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LETTER 58Th Vacuum Symposium Proceedings LETTER 58th Vacuum Symposium Proceedings Another Possible Origin of Temperature and Pressure Gradients across Vane in the Crookes Radiometer* Kazuki DENPOH*1,a) *1Individual Participant, Kai-city, Yamanashi 400–0104, Japan (Received July 12, 2017, Accepted September 15, 2017) In conventional studies on the Crookes radiometer, vane temperature was presumed to be higher at the black side than at the shiny side. In this study, a new hypothesis – vane is isothermal but accommodation coe‹cients are diŠerent at the black side and at the shiny side – has been proposed and examined using heat transfer and Direct Simulation Monte Carlo (DSMC) simu- lations. The results prove that the vane is indeed isothermal under the sunlight and that gas temperature and pressure at the black side with the perfect accommodation coe‹cient become higher than those at the shiny side with a small accommodation coe‹cient. The pressure gradient across the vane acts as an area force to push the vane from the black side. It is also found the glass bulb temperature strongly aŠects the revolution of the vanes. side of the vane must be larger than that at the shiny side 1. Introduction since the mica vane in the Crookes radiometer was black- The Crookes radiometer1,2) wasinventedinthe19th ened with soot on one side13), and in experiments by Ota, century and even today still attracts many researchers as et al.3,4) one side of aluminum (Al) vane was coated with an interesting subject for study because of complicated carbon-black powder. Porous and rough12) surfaces of physics involved in its simplest machinery. Therefore, such soot or carbon-black powder could make the eŠec- signiˆcant eŠorts have been made via numerical tive accommodation coe‹cient nearly unity due to fre- simulations3–10) to reveal forces on vanes in the radiome- quent interactions between molecules and those surfaces. ter. This article is organized in the following manner. In those studies, the vane temperature was supposed Time variations of the vane temperature are estimated to be much higher at the black side due to larger photon through heat ‰ux balance equations in the next section. energy absorption than at the shiny side. In addition, ac- Then, in section 3, eŠects of accommodation coe‹cients commodation coe‹cient was assumed to be the same at on rareˆed ‰ow ˆelds formed around four vanes in a the both sides of the vane, and in many cases the perfect radiometer are investigated using a DSMC software14), accommodation coe‹cient was speciˆed. Their simula- which runs on the Microsoft Excel. Finally, we give con- tion results showed thermal transpiration (or thermal clusions of the present study. creep force) was produced due to temperature diŠerence 2. Evaluating Vane Temperature DT given between both sides of the vane and an area force by pressure diŠerence Dp induced across the vane. Figure 1 illustrates a one-dimensional (1D) model of Those forces push the vanes from the black side. heat ‰uxes coming into and going out from a vane. The However, one might wonder whether such that large heat ‰ux qin incoming from the sunlight to the vane temperature diŠerence DT can really exist inside the thin through the glass bulb is perfectly absorbed only at the vane. Therefore, in this study, a new hypothesis – the black side of the vane. And outgoing ‰uxes by gas heat thin vane heated by the sunlight is isothermal but accom- conduction and radiation are emitted from the both sides modation coe‹cients are diŠerent at the black side and of the vane. The perfect accommodation coe‹cient is as- at the shiny side – has been proposed and examined. Accommodation coe‹cient a11,12) is a fraction indicat- ing how much gas molecules impinging a wall accommo- date to the wall temperature Tw and described as a＝(Ti －Tr)/(Ti－Tw), where Ti and Tr are the temperatures of incident and re‰ected molecules, respectively. The re‰ec- tion is perfectly specular for a＝0 and diŠuse one for a＝ 1. It is well known that accommodation coe‹cient sig- niˆcantly aŠects gas heat conduction, especially in the transition regime to the free molecular ‰ow regime12). Naturally, the accommodation coe‹cient at the black * Presented as a contributed poster at the 58th annual symposium of the Vacuum Society of Japan, Aug. 18, 2017. Fig. 1 1D model of heat ‰uxes coming into and