Fundamentals of Vacuum Technology (1248463), страница 6
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The continuum theory and the summarization of the gas laws whichfollows are based on experience and can explain all the processes in gasesnear atmospheric pressure. Only after it became possible using ever bettervacuum pumps to dilute the air to the extent that the mean free path rosefar beyond the dimensions of the vessel were more far-reachingassumptions necessary; these culminated in the kinetic gas theory. Thekinetic gas theory applies throughout the entire pressure range; thecontinuum theory represents the (historically older) special case in the gaslaws where atmospheric conditions prevail.a, b = constants (internal pressure, covolumes)Vm = Molar volumealso: Equation of state for real gasesClausius-Clapeyron EquationL =T·dp· ( V − Vm, l )dT m, vL = Enthalpy of evaporation,T = Evaporation temperature,Vm,v, Vm,l = Molar volumes of vapor or liquid1.3.2 Kinetic gas theorySummary of the most important gas laws (continuum theory)Boyle-Mariotte Lawp ⋅ V = const.for T = constant (isotherm)With the acceptance of the atomic view of the world Ð accompanied by thenecessity to explain reactions in extremely dilute gases (where thecontinuum theory fails) Ð the Ókinetic gas theoryÓ was developed.
Using thisit is possible not only to derive the ideal gas law in another manner but alsoto calculate many other quantities involved with the kinetics of gases Ð suchas collision rates, mean free path lengths, monolayer formation time,13HomeVacuum physicsdiffusion constants and many other quantities.The mass of the molecules isModel concepts and basic assumptions:1. Atoms/molecules are points.2. Forces are transmitted from one to another only by collision.3. The collisions are elastic.4. Molecular disorder (randomness) prevails.mT = M = Mass / molNA Molecules / molA very much simplified model was developed by Kršnig.
Located in a cubeare N particles, one-sixth of which are moving toward any given surface ofthe cube. If the edge of the cube is 1 cm long, then it will contain n particles(particle number density); within a unit of time n á c á Æt/6 molecules willreach each wall where the change of pulse per molecule, due to thechange of direction through 180 ¡, will be equal to 2 á mT á c. The sum ofthe pulse changes for all the molecules impinging on the wall will result in aforce effective on this wall or the pressure acting on the wall, per unit ofsurface area.1n· c · 2 · m T · c = · n · c2 · m T = p36Avogadro constantNA = 6.022 ⋅ 1023 molÐ1For 1 mole,mT=1MandV = Vm = 22.414 l (molar volume);Thus from the ideal gas law at standard conditions(Tn = 273.15 K and pn = 1013.25 mbar):p ·V =m· R ·TMFor the general gas constant:Nn=Vwherewhere NA is AvogadroÕs number (previously: Loschmidt number).R=Derived from this isp ·V =1· N · mT · c231013.25 mbar · 22.4 ` · mol −1=273.15 K= 83.14mbar · `mol · KIdeal gas law (derived from the kinetic gas theory)–If one replaces c2 with c2 then a comparison of these two ÒgeneralÓ gasequations will show:p ·V =m1· R · T = · N · m T · c2M3p·V = N · (mT · RM) ·T =orm · c22·N·( T)23The expression in brackets on the left-hand side is the Boltzmann constantk; that on the right-hand side a measure of the moleculesÕ mean kineticenergy:Boltzmann constantk=mT · RM= 1.38 · 10 −23JKMean kinetic energy of the moleculesE kin =thusmT · c 22p · V = N· k · T =2· N · E kin31.4The pressure ranges invacuum technology and theircharacterization(See also Table IX in Chapter 9.) It is common in vacuum technology tosubdivide its wide overall pressure range Ð which spans more than 16powers of ten Ð into smaller individual regimes.
These are generallydefined as follows:Rough vacuum (RV)Medium vacuum (MV)High vacuum (HV)Ultrahigh vacuum (UHV)1000 Ð 11 Ð 10Ð310Ð3 Ð 10Ð710Ð7 Ð (10Ð14)mbarmbarmbarmbarThis division is, naturally, somewhat arbitrary. Chemists in particular mayrefer to the spectrum of greatest interest to them, lying between 100 and1 mbar, as Òintermediate vacuumÓ.
Some engineers may not refer tovacuum at all but instead speak of Òlow pressureÓ or even ÒnegativepressureÓ. The pressure regimes listed above can, however, be delineatedquite satisfactorily from an observation of the gas-kinetic situation and thenature of gas flow. The operating technologies in the various ranges willdiffer, as well.In this form the gas equation provides a gas-kinetic indication of thetemperature!14HomeVacuum physics1.5Types of flow and conductance∆pmbarThree types of flow are mainly encountered in vacuum technology: viscousor continuous flow, molecular flow and Ð at the transition between these twoÐ the so-called Knudsen flow.qm%110021.5.1 Types of flowqm∆p50Viscous or continuum flowThis will be found almost exclusively in the rough vacuum range.
