Methods of Determining Thermal Accommodation Coefficients from Free Molecular Flow Heat Transfer Data

Transcription

1 The Space Cngress Prceedings 1969 (6th) Vl. 1 Space, Technlgy, and Sciety Apr 1st, 8:00 AM Methds f Determining Thermal Accmmdatin Cefficients frm Free Mlecular Flw Heat Transfer Data R. K. Irey University f Flrida D. E. Klett University f Flrida Fllw this and additinal wrks at: Schlarly Cmmns Citatin R. K. Irey and D. E. Klett, "Methds f Determining Thermal Accmmdatin Cefficients frm Free Mlecular Flw Heat Transfer Data" (April 1, 1969). The Space Cngress Prceedings. Paper 5. This Event is brught t yu fr free and pen access by the Cnferences at ERAU Schlarly Cmmns. It has been accepted fr inclusin in The Space Cngress Prceedings by an authrized administratr f ERAU Schlarly Cmmns. Fr mre infrmatin, please cntact cmmns@erau.edu.

2 METHDS FR DETERMINING THERMAL ACCMMDATIN CEFFICIENTS FRM FREE MLECULE FLW HEAT TRANSFER DATA D. E. Klett and R. K. Irey Department f Mechanical Engineering University f Flrida Kn Area NMENCLATURE Specific heat at cnstant vlume Average speed Energy Energy flux Diffuse shape factr f surface i with respect t surface j Knudsen number (hereafter abbreviated fmf heat transfer) may be a large percentage f the ttal heat flux in a system with Kn in the range 3 t 100. ne criteria fr the imprtance f fmf heat flux in a system is the value f the radiatin heat flux. The radiatin decreases with surface temperatures as the furth pwer. Hence it is primarily at lwer temperatures, such as ccur in crygenic insulatin applicatins, that the fmf heat flux is f mst imprtance. T calculate the fmf heat flux in a system it is necessary t have sme knwledge f the thermal accmmdatin cefficients fr the gases and surfaces invlved. The thermal accmmdatin cefficient, a, is defined by the relatin Bltzmann cnstant m Mlecular mass n Mlecular density r number f surfaces n" Mlecular flux p Pressure Free mlecule flw heat flux Ttal free mlecule flw heat flw Gas cnstant Radius r radial displacement Temperature Accmmdatin Cefficient 0,4) Spherical crdinates # Rati f specific heats INTRDUCTIN Heat transfer in rarefied gases, a previusly little investigated subject, has undergne advances in the past few years due t increased areas f applicatin. Amng new areas f applicatin are the insulatin f bster prpellant strage vessels and heat transfer t exterir surfaces f space vehicles. Lw density heat transfer is subdivided int several regimes with the Knudsen number, Kn (The ratin f mean free path t charac teristic dimensin f the system) serving as the criteria fr designatin.' At sufficiently lw pressures Kn becmes large cmpared with unity. Fr Kn greater than abut 3, intermlecular cllisins in a gas becme negligible cmpared with gas mlecule - bundary cllisins. This is referred t as the free mlecule flw regin f gas dynamics and heat cnductin. Heat cnducted between surfaces separated by a gas with Kn>3 ccurs predminately by the mechanism f thermal exchange by direct mlecule - wall cllisins. Free mlecule flw heat transfer E. - E r E i - Ew CD where E^ is the energy carried by an incident mlecule, E r is the energy carried away by a reflected mlecule and EW is the energy that a mlecule wuld pssess if in thermal equilibrium at the wall temperature, T. Thus a represents the degree t which an incident mlecule appraches thermal equilibrium with a surface during a cllisin. A value f unity fr a represents cmplete accmmdatin. The thermal accmmdatin cefficient appears t be a functin f the nature f the gas, the nature and temperature f the surface, the Knudsen number and the energy difference between incident mlecules and the fully accmmdated energy.^ The purpse f this paper is t describe and cmpare tw techniques fr determining the thermal accmmdatin cefficient fr a given gas and surface at a specified thermdynamic state. ANALYSIS Tw techniques are t be cmpared; ne due t Klett and Irey described in an earlier paper^, which utilizes the classical lumped analyses apprach, and a methd which stems frm the analytical wrk dne by Yau Wu3>4,5-[ n frmulating an integral apprach t the prblem. Necessary fr the applicatin f bth techniques is a cllectin f heat flux data fr a gas in a free mlecule flw enclsure. The data used fr this wrk were btained by the use f a tw-directinal guarded calrimeter. The experimental apparatus and techniques used in cllecting the data are described in references 2 and 6. The data cnsist f values fr the fmf heat flw between tw cncentric cpper cylinders separated by a gas at varius pressures in the Knudsen number range 2 t 30. The surfaces were held at cnstant temperatures crrespnding t the nrmal biling pints f liquid nitrgen and liquid Fren 12, i.e K and 243 K respectively. Data were btained fr rientatin f the heat flux vectr in bth the inward and utward nrmal directins. The heat flw data fr air, nitrgen and helium 1-1

