^p^jAA^O\ i5. M. LEADON o CO < NATIONAL ADVISORY COMMITTEE § FOR AERONAUTICS TECHNICAL MEMORANDUM 1301 THE FLOW OF GASES IN NARROW CHANNELS By R. E. H. Rasmussen Translation of "Uber die Stromung von Gasen in engen Kanalen." Annalen der Physik, Band 29, Heft 8, August 1937 Washington UNIVERSITY OF FLORIDA August 1951DOCUMENTS DERARTMENT 120 MARSTON SCIENCE LIBRARY RO. BOX 117011 GAINESVILLE, FL 32611-7011 USA cf^ V, (V + V"), V ' 3 cm V " cm cm-^ 1298.5 1301.0 1298.0 1368.0 1366.1 1370.6 2669. T 2670.2 2669.3 1299.2 1368.2 V averages 2669.7 Since the equation Vq' + Vq" = (V + V")^ is to be fulfilled, the values Vg' = I3OO cm^, V^" = I369 cm^, (V + V")^ = 2669 cm3 were U - J O -^-^ ? \- 'O involved. Subscript o indicates that the values apply when the pressure in the apparatus is zero. Owing to motion of the mercury in the gages, the volume varies with the pressure, hence V ' = V ' + ap and analogousl for Vp" . The inside diameter of the manometer tubes was around I.5 cm, hence, the area was about 1.8 cm^ and a = 0.9 if P is given in cm Hg. MCA TM 1301 The formula (l) is applicable when assuming constant volumes. Gaede-^ has shown how the variation of the volume should be allowed for in the calculation, and that it yields a correction factor ■^ = ^ * 1(77 - V7] V ' V " by which the quantity — 2 °__. must be multiplied. He further called 'o *o attention to the fact that the small volume v between the flow channel and the stop cock in the feed line yields a correction factor P ' - P " V V 11 k,. = 1 - — - — with which the ratio -^ =Tr must be multiplied. -v V' V" Pg - Pg Gaede's corrections were taken into consideration in all measurements. Every alteration of the apparatus, every exchange of the flow channels and so forth, was followed by a determination of the volumes, and the insertion of the correct values in the formula. Care was taken to •keep V from getting greater than necessary, as a rule, only a few cm3 (determined by weighing out with water or mercury) . All tests were run at room temperature. The temperature t of the K flow channel was read on a thermometer mounted as near as possible to the channel, while temperature ty, was read on a second thermometer mounted on a level with the receptacles of the McLeod gages. Because of the difference in height and the natural temperature distribution in the room, t^ was usually several tenths of a degree higher than t . This difference was sufficient to prevent a distillation of the mercury from the manometers up to the flow channel. All pressure measurements were, in addition, corrected for this temperature difference. Lastly, all test data were reduced to the same temperature, 20° C, by means of Knudsen's and Sutherland's formulas for the temperature relationship between mole- cule flow and internal friction. All of the temperature corrections were very small, usually less than I/2 percent. For one part of the measurements, a pair of gas traps (locks), cooled with ice or liquid air, were blown into the feed lines AK and BK (fig. 1) to prevent the mercury vapor and any other condensable impurities from passing into the flow channel. These measurements were corrected for the apparent volume increase caused by the cooling. They amounted to about 20 cm-^ at the most. \. Gaede, Ann. d. Phys. i+1, p. 289, 1913. NACA TM 1301 7 All measurements had to be corrected for the flow resistance of the intake line. This resistance was unusually high for the prism apparatus I because of unsuitable design. In this case, the correction was so deter- mined that, while widening the slit from 1 to 2 mm, the resistance was lowered to a disappearingly small value. After that, T was measured as usualj for these measurements, the resistance of the inlet line alone determines the measured values, which are indicated by To. The uncor- rected value of the flow is indicated by Tpj^j^ and the corrected value for the inlet pipe resistance by Tpr. Therefore, for each pressure, according to the definition of T Vp- -p")h = ^(p' -p")k = ^r.k(^' -p") together with p- - p" = (p- - p")r + (P' - P")j^ the partial pressure decrements through the inlet lines and the channel carry the subscripts R and K. From these equations follow _ Tp % - Tr+k (2) R ' R+K All measurements were corrected in the same manner. For prism apparatus I and for the porous filter plate, the corrections at the lowest pressures amounted to more than 50 percent. For prism apparatus II and the other channels, the corrections were