficit as the real distribution. The mean radius is r¼ ZN ar2 dr= 0 ZN ardr (C.1) 0 For a uniform top-hat distribution of radius r1, equation (C.1) gives a mean radius of (2/3)r1. Thus the radius, r1, of the top-hat distribution with the same mean radius as a is 3 r1 ¼ 2 ZN 0 ar2 dr= ZN 0 ardr and the appropriate uniform a value within the top hat (i.e. for r < r1) is: a¼2  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eb ¼ E00 1 þ 3 acrit =a (C.2) 19 ZN 2 ardr=r1 0 13 0 N 12 0N Z Z 8@ 2A a r dr A = @ a r dr ¼ 9 0 (C.3) 0 The actual distribution may be represented by a power law such as aðrÞ ¼ a0 1 À ðr=r2 Þn ; r À Á r2 : aðr Þ ¼ 0; r > r2 (C.4) Substituting this in equations (C.2) and (C.3) gives: r1 ¼ r2  nþ2 nþ3  (C.5) and ðn þ 3Þ2 a ¼ a0 n (C.6) ðn þ 2Þ3 The power law distribution approaches a top hat as n tends to infinity and, consistent with this, in this limit (C.5) reduces to r1 ¼ r2 and (C.6) to a ¼ a0 . Alternatively, the charge distribution with an earthed backing has been fitted [23] to an expression of the form:  aðrÞ ¼ a0 1 À ðr=r2 Þ2 2 ;r r2 : aðr Þ ¼ 0; r > r2 (C.7) With this function equations (C.2) and (C.3) evaluate to: r1 ¼ 24 r ¼ 0:686r2 35 2 (C.8) and a¼ 1225 a ¼ 0:709a0 1728 0 (C.9) Expressions (C.5–C.9) illustrate how to convert more detailed representations of the charge distribution in a discharged patch to approximate top-hat distributions for use in our simple model. References [1] U. von Pidoll, E. Brzostek, H.-R. Froechtenigt, Determining the incendivity of electrostatic discharges without explosive gas mixtures, IEEE Trans. Ind. Appl. 40 (2004) 1467–1475. [2] J.M. van der Weerd, Electrostatic charge generation during tank washing. Spark mechanisms in tanks filled with charged mist, J. Electrostat. 1 (1975) 295. [3] J.S. Mills, E.J. Haighton, Electrostatic discharges: charge-transfer measuring techniques, J. Electrostat. 13 (1982) 91–97. [4] B. Makin, P. Lees, Measurement of charge-transfer in electrostatic discharge, J. Electrostat. 10 (1981) 333–339. [5] J.N. Chubb, G.J. Butterworth, Charge-transfer and current flow measurements in electrostatic discharges, J. Electrostat. 13 (1982) 209. [6] L.G. Britton, T.J. Williams, Some characteristics of liquid-to-metal discharges involving a charged ‘‘low risk’’ oil, J. Electrostat. 13 (1982) 185–207. [7] J. Chubb, Measurement of charge-transfer in electrostatic discharges, J. Electrostat. 64 (5) (2006) 301–305. [8] H.L. Walmsley, Threshold potentials and discharge charge-transfers for the evaluation of electrostatic hazards in road tanker loading, J. Electrostat. 26 (1991) 157–173. [9] N. Gibson, D.J. Harper, Parameters for assessing electrostastic risk from nonconductors – a discussion, J. Electrostat. 21 (1988) 27–36. [10] H.L. Walmsley, J.S. Mills, Electrostatic ignition hazards in road tanker loading: part 1. Review and experimental measurements, J. Electrostat. 28 (1992) 61–87. [11] EN 13463–1:2001. 20 H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 [12] IEC 60079-0, Electrical Apparatus for Explosive Gas Atmospheres, Part 0: General Requirements, Section 26.14, 2006. [13] E. Heidelberg, Test and judgment of ignition hazards caused by electrostatic spray guns and charged surfaces without the use of explosive mixtures, J. Electrostat. 