High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
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elation: willresult.AgivensurfaceofthebodydSwhoselocaltemperatureisTswillradiate<strong>and</strong>losepower atarateproportionaltoT4dS.Moreprecisely,thepowerradiatedisgivenbyStefan-Bolztman incorporatedinthecomputations.Typicallyforagivendepositedpower,atemperaturegradient<br />
atatemperatureassumedtobe300Khenceforth). Tocomputethetemperatureriseduetopowerdepositioninthesteadystatecase,weusethe T0istheambienttemperatureofthecanonicalsystemthebodyislocatedin(inourcasevacuum whereistheStefanconstant(=5:670108Wm2K4),istheemissivityofthebody,<strong>and</strong> followingnumericaliterativemethod.Firstly,let'sassume(<strong>and</strong>thisisindeedthecase)thatthe TRradiatorconsistsinacircularfoilofradiusr<strong>and</strong>thicknesstwhosenormalmakesanangle dPS=2dS(T4ST40) (4.13)<br />
withthebeamaxis.Inasuchcase,thepowerisevacuatedradiallyviatheconductionmechanism. Wec<strong>and</strong>ividethefoilbyaseriesofannuliofouter<strong>and</strong>innerradiiri<strong>and</strong>ri+1(seeg.4.2). Thereforethetemperatureoftheithannulusisrelatedtothetemperatureofthei-1thcrownby:<br />
whereRi=1 <strong>and</strong>titsthickness,<strong>and</strong>Piisthetotalpowercomingfromtheithcrown.Partofthepowerisheat radiated<strong>and</strong>theremainingistransmittedviaconductiontothenextelement.Theradiatedpower Figure4.2:MethodologytocomputetemperatureriseinacylindricallysymmetricTRradiator.<br />
is: 2tlnri+1 riisthethermalresistance(isthethermalconductivityofthefoilmaterial Pi=Pi1+2(T43004)(r2ir2i1) Ti=Ti1+RiPi (4.14)<br />
Tocomputethetemperatureriseatthebeamedgecrown,weintroducedtheparameterPext current)usingtheequations[29]:Tcenter=Tn+1 thatrepresenttheevacuatedpower.Hencevaryingthevalueofthisparameter<strong>and</strong>iteratingthe Eqns.(4.14)<strong>and</strong>(4.15)fromtheedgeofthefoiluptotheedgeofthebeamallowstodetermine thetemperatureatthebeamcenter<strong>and</strong>thedepositedpower(fromwhichwecangetthebeam (4.15)<br />
4tPn (4.16)<br />
beam <strong>and</strong> screen center<br />
rxry<br />
rn-1<br />
beam edge<br />
R n R n-1 R n-2<br />
T n<br />
rn-2<br />
T n-1 T n-2<br />
R 3 R 2 R 1<br />
T 3 T 2<br />
screen edge<br />
T=300K<br />
P ext
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