going out from a) E–mail: denpoh＠mvi.biglobe.ne.jp avane. Vol. 60, No. 12, 2017 ―()13 ― 471 sumed at the both sides of the vane as most conventional 0.1 mm for the base material and 0.01 mm for the soot studies did. Presuming the Biot number Bi is much less layer as shown in Fig. 1. The sunlight heat ‰ux19,20) is as- 2 than unity (Bi91), heat ‰ux balance equations to solve sumed to be 700 W/m ,andTg is ˆxed at 298 K (25 deg. the vane temperature TV at time t can be simply written C). as lumped ones, i.e., As plotted in Fig. 2, TV rises with time from Tg to 363 &T t K (90 deg. C) for Al and 347 K (74 deg. C) for mica, re- q －(qt ＋qt )＝－k V,(1)spectively. The temperature diŠerence between the black in g,B r,B &x side (soot) and the shiny side is quite small and at most &T t＋Dt 0.12 K. The actual Bi calculated is less than 0.01 for each rL C V ＝q －(qt ＋qt )－(qt ＋qt ),(2) b p &t in g,B r,B g,S r,S Al and mica vane, and therefore solving the lumped equ- ations (1) and (2) is reasonable. Consequently, the results where, k is the thermal conductivity, r the density, L the b suggest that thin vane should be almost isothermal under thickness, and C the speciˆc heat at constant pressure of p the sunlight. the vane, respectively. The superscriptions t and t＋Dt indicate time, Dt being the time step. The heat ‰uxes out- 3. Simulation of rareˆed ‰ows in radiometer going from the vane, i.e. q by gas heat conduction and g Next, rareˆed ‰ows induced under the new hypothesis q by radiation heat transfer are described as, r have been investigated using the DSMC software14). 1 1 8kTg Figure 3 depicts thermal boundary conditions of a two- qt ＝ n švDE＝ nk (T t －T ), (3) g,B/S 4 2 pm V,B/S g dimensional (2D) radiometer. The temperatures of the vane and the glass bulb are kept at TV and TG, respec- t t 4 4 qr,B/S＝eB/Sss(T V,B/S) －T gt,(4)tively. Each accommodation coe‹cient is aB at the black side and a at the shiny side of the vane. The perfect ac- where, n is the gas density, šv the mean thermal velocity S commodation is assumed at the glass bulb surface. The of gas molecules, m the mass of a gas molecule, k the length L of the vane is 13 mm. The thickness L is 2 Boltzmann constant, T the ambient gas temperature, e a b g mm, which is thicker than actual one (～0.1 mm) be- the emissivity, and s the Stefan-Boltzmann constant, re- cause of a limitation of the software. The inner radius of spectively. The subscriptions B and S denote the black the glass bulb is 25 mm. side and the shiny side of the vane. The gas in the radiometer is air with its molecular Figure 2 shows the vane temperature T at the black V weight of 28.97 g/mol and two rotational degrees of side and at the shiny side as a function of time, obtained freedom. The inter-molecular collision diameter is calcu- by solving equations (1) and (2) alternately. In this calcu- lated from the viscosity at the reference temperature T lation, Al and mica are chosen as a base material of the ref ＝ T T . The Maxwell molecule21) is speciˆed as a vane, and soot as its blackside.Theirthermal V G molecular model, and collisions are treated using the properties15–18) are tabulated in Table 1. The thickness is variable hard sphere model22) and the Larsen-Borgnakke model23). As initial conditions, the radiometer is ˆlled with the stationary air at a reference pressure pref and Tref. Figure 4 shows ‰ow ˆelds obtained for pref＝1Pa,TV ＝348 K (75 deg. C), TG＝298 K (25 deg. C), aB＝1(per- fect accommodation), and aS＝0.01. The gas tempera- ture and thus the pressure which corresponds to the total kinetic energy of gas molecules24) become higher at the black side of the vane than at the shiny side. As a conse- Fig. 2 Time variations of vane temperature for two diŠerent base materials, Al and mica. Each vane has a soot layer as the black side. Table 1 Thermal properties of vane materials15–18). 3 r (kg/m ) Cp (J/kg–K) k (W/m–K) e Al 2688 905 237 0.17 Mica 2100 880 0.5 0.72 Soot 100 1000 0.05 0.95 Fig. 3 Thermal boundary conditions in 2D radiometer model. 472 ―()14 ― J. Vac. Soc. Jpn. Fig. 6 Air ‰ow ˆelds simulated for p ＝1Pa,T ＝348 K, T ＝ Fig. 4 Air‰owˆeldssimulatedforpref＝1Pa,TV＝348 K, TG＝ ref V G 348 K, a ＝1, and a ＝0.01. 298 K, aB＝1, and aS＝0.01. B S servations. It is said that the revolution slows down and some- times stops when the radiometer is exposed to the sun- light for a long time.
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