21 (1988) 1–17. [14] E. Heidelberg, Entladungen an elektrostatisch aufgeladenen nichtleitfahigen metallbeschichtungen, PTB-Mitt. 80 (1970) 440–444. [15] A. Pedersen, On the electrical breakdown of gaseous dielectrics, IEEE Trans. Electr. Insul. 24 (1989) 721–739. [16] J.P. Donohoe, Physical characteristics of criteria governing the computation of air gap breakdown voltages for slightly divergent fields, IEEE Trans. Dielectr. Electr. Insul. 5 (1998) 485–492. [17] E.U. Landers, Distribution of charge and field strength due to discharge from insulating surfaces, J. Electrostat. 17 (1985) 59–68. [18] A. Bouwers, Elektrische Hochstspannungen, Springer, Berlin, 1939, p. 153. [19] H. Steinbigler, Anfangsfeldstarken und Ausnutzungsfaktoren Rotationssymmetrischer Elektrodenanordnungen in Luft, Diss (1969) Technische Universitat Munchen. [20] J.L. Davidson, T.J. Williams, A.G. Bailey, Electrostatic discharges between charged insulators and grounded spheres, J. Electrostat. 56 (2002) 29–42. [21] N. Gibson, F.C. Lloyd, Incendivity of discharges from electrostatically charged plastics, Br. J. Appl. Phys. 16 (1965) 1619–1631. [22] E. Heidelberg, Generation of igniting brush discharges by charged layers on earthed conductors, in: Static Electrification Conf., 1967, pp. 147–154. [23] J.L. Davidson, T.J. Williams, A.G. Bailey, G.L. Hearn, Characterisation of electrostatic discharges from insulating surfaces, J. Electrostat. 51–52 (2001) 374–380. [24] J.L. Davidson, T.J. Williams, A.G. Bailey, R.P. 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Glossary8 a: Sphere radius, m acrit: Characteristic radius used in breakdown criterion (¼10À2 m) g: Discharge gap (g ¼ D À a or g0 ¼ D0 À 1), m gq: The quenching distance, m p: Pressure (used in breakdown criterion), bar rd: Radius of patch affected by discharge, m 2 2 reff: Effective patch radius ðreff ¼ ad rd Þ, m z: Distance along the axis of symmetry from the sphere centre, m B: Constant in Pedersen breakdown criterion, (¼2.42 kV/mm bar) C: Constant in Pedersen breakdown criterion (¼2.08 kV/(mm bar)0.5) D: Separation of disc and sphere centres, m E: Electric field component along the axis of symmetry, V/m E0: Actual electric field at the surface of the sphere, V/m E00: Characteristic electric field in Steinbigler breakdown expression (2.2106 V/m) Eb: Breakdown field at the surface of the sphere, V/m Ebs: Breakdown field at the surface of a plastic sheet, V/m E*: Characteristic field (taken as Eb), V/m K: The ratio of reff to D L*: Characteristic length (taken as sphere radius, a), m Qi: Induced charge on the sphere, C Q*: Characteristic charge (pa2s*), C R: Disc radius, m ad: Fraction of initial charge removed from patch affected by discharge 30: Permittivity of free space, F/m 3a: Relative permittivity of air (w1) 3m: the charge-transfer collection efficiency l: Mean free path of an electron in air, m n: The ionization potential of O2, V s: Charge density on disc (uniform), C/m2 s*: Characteristic charge density (23a30E*), C/m2 smax: The maximum charge density that can exist on a plastic surface in air without causing breakdown, C/m2 DQg: The charge transfer across the gap, C DQi: The change in induced charge on the sphere during the discharge, C F: Potential, V Fb: Breakdown potential F*: Characteristic voltage (¼E*L*), V 8 Dimensionless quantities normalized by the characteristic values are indicated by primes. ...
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