less than 10 percent. The Flow Channels The prism apparatus I is shown in figure 2. The top picture shows the prisms (in part in vertical section) mounted in a clamping device, the arrangement of which is readily apparent. The middle figure shows the prisms from above; one, P-., is seen in horizontal section. The bottom picture shows the hypotenusal faces of the two prisms. In the face of the one prism P-, three concentric channels K-i , K , and K have been ground, each one about 1 mm wide and 1 mm deep. Three conical 8 MCA TM 1301 holes H,, Hp, and H^, into which the glass tubes 1, 2, and 3 were ground, and sealed with apiezon grease, serve as leads to the channels. Two of these tubes are inlets, the third was closed and merely served as a stopper. For the series of measiirements, recorded in figure 10, channel K^ was connected with volume V ' and Kg with V" . The flow was along Ko -^ Kg. The spacing of the prisms was gaged by placing a strip of tin foil 5 nnn wide between the faces around the outermost channel Ko Correct tightening of the wing nuts F makes the ring lie flat between the faces, and become parallel. The apparatus was sealed with shellac dissolved in pure alcohol, which was made to run down in the groove between the prisms. This seal was perfectly satisfactory after the alcohol evaporated. The radii of the two annular slits were R-^^ = 1.1952 cm R = 1.8897 cm in the wide annular slit between Kp and Kn and ^1 = 0. 899^ cm Rg = 1 .0919 cm ^1 and Kp . in the narrow slit between The high inlet resistance of prism apparatus I and the subsequently large corrections made it desirable to make a number of measurements with another prism apparatus of more adequate inlet design (fig. 3)- This apparatus had only one annular slit between the radii R-, = 0.8123 cm and Rg = 1.598^ cm. The inlet lines were no less than 5 nmi in diameter at any point. The circular channel in the hypotenuse surface of the prism P^ had a trapezoidal cross section of about 2 mm in depth; it was h mm wide at the base and 8 ram on top. The clamping device had an unusual feature. MCA TM 1301 The tension was elastic; the wing nuts F did not press directly on the beam L, but on a pair of helical springs S which transmitted the pres- sure on L. The stiffness of the springs was measured and the compression read from scales etched in the blocks C. Thus, the pressure on the prisms was known and it was possible to obtain equal pressure in both springs. The distance of the prism surfaces was determined by Christiansen's method by measuring the angle between the Herschel interference fringes. When monochromatic light of wave length X is so reflected that the angle of incidence i is very near to the boundary angle of the total reflection, the reflecting power of the surface will be very high, and the light is reflected several times (fig. h) . The effect of the inter- ference of the reflected beams J>, . Jv. ... is such that the reflected o / intensity is 0, when cos b = rr-JO-i (a- = distance of the surfaces, n^ — m = a whole number). Bearing in mind that cos b = l/l -( — | sin^i and that in the case of total reflection m = 0, cos b = 0, sin i-, = -2. , \2 / . \2 one obtains the equation sin^l^ - .i„=^i^ = [^j ^^ ! X .' ^ (3) for defining the directions fi^"] of zero reflected intensity. In addition i + P = p, n sin a = n-L sin p {k) As the angles between the dark fringes in the reflected light are 2 2 2 small, it is permissible to put sin i - sin i^j^ = sin i^di^. Differentiation of the equations (h) leaves n cos a^ da = -n-, cos 3q di, which, when introduced in equation (3) and resolved, gives ^o , / '^o COS Pp A^ - mi^ ^^ ./T \/2n cos a^ cos i^ \/ da^ _ ^^ " ^(^q) ^x ' (5) 10 MCA TM 1301 where da^ _ j^ is the angle between the m"^" and the m-|_^ dark line. If specifically m - m-, = 1, it is seen that the quantity x = ^ r-^ where Aa indicates the angle between two adjacent fringes, is to be constant. Thus, the angles Zkx act as the digits 3^ 5^ 7 • • • In transmitted light, the complementary image, light stripes on dark back- ground, are "'isible, but the spacing of the stripes is unchanged (fig. 5)- The fringe system is observed in a telescope set to infinity, the angular distances are measured on a micrometer. As a rule, the first three to six fringes were used. Quantity x was defined as average value or else computed by a different equalization of the individual values, after which a is calculated by formula (5). Since the quantity K/, N