3 are shwn in Figure 1. MPED ANALYSIS cylinder gemetry t be given by the expressins 1 > 6 Frm the classical derivatin f the fmf heat flux between tw cncentric cylinders at cnstant temperatures the fllwing equatin is btained. l» 6 and r 1 /r 2 )(a 1 a 2 -a 1 )-a 2 (5) where p is the pressure f the gas separating the cylinders, T is the mean temperature f this gas, Uj, Tj,, A, and a 2, T 2, A 2 are the accm mdatin cefficients, temperatures and areas f the inner and uter cylinders respectively. It can be seen that fr given temperatures, pressure and heat flux and unknwn accmmdatin cefficients the. equatin has tw unknwns, a,, and a 2. Thus the necessity fr btaining data fr the heat flux vectr in tw directins; this prvides tw equatins in the tw unknwns. Sme discussin n the mean temperature, T, f the gas is nw necessary. In a free mlecule flw situatin a gas cntained between tw cnstant temperature surfaces will be cmpsed f tw streams f mlecules, ne leaving surface 1 with sme reflected temperature Tr-, and ne leaving surface 2 with sme reflected temperature Tr 2. The mean temperature, T, then represents sme average f these tw reflected temperatures and will, in general, be spatially dependent. By emplying a radiatin analgy and assuming the surfaces t be diffuse reflectrs and the tw streams f mlecules t be Maxwellian, the expressin fr the mean temperature, T, as a functin f r was fund t be 6 L l = h\ where r-, and r? are the radii f the inner and uter cylinders and r frm the center line. is the radial distance Taking the average f this expressin by integrating ver the range frm TJ t r 2 and dividing by fr the averaged mean temperature. In 1 2 (2) r^-r 2 yields the fllwing expressin 2(rr r l ) 2(r 2 -r 1 ) If the values f r, and r~ fr the experimental apparatus, i.e ana respectively, are put int (3) it becmes VT ,5801 The reflected temperatures T and T are fund by the classical derivatin Mr the cncentric C3) (4) (r/rj(a a -a )-a 1 When equatins (4), (5), and (6) are cmbined with the tw equatins resulting frm (2) fr the tw directins f heat flux, under the cnstraint that the heat fluxes be cmpared at equal Knudsen numbers, the result is a set f five equatins in the five unknwns a-,, «2, Tr ^, T and T. These equatins have been slved n an IBM 360 cmputer using an iterative scheme. These previusly reprted results 2 fr the three gases air, nitrgen and helium are shwn pltted in Figures 2 thrugh 7 alng with the results btained by the techniques described belw. INTEGRAL ANALYSIS Wu shws that the mlecular flux, n", in a free mlecule flw enclsure with arbitrary wall temperature distributin is an invariant fr the system. 3 It may be evaluated frm the expressin where Tp L is the temperature f the mlecules leaving ~he surface element which subtends the slid angle element sine)) dcf> d at the pint r. The temperature at a pint in the system is defined by the average kinetic energy f a mlecule E=l kt = ^1 2 2 where c is the average speed f the mlecule. Frm this T(r) = 3k Upn assuming a Maxwell Bltzmann velcity distributin the expressin fr the temperature evlves as T(r) = Jr.fr. 1* VVT0+ sin( >d<j)d<f> Writing an energy balance fr sme pint S n the enclsure gives E'.f (S) = // E"( 1 A r (6) (7) (8) K(S',S) ds 1 (10) where the subscripts i and r refer t incident and reflected, and K(S',S) is the prbability 1-2