Thecharacter of this type of flow is determined by the interaction of themolecules. Consequently internal friction, the viscosity of the flowingsubstance, is a major factor. If vortex motion appears in the streamingprocess, one speaks of turbulent flow. If various layers of the flowingmedium slide one over the other, then the term laminar flow or layer fluxmay be applied.Laminar flow in circular tubes with parabolic velocity distribution is knownas Poiseuille flow. This special case is found frequently in vacuumtechnology.
Viscous flow will generally be found where the moleculesÕ meanfree path is considerably shorter than the diameter of the pipe: λ Ç d.A characteristic quantity describing the viscous flow state is thedimensionless Reynolds number Re.Re is the product of the pipe diameter, flow velocity, density and reciprocalvalue of the viscosity (internal friction) of the gas which is flowing. Flow isturbulent where Re > 2200, laminar where Re < 2200.The phenomenon of choked flow may also be observed in the viscous flowsituation. It plays a part when venting and evacuating a vacuum vessel andwhere there are leaks.Gas will always flow where there is a difference in pressureÆp = (p1 Ð p2) > 0. The intensity of the gas flow, i.e. the quantity of gasflowing over a period of time, rises with the pressure differential.
In the caseof viscous flow, however, this will be the case only until the flow velocity,which also rises, reaches the speed of sound. This is always the case at acertain pressure differential and this value may be characterized asÒcriticalÓ: p ∆pcrit = p1 1− 2 p1 crit 751000470250sventing time (t)1 Ð Gas flow rate qm choked = constant (maximum value)2 Ð Gas flow not impeded, qm drops to Æp = 0Fig. 1.1Schematic representation of venting an evacuated vesselMolecular flowMolecular flow prevails in the high and ultrahigh vacuum ranges.
In theseregimes the molecules can move freely, without any mutual interference.Molecular flow is present where the mean free path length for a particle isvery much larger than the diameter of the pipe: λ >> d.Knudsen flowThe transitional range between viscous flow and molecular flow is known asKnudsen flow. It is prevalent in the medium vacuum range: λ Å d.The product of pressure p and pipe diameter d for a particular gas at acertain temperature can serve as a characterizing quantity for the varioustypes of flow.
Using the numerical values provided in Table III, Chapter 9,the following equivalent relationships exist for air at 20 ¡C:Rough vacuum Ð Viscous flowλ<d⇔ p ⋅ d > 6.0 ⋅ 10Ð1 mbar ⋅ cm100(1.22)Medium vacuum Ð Knudsen flowA further rise in Æp > Æpcrit would not result in any further rise in gas flow;any increase is inhibited. For air at 20,¡C the gas dynamics theory revealsa critical value of p2 . = 0528 p1 crit(1.23)The chart in Fig. 1.1 represents schematically the venting (or airing) of anevacuated container through an opening in the envelope (venting valve),allowing ambient air at p = 1000 mbar to enter.
In accordance with theinformation given above, the resultant critical pressure isÆpcrit = 1000 ⋅ (1Ð 0.528) mbar Å 470 mbar; i.e. where Æp > 470 mbar theflow rate will be choked; where Æp < 470 mbar the gas flow will decline.dd<λ<2100⇔⇔ 6 ⋅ 10Ð1 > p ⋅ d > 1.3 ⋅ 10Ð2 mbar ⋅ cmHigh and ultrahigh vacuum Ð Molecular flowλ>d2⇔ p ⋅ d < 1.3 ⋅ 10Ð2 mbar ≠ cmIn the viscous flow range the preferred speed direction for all the gasmolecules will be identical to the macroscopic direction of flow for the gas.This alignment is compelled by the fact that the gas particles are denselypacked and will collide with one another far more often than with theboundary walls of the apparatus.
The macroscopic speed of the gas is a15HomeVacuum physicsÒgroup velocityÓ and is not identical with the Òthermal velocityÓ of the gasmolecules.In the molecular flow range, on the other hand, impact of the particles withthe walls predominates. As a result of reflection (but also of desorptionfollowing a certain residence period on the container walls) a gas particlecan move in any arbitrary direction in a high vacuum; it is no longerpossible to speak of ÓflowÓ in the macroscopic sense.It would make little sense to attempt to determine the vacuum pressureranges as a function of the geometric operating situation in each case. Thelimits for the individual pressure regimes (see Table IX in Chapter 9) wereselected in such a way that when working with normal-sized laboratoryequipment the collisions of the gas particles among each other willpredominate in the rough vacuum range whereas in the high and ultrahighvacuum ranges impact of the gas particles on the container walls willpredominate.In the high and ultrahigh vacuum ranges the properties of the vacuumcontainer wall will be of decisive importance since below 10Ð3 mbar therewill be more gas molecules on the surfaces than in the chamber itself.
Ifone assumes a monomolecular adsorbed layer on the inside wall of anevacuated sphere with 1 l volume, then the ratio of the number ofadsorbed particles to the number of free molecules in the space will be asfollows:at 1at 10Ð6at 10Ð11mbarmbarmbar10Ð210+410+9For this reason the monolayer formation time τ (see Section 1.1) is used tocharacterize ultrahigh vacuum and to distinguish this regime from the highvacuum range. The monolayer formation time τ is only a fraction of asecond in the high vacuum range while in the ultrahigh vacuum range itextends over a period of minutes or hours.
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