is constant for a certain color, it can be found when the refraction ^ O'^ conditions and the prismatic angle are measured. The distances were measured at several points on the prism surfaces. An illustrative example of such a measurement is given in figure 6. Decisive for the molecule flow is the average value of the second power of the distance, and for the laminar flow, the average value of the third power of the distance. In the cited example a ~1/ a ~ \/ar 9.21 ± 0.02(1 The fluctuations of the room temperature had no measurable effect on the prism spacing; but a increases a little with the pressure in the apparatus so that a = aQ(l + cp). In the foregoing example c = 1.5 X 10" (p in cm Hg. ) Since p < 40 cm Hg, it is seen that a becomes, at the most, 0.6 percent higher than a.^. Since the annular slits of the prism apparatuses were formed between fine optical surfaces on which, according to Knauer's and Stern's investigations, at grazing incidence, a part of the molecules is mirror reflected, it was of great interest to measure the molecule flow in slits between rough surfaces for which the cosine law is rigorously applicable. For this purpose, the apparatus represented in figure 7 was resorted to. The slit was formed between the ground surfaces of two thick round pieces of plate glass P-, and V which were mounted in the clamping device as indicated. The lower surface of the top plate carried the three channels K , K , K_, and the three conical openings with the glass tubes 1, 2, and 3. Tin foil with rectangular sectors 1, 2, 3^ ^ ^^^ placed between the glass plates, thus forming two rectangular slits, one between K2 and Ko, the other between K-]^ and Ko. The width a of the slit is governed NACA TM 1301 11 by the thickness of the tin foil. The dimensions L - the length of the slit in flow direction - and b were measured with the cathetometer. Only the slit between K and K was used for flow measurements j its dimensions were L = 0.97 cm and b = 1.32 cm. Owing to the mat sxirface the slit width a could not be measured optically, but was determined on the basis of the laminar-flow data. A measurement of the thickness of the tin foil produced no appropriate determination of a because the large surfaces were not level enough. The grainy irregularities of the surface were of the order of magnitude of 0.01 mm. To increase this roughness still more, they were blackened in some tests with soot from a wide flame of burning turpentine oil. The apparatus was sealed with Picein, as Indicated in figure 7- The measurements indicated that the magnitude T v/m = f / .. for a given channel is solely a function of the mean path length and independent of the gas. For a more accurate check of this rule, flow channels were used which first remained geometrically constant during the time interval of the test series with different gases. Two channels, both satisfying the cited demand, were used for this purpose: a cylindrical slit between two coaxial cylindrical surfaces and a porous body, a filter plate of sintered glass. The cylindrical slit was obtained by fitting an accurately milled brass plug (compare fig. 8) in an accurately drilled hole of a solid brass block. Figure 8 represents a cut through the axis of the apparatus. The cylindrical surfaces A- and B fit the holes exactly, while the diameter of the surface C is about O.O3 mm less. The tubes R-|_ and R2 serve as inlets; so the flow in the cylindrical slit is parallel with the axis. The slit was L = 0.^7 cm in length, the diameter of the s\irface C was 1.9997 ± 0.0001 cm. The diameter of the hole itself was not measiired, but that of the employed drill was 2.0019 cm. The apparatus was sealed by means of stiff stop- cock grease applied to the groove R. The channels h were drilled to prevent an almost closed dead-air space from forming below the paz-t B. The porous filter plate was fused in a glass tube of the form depicted in figure 9. The diameter of the plate was about 1.9 cm, its thickness about 0.2 cm. The size of the pores was not known from the start. An upper limit is obtained, according to Weber by recording the pressure required to force air bubbles through the filter when it is wet and coated with a layer of water. At around 13 cm Hg, the air started to penetrate at a certain point, but an increase of the pressure to about 17 cm already forced air through at many places. This proved that the filter was homogeneous to some extent. On computing the surface tension of the water at 73 dynes/cm and assuming complete wetting, it is found 6 S, Weber, Teknlsk Tidsskrift I917, Nr. 37. 