Authorea

Jari Tapani Kolehmainen added file MonodisperseCharges/introduction.tex over 8 years ago

Commit id: 020888c7218e1ec0779a304fde9486fb96d88a82

deletions | additions

%!PS-Adobe-2.0 EPSF-2.0 %%Title: my-plot.eps %%Creator: gnuplot 4.2 patchlevel 6 %%CreationDate: Wed Oct 14 16:46:44 2015 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%EndComments %%BeginProlog /gnudict 256 dict def gnudict begin % % The following 6 true/false flags may be edited by hand if required % The unit line width may also be changed % /Color true def /Blacktext false def /Solid false def /Dashlength 1 def /Landscape false def /Level1 false def /Rounded false def /TransparentPatterns false def /gnulinewidth 10.000 def /userlinewidth gnulinewidth def % /vshift -50 def /dl1 { 10.0 Dashlength mul mul Rounded { currentlinewidth 0.75 mul sub dup 0 le { pop 0.01 } if } if } def /dl2 { 10.0 Dashlength mul mul Rounded { currentlinewidth 0.75 mul add } if } def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def Level1 {} { /SDict 10 dict def systemdict /pdfmark known not { userdict /pdfmark systemdict /cleartomark get put } if SDict begin [ /Title (my-plot.eps) /Subject (gnuplot plot) /Creator (gnuplot 4.2 patchlevel 6 ) /Author (Jari T. 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bind def /C11 {BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath} bind def /C12 {BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath} bind def /C13 {BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath} bind def /C14 {BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc} bind def /C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath} bind def /Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath} bind def /Square {dup Rec} bind def /Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def /S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def /S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def /S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill 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sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare} bind def /S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def /S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def /D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def /D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def /D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def /D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def /D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def /D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def /D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def /D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def /D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def /D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def /D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def /D11 {gsave translate 45 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M 4 -4 L 0 12 M 8 -4 L 4 12 M 10 0 L stroke} >> matrix makepattern /Pat6 exch def << Tile8x8 /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} >> matrix makepattern /Pat7 exch def << Tile8x8 /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L 12 0 M -4 8 L 12 4 M 0 10 L stroke} >> matrix makepattern /Pat8 exch def << Tile8x8 /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L -4 0 M 12 8 L -4 4 M 8 10 L stroke} >> matrix makepattern /Pat9 exch def /Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def /Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def /Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def /Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def /Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def /Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def /Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def } def % % %End of PostScript Level 2 code % /PatternBgnd { TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse } def % % Substitute for Level 2 pattern fill codes with % grayscale if Level 2 support is not selected. % /Level1PatternFill { /Pattern1 {0.250 Density} bind def /Pattern2 {0.500 Density} bind def /Pattern3 {0.750 Density} bind def /Pattern4 {0.125 Density} bind def /Pattern5 {0.375 Density} bind def /Pattern6 {0.625 Density} bind def /Pattern7 {0.875 Density} bind def } def % % Now test for support of Level 2 code % Level1 {Level1PatternFill} {Level2PatternFill} ifelse % /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont pop /MFshow { { dup 5 get 3 ge { 5 get 3 eq {gsave} {grestore} ifelse } {dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 get exch 4 get {show} {stringwidth pop 0 R} ifelse }if dup 5 get 0 eq {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 get 1 eq {dup 2 get exch dup 3 get exch 6 get stringwidth pop -2 div dup 0 R} {dup 6 get stringwidth pop -2 div 0 R 6 get show 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop pop aload pop M} ifelse }ifelse }ifelse } ifelse } forall} bind def /MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 6 get stringwidth pop add} {pop} ifelse} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def /XYsave { [( ) 1 2 true false 3 ()] } bind def /XYrestore { [( ) 1 2 true 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\section{Introduction} It has been known for centuries that during a mechanical contacts materials may develop static charges. This phenomenon is observed when one rubs a balloon against wool cloth, and observes the balloon sticking to the cloth and is known as tribocharging. It has also been known for decades that granular material in transport lines and fluidized beds tend to create charges \cite{lacks_contact_2011}. This charging may cause spark generation leading even to powder explosions \cite{jones_king_1991}. Triboelectric charging is generally been considered unwanted phenomenon, such as particle wall fouling in polyethylene reactors \cite{hendrickson_electrostatics_2006}, but is also crucial in certain applications such as photocopying and laser printing where toner particles are charged via triboelectric charging. Triboelectric charging is still very poorly understood phenomenon and multiple different mechanism have been proposed for this in past decades \cite{lacks_contact_2011}. The most widely cited mechanism for the tribocharging is electron transfer \cite{harper_1967}, where the charge transfer is believed to happen due to electrons transfering from material surface to another material surface. While this mechanism is found to be in good agreement with experiments for metals, there is some debate weather it is applicable to insulators \cite{mccarty_electrostatic_2008}. In electron transfer model materials tendency to pick up charges is described by a work function value. For metals this work function value is defined as the energy needed to remove one electron form the metal surface \cite{harper_1967}. For insulators this work function value correlates poorly with the tribocharging behavior, and instead effective work function value is often used to describe the charging behavior of insulators in numerical simulations \cite{laurentie_numerical_2010}. Unfortunately, there is no direct way to measure the effective work function value due to its vague definition. Moreover, it is well-known that the triboelectric charging of insulators depends on the ambient humidity \cite{gouveia_electrostatic_2009} and on the particle size \cite{sowinski_investigation_2010,forward_charge_2009,zhao_bipolar_2003}. It is not straght forward to determine these effects and the effective work function would need to modelled for these parameters. There has also been more direct simulation approaches to take the insulator charge size dependency into account by modelling the electrons on the particle surface \cite{duff_particle_2008} or by introducing high and low energy electrons \cite{kok_electrification_2009}. While these models can capture the size dependency to some extent, it is hard to incorporate charge transfer of different materials into these models. Furthermore, modelling the electrons in the particle surface is computationally very demanding and not suitable for fluidized bed simulations with more than thousand particles. It was proposed in \cite{laurentie_discrete_2013} that the effective work function value could be determined from macroscopic charging behavior. Laurentie et. al. charged the particles by a vibrated bed, and determined run series of simulations with various effective work function values to match the simulations and experiments. The determined work function value was validated by similar experiments, and the results showed very good agreement. For these reasons, this study will also base the triboelectric charging behavior to effective work functions as they seem a promising tool for simulating triboelectric charging. There has been multiple studies concentrating in electrostatic effects on fludized beds. Earlier computational studies were based on Eulerian-Eulerian simulations that model the solid and gas phase as continuum \cite{rokkam_computational_2010,jalalinejad_effect_2012}. These studies assumed constant charge on particles, and solved the electric field by solving a Poisson equation for electric potential. If the permittivity of the varying solid content is taken into account the large scale electric field can be solved accurately \cite{rokkam_computational_2010}. These studies found that the electrostatics altered the bubbling behavior of fluidized beds by squeezing bubbles at the center of the bed \cite{rokkam_computational_2010,jalalinejad_electro-hydrodynamics_2013,jalalinejad_effect_2015}. The shortcoming of the Eulerian-Eulerian approach is that it require additional modeling for particle drag and particle stresses since the local electrostatic effects may alter these parameters. These local effects have not been addressed in any of these simulations. Furthermore, it is not easy to simulate non uniform charge distributions with Eulerian-Eulerian simulations. To overcome this recent article \cite{hassani_numerical_2013} simulated bubbling fluidized bed by employing four-way-coupled CFD-DEM simulations that model particles individually, and use Eulerian modeling for the solid phase. The study used also predefined charges on particles and considered both monodisperse (same charge on all the particles) and bidisperse case where particles had different prefixed charges. In the article \cite{hassani_numerical_2013} concluded that the bubble size decreased with introduction of charge on particles in mono charged case that is in-line with the Eulerian studies of Jalalinejad \cite{jalalinejad_effect_2012,jalalinejad_electro-hydrodynamics_2013,jalalinejad_effect_2015}. In the bidisperse case, the oppositely charged particles formed chains inside the bubbling bed, and interestingly caused the bubbles that were similar to the neutrally charged case. The aim of this study is to inspect the interplay between the triboelectric charging and electrostatic effects. The triboelectric model chosen was similar to \cite{laurentie_discrete_2013} while the electrostatic force was modelled in a similar way to \cite{hassani_numerical_2013}. The fluidization regime was chosen slightly above the bubbling regime as its more relevant for polyethylene reactors.