4 that a mlecule leaving pint S' n the bundary will impinge upn pint S. The integral extends ver the entire surface, A, f the enclsure. where F^- is the diffuse angle factr, the ttal heat flw t surface i frm all surrunding surfaces becmes Assuming cnstant specific heats, the fllwing expressin fr the reflected and incident temperatures may be written directly frm (10) Q = m ri" (c a. A. // T (S') K(S',S) ds' A r (11) F.. T. - T ), ij rj wi (14) Equatin (1), the definitin f a, may be written in the frm, prviding cy is cnstant Tr (S) = (1 - a) T. + a TW Nw using (11) fr T^ this may be written as Tr (S) = (1 - a) // T (S') K(S',S) ds ' A r a Tw (S) (12) where a is the accmmdatin cefficient at the pint S. surface is The net heat flux t the pint S n the M _ pn _ pn q fmf i r r in terms f temperatures = m n " ( c - Tr (S)) Rewriting equatin (12) fr n cnstant temperature surfaces and integrating ver A- in a similar fashin yields an expressin fr the reflected temperatures. = (1 - cx) + a. T i w (15) In the present case f cncentric cylinders with flat bttms the enclsure was divided int fur lumped cnstant temperature surfaces. Thus (15) becmes a set f fur linear algebraic equatins. Cmbining these with (14) fr the tw cases f reversed heat flux gives a set f six equatins in the fur unknwn reflected temperatures and tw unknwn accmmdatin cefficients.* Since n" is invariant in a free mlecule flw enclsure it may be evaluated at the pint f pressure measurement s lng as this pint is a part f the free mlecule flw system. Fr instance, if the pressure is measured externally by a McLed gage cnnected t the system by a free mlecule flw passage, the pressure and temperature f the gage, p and T respectively, are used in (7) which then becmes Thrugh the use f (11) and (12) this may be written as n" = ~\/2TTm k T V nr (16) q"(s) = m n" (cy + ^R) a Tr (S') K(S',S) ds' - TW (S)] (13) Nw, if the enclsure cnsists f n cnstant temperature segments, the heat flux t the i tn segment is q" - m n"(c + ^R) a. i v i I // T K(A A ) da - T j=l Aj rj J X J W Under the assumptin that the surface A. is unifrmly bmbarded the temperature T r j may be cnsidered cnstant and mved utside the integral. If this expressin is integrated ver the surface area A^ and it is nted that fr a diffuse reflectin system // // K(A A ) da da = At F Ai Aj ' J J Hwever this frtunate situatin was nt present in the experimental arrangement. The pressures recrded in the data were the average pressures within the fmf enclsure. It is there fre necessary t knw the mean temperature, T, as befre. Perfrming the integrals in equatin (9) at the pint r~ crrespnding t the averaged value f T given in equatin (4) results in an expressin fr the mean temperature frm the integral apprach. T = r2 (17) Equatins (4) and (17) are bth expressins fr the mean temperature at a pint in the gas fr the specific gemetry at hand. Implicit in the derivatin f (4) are a pair f lgical but arbitrary definitins f the average mlecular number density and the average mlecular flux in terms f the tw types f mlecules present, i.e. thse with temperature T. and thse with *The accmmdatin cefficients are assumed t be the same n like surfaces at the same temperature. 1-3