12 MCA TM 1301 that the narrowest spot of the widest pore of the filter has a diameter of about 17m when the pore is regarded as circular. 3. TEST DATA Flow measurements were made with hydrogen, oxygen, carbon dioxide, and argon. Hydrogen was produced in the Kipp apparatus from zinc and dilute hydrochloric or sulfuric acid. It was purified and dried in a solution of potassium hydroxide, a saturated alkaline solution of potassium permanganate, concentrated sulphuric acid, ajid lastly, in phosphorous anhydride. Oxygen was taken from a commercial tank and then dried in svilfuric acid and phosphorous anhydride. Carbon monoxide was manufactured in Kipp's apparatus from marble and dilute hydrocholoric acid, washed in distilled water and then dried like the other gases. An assay with absorption in potassium solution indicated 99-9 percent purity. Argon was obtained from the factory guEiranteed 99.5 percent pure; it was used without being treated. The Measurements with the Annular Slits The principal test data are correlated in table 1. The lines 2 to h give the slit dimensions with the optically defined values of the slit width a. Line 6 gives the minimum values of T y/M (M = molecule number of gas). Line 7 gives the values of T V^ at the lowest pressures at which tolerably reliable measurements could be made. The other lines eire discussed below. The accuracy of the data is characterized by the digit number, which is such that the uncertainty is one or several units of t] the last digit. The several measurements are represented in figures 10, 11, 12, and 13. The quantity T v'^ is plotted against the log-.^ ■=• = ^og -,q ^J* X = mean path length measured at 1 cm Hg, 20 in units In. The reason for this choice of representation is found in the following paragraphs. MCA TM 1301 13 At high pressures - in the laminar -flow region - the internal friction of the gas governs the flow. Assuming that the speed of the gas along the surface is zero rta^ T = — r;p 6ti2 — where t] = coefficient of internal friction, Z = natural log. Bearing in mind the slip, one obtains the boundary condition UqI = Ti— , with Uq = velocity along the surface, | the coefficient of ex- ternal friction, and ^ the velocity gradient. With ^ = ^ = slip coefficient T t "(^^ ^ ^^^S) P = A^p + B. where Ai and Bi are constants inside of a press\ire area in which ^p is to be regarded as constant. By kinetic theory, t] = cpfiX,, with p = gas density, ^ = average value of thermal velocities, and c = numerical factor. The definition of X, is that given in Landolt-Bornstein' s tables' x^(0=T6 c. H«) = I gi^ j^^ c. (7) where ^2 = « 273 -; R = 83.15 X 10^ ^ ^^ — - = gas constant o 8 M degree mol 7 'Landolt-Bornstein, Tables 1, 1923, p. 119. li^ NACA TM 1301 The employed numerical values were: Gas X2 (0°76 cm Hg) Xj_ (20°1 cm Hg.) ^20° c E2 °2 co^ A 1123 X 10"^ cm 61+7 X 10"^ cm 397 X 10"^ cm 635 X 10"^ cm 8.65^ 5.03^i 3.12H i+.96^ 0.881 X 10-^ 2.0I+0 X 10"^ 1.1+85 X 10"^ 2.21+7 X 10"^ 72 128 27 1+ 170 C = Sutherland's Constant With these numerical values ti^qO = 0.717 X 10 X x.-^ X \/M (7a) for all gases. Inserting (7a) in (6) and bearing in mind that 1 cm Hg = 13,296 dynes per cm2 one obtains T ^ = 0.971 X lo9 X -|^ ^ + 5.83 X lo9 _|_ Lp = A i + B, (8) R2 \i ^— ^ '■ " R2 H if p is measured in cm Hg and X in l-i. From it, it is seen that T v^ at high pressure becomes the same function of the mean path length for all gases as a result of the definition of X.. From figure ll+, where the measurements with the annular slit in prism apparatus I are reproduced, it then also follows that T \fM at high pressure is linearly dependent on p/a. , and that the dependence is the same for all gases, which simply implies that the measurements are in relative agreement with the employed values ^20* Starting from the concepts of the molecule flow, it must be assumed that the rule T \1M = f/^ 'i must hold also at the lowest pressures, where MCA TM 1301 15 the internal friction has either only little or else no significance at all for the flow. And so figure 10 actually indicates that the dependence of T \/M' on p/x, is nearly the same for all gases, not only at the high pressures, but also in the entire pressure range explored. The discrepancies lie at the limit of instrumental accuracy. The foregoing indicates that the minimum of T lies at the pressure where the mean path length X is equal to the slot width a. The height of the minimum can also he approximated quite simply in ;l£ ,8 view of the validity of the relation T„. ~ T„ , T„ being calculated "' min Kn' Kn from Knudsen's general formula*- ^Kn - V M w' B which, applied to a circular slit, gives ^dZ . (9) S2 The values 1-^^ \jW computed from equation (lO) are given on line 8 of table 1. At the lowest pressures, T /W becomes constant independent of the pressure, as is plainly seen in figure 13 (hydrogen in prism apparatus II), where the measurements have been carried out to about p = 3 X 10-5 cm Hg. Throughout the entire pressure zone in which the mean path length was relatively large compared to the slit width R2 " ^i (marked on all graphs), T is practically constant (T ) . In these measurements, the inlet lines were cooled by liquid air, so that the flowing hydrogen was not contaminated by mercury vapor. Another effect of the cooling was that the McLeod measurements were more reliable and this made it possible to extend the measurements so far into the low pressure zone. In the other test series, the boundary values T„ were less accurately determ ined . o Ann. d. Phys. 28, p. 76, I909. l6 MCA TM 1301 A calculation of the free molecule flow in slits of the form employed here has never been attempted, to the writer ' s knowledge . An attempt at an accurate treatment of this problem resulted in difficult calculations and will not be mentioned. Professor N. Bohr pointed out that an approximate solution could be obtained by appropriate use of Clausing' s formula for the molecule flow in a rectangular slot. This formula, which is valid for b » L » a, can be written as T ^M = i#V^fl + zL)a£b (11) Cl 2 ■/ n \2 a/ L For example, putting L = Rp - Ri^ b = ^1(^2 ■•" %) gives for the flow in the circular slot ^^ 1 '\r~ -,r~ fi ^2 - ^i^^^fe + %) Tci Vm = i y^n /r^ (i . Z-^^J-A-2_1A ^ (12) and this expression must become exact when (R2 + R2_) » Ro - R-i » a. The values of line T^, v/^ in table 1 are computed from formula (12); Ro + Ri C:i the ratio — ^ i is also given in the table for the various circular ^2 -^1 slits. It is seen that the computed values T \/m are greater every- where than the observed T V^. This is due, in part, to the fact that the measurements were not extended to sufficiently low pressures in all test series, and in part to the fact that the mercury vapor was frozen in only one of the aforementioned series and therefore had a retarding effect on the flow in the other series; lastly, the condition R + R » R - R is not fulfilled and it is easily seen that the very noncompliance with this condition must result in a divergence in the R^ + Rn direction of computed value > observed value. The ratio '2 " ^^1 Rg - R^ is greatest for the small circular slit, namely, 10. U, and here also the agreement between observation (I.6) and calculation (I.72) is fairly good. It was found that all test data can be represented in close approximation by MCA TM 1301 17 T\/M = Ai + B + C log A. ^ + D (13) ^2 - ^1 + D The values of the constants A, B, C, and T), which are dependent on the slit, but not on the nature of the gas, were determined graphically and entered in lines 9 to 12 of table 1. The curves of figures 10 to ik are computed from equation (13) with these constants. At high pressures, the last term (c) in equation (13) is dis- appear ingly small and by comparison with equation (8) it is seen that A = 0.971 X 10^-%-. From this equation, one of the quantities A or a can be computed when the other is known. Computing a from the A values gives the values indicated with a in table 1; they are str all smaller than those defined optically. The difference Aa = a . - a , is in all cases about O.I5 - 0.2|i. No satisfactory explanation of this discrepancy could be found. It was possible to attribute it to the inertia forces which were not allowed for in the calculation of equations (6) and (8); but the calculation indicated that they were only about 1/10° of the friction forces and hence were altogether insignificant. On computing the coefficient A-|_ of formula (6) for each gas and then a , by means of the value il2oO^ almost the same value is obtained for all three gases Hp, Og, and CO2, which, at about 9-01-1, is in good agree- ment with the value 9-02M. obtained in this case from equation (8). This implies that the employed gases were sufficiently pure. To be completely sure in this respect, a series of measurements with atmospheric