5 temperature Tr2. Equatin (17) is based slely n the definitin f the mean kinetic energy f a mlecule and is therefre less arbitrary and mre nearly crrect. The nly deviatin frm exactness in this expressin is the apprxi matin in the reflected temperature distribu tin used in perfrming the integratin. The slutin f equatins (14) and (15) is accmplished n the cmputer by assuming starting values fr the accmmdatin ceffi cient, slving the set (15) and using the values f Trj, s fund, t calculate new values f a frm (14). T separately investigate the influences f the mean temperature and the number f lumped surfaces several variatins in data reductin were tried. The previusly reprted values f the accmmdatin cefficients which were btained frm equatin (2) with equatin (4) fr the mean temperature are cmpared in Figures 2 thrugh 7 with the values btained by the fllwing methds: A) Equatin (14) cnsidering fur lumped surfaces was used with equatin (17) fr the mean temperature. B) Equatin (14) cnsidering fur lumped surfaces was used with equatin (4) fr the mean temperature. C) Equatin (14) cnsidering tw lumped surfaces was used with equatin (17) fr the mean temperature. D) Equatin (14) cnsidering tw lumped surfaces was used with equatin (4) fr the mean temperature. DISCUSSIN F RESULTS It may be seen frm Figures 2 thrugh 7 that the previusly reprted values differ by as much as 0.10 frm the values calculated by methd (A), i.e., by using the equatins derived frm the integral analysis and assuming fur lumped surfaces. The largest difference ccurs with helium when Tw=243 K. The same trend with surface temperature, i.e. a decreases with increasing T, ccurs in bth cases as it des, in fact, in all f the cases tried. The trend with Knudsen number is the same fr bth these techniques with the exceptin f nitrgen fr 243 K. In this instance the previusly reprted values decrease with Kn whereas the methd (A) values increase with Kn. Studying the results frm the ther reductin schemes reveals that increasing the number f lumped surfaces tends t decrease the difference in the cmputed values fr the tw surface temperatures. A similar effect is nted fr the influence f the mean temperature. The use f equatin (3) reduces the differences in the accmmdatin cefficient due t surface temperature ver thse fund by the use f equatin (17), hlding cnstant the number f lumped surfaces cnsidered. Cmparing the previusly reprted values with thse fund by methd (D) illustrates the fact that equatin (14) fr tw lumped cylindrical surfaces becmes equivalent t the classical expressin, equatin (2). CLSURE Tw appraches t the prblem f fmf heat flux between tw cncentric cylinders have been presented. Discussed first were the results f a lumped analysis fr the particular gemetry and then a general integral analysis gd fr any cnfiguratin. The integral analysis prvides the mst useful tl fr making heat transfer calculatins and, in a reverse sense, fr determining thermal accmmdatin cefficients frm heat flux data. It allws any system gemetry and surface temperature distributin althugh fr a very cmplicated cnfiguratin the determinatin f the shape factrs may becme intractable. Fr the system gemetry emplyed in this wrk the integral technique must be cnsidered mre inherently accurate due t the finite length f the cylinders giving rise t end effects nt accunted fr in the lumped analysis. Its use is therefre recmmended fr making calculatins and fr data reductin. If fmf heat flw data are t be btained fr purpses f determining thermal accmmdatin cefficients it is recmmended that the pressure sensing device, if external t the system, be cnnected by a free mlecule flw passage. Thus the necessity f evaluating equatin (9) with the attendant intrductin f sme degree f apprxi matin and uncertainty is avided. REFERENCES 1. Devienne, F. M., "Lw Density Heat Transfer," Advances in Heat Transfer, Academic Press, New Yrk (1965). 2. Klett, D. E. and Irey, R. K., "Experimental Determinatin f Thermal Accmmdatin Cefficients," Paper K-l presented Crygenic Engineering Cnference, Cleveland, hi, Aug T be published Adv. Cry. Eng. Vl. 14, (1969). 3. Wu, Y. "Theries f Thermal Transpiratin and Mlecular Effusin," United States Air Frce ffice f Aerspace Research, AFSR , AD (1965). 4. Wu, Y., "Kinetic Thery f Thermal Cnductin in a Cllisinless Gas," United States Air Frce ffice f Aerspace Research, Aerspace Research Labratries, ARL , AD (1966). 5. Wu, Y., "Thermal Cnductances in a Cllisin- less Gas Between Caxial Cylinders and Cncentric Spheres," United States Air Frce ffice f Aerspace Research, Aerspace Research Labratries, ARL , AD (1966). 6. Klett, D. E., "Experimental Determinatin f Thermal Accmmdatin Cefficients Using a Tw Directinal Guarded Calrimeter," Masters Thesis, University f Flrida, Depart, f Mechanical Engineering (1968). 1-4

6 NTE: USE THE LEFT RDINATE FR NITRGEN AND HELIUM AND THE RIGHT RDINATE FR AIR I i C X 5 -I C I X LL. g 2 UJ <D a: i U Lu 0.2 i UJ CASE I CASE ii AIR: CASE i 0 AIR: CASE ii INNER SURFACE AT UTER SURFACE AT INNER SURFACE AT UTER SURFACE AT NITRGEN: CASE i D NITRGEN: CASE ii v HELIUM: CASE i A HELIUM: CASE ii 77.4 K 243 K 77.4 K K 1.0 X i I LJ ijj tr.i 10" 5 I0~3 2 PRESSURE (mm f Hg) 10" 0.05 Fig. I Free Mlecule Flw Heat Flux Versus Pressure 1-5

7 T r - i i I I i r it: itj ' <>- * I * a Fig < Fig KNUDSEN NUMBER PREVIUS DATA METHD A METHD B METHD C METHD D Accmmdatin Cefficient Versus Knudsen Number Fr Nitrgen Cpper At 77.4 K KNUDSEN NUMBER Accmmdatin Cefficient Versus Knudsen Number Fr Nitrgen n Cpper At 243 K nr 14 n 16

8 I.I Si H u: KNU08EN NUMBER PREVIUS DATA METHD A METHD B METHD C - METHD D Fig. 4 Accmmdatin Cefficient Versus Knudsen Number Fr Cpper At 77.4 K KNUD8EN NUMBER 12 I 14 Air n. 5 Accmmdatin Cefficient Versus K nds en Number Fr Air n Cpper At 243 K H

9 0.5, u: 0.6 H UJ u KNUSEN NUMBER PREVIUS DATA METHD A METHD B METHD C METHD D Fig. 6 Accmmdatin Cefficient Versus Knudsen Number Fr Helium n.5 <X4-0.3-) Cpper At 77.4 K KNUDSEN NUMBER Fig. 7 Accmmdatin Cefficient Versus Knudsen Number Fr Helium n Cpper At 243 K