air were carried out. The air was dried in sulphuric acid and phosphorous anhydride. The inlet tubes were cooled with ice in order to reduce the mercuiy -vapor pressure in the slit to a low value. The measurements were carried out in such a way that for each value of the pressure p (average values) several (2, 3^ or k) independent measurements of the quantity T were made at decreasing values of the pressiore difference. The value of T for the same p never indicated a systematic course, which confirms that the inertia forces are disappear ingly smallj hence, the use of the average value of T for each p. The measurements were limited to the high-pressure range. The results are represented in the following table: 18 NACA TM 1301 Atmospheric Air in Prism Apparatus IIj ^pt ~ ^•^'?^ p T ^ C (T - C) (T - C) A ^ ODS obs ber o-b 2.15 0.2251 0.0085 0.2166 0.2167 -0.0001 5.21 .3061 .0037 .3021^ .30U5 -.0021 9.36 Mil .0020 .i+251 .i|236 +.0015 13.79 .55U5 .0014 •5531 .5508 +.0023 20.08 .730i^ .0010 •729i^ .7313 -.0019 2i+.82 .8616 .0007 .8609 •8673 -.0061^ 28.59 .9828 .0007 .9821 .9755 +.0066 32.39 1.0810 .0006 i.o8oi+ 1.081+5 -.0041 36.50 1.2150 .0005 1.2lil5 1 . 2026 +.0119 In order to be able to use all the measurements for determining the slope coefficient taken into account to some extent. A-[_, the curvature of the T, p curves must be formula (13) was computed and subtracted from constants For this reason, the C term of obs The values of the 1.5, c ^ 0.67 1/m \f29 0.12 J X,-L = 4.7|a vere utilized. The correction term is indicated with C in the table. The corrected term (T - C')qi_|3 is to be linearly dependent on p. Smoothing resulted in T - C = 0.0287p + 0.1550j the values computed from it along with the differences A ■, are reproduced in the table. -. Q-Q^Q^ and Ti2o = 1-820 X 10" in equation (6) gives Inserting ^1 = str 7.98^. An optical measurement of the slit width accompanied the flow measurements. The results were agpt ~ 8.17m measured with green mercury light and aopt = 8.13m. measured with yellow mercury light j average value aQp-^ = 8.15 ± 0.02m . The difference agpt " ^str~ 0.17M. This example proves without a doubt that the nonagreement of the two methods is real, and that it cannot be explained by insufficient purity of gas and subsequent uncertainty of the value of T[. The optical test method is not gone into any further. Unless stated otherwise, the optically determined values of the slit widths are used in this report. Returning to the discussion of formula (13), it is noted that, by comparing the expressions (12) and (13) for X. = » NACA TM 1301 19 2 . — a (Rp + Bi ) v = b\/RT — — ^ i- = ^ C ^2 - \ c where c and b are pure numbers. From this, it is seen that the ratio 1. = £ should be constant, which (compaire table 1) is not C c altogether true. The discrepancies are largely due to the previously cited insufficient fulfillment of the condition R + R » R^ - R-,. As stated before X. ^ a, T . ~ T„^ mln ' mm Kn is valid for the minimum. Knudsen's general formula L ^^-n\- -=i^i;? dZ is obtained by computing the tangential motion quantity B, given off on the channel wall per unit time per unit area, and by putting the total motion quantity Jo B-j^o dZ equal to the driving force (p' - p")S. A premise of this calculation is that B-|_ can be proportional to the average value of the flow velocities over the channel cross section. In consequence, the calculation can be valid only for channels with somewhat homogeneous cross sections, for example, slits of constant width, circular-cylindrical pipes, and so forth, as already noted by v. Smoluchowskl and Clausing. Furthermore, these researchers have indicated that an exact solution on the assumption of the cosine law for X = 00 produces different and greater values of the quantity T than Knudsen's calculation. Then it is readily apparent (for instance, on examining the Smoluchowskl - Clausing calculation, or in the calculation of the pure effusion in a slit J compare on the next page) that a very large part of the free 20 NACA TM I30I molecule flow in the slit is due to molecules which fly approximately parallel to the slit walls and therefore cover great distances between two collisions against the wallsj this holds for infinitely small pressures. If the pressure has such a value that a (small) number of collisions occior (in the zone L > X > a), these far-reaching molecules will con- tribute much less by reason of the mutual collisions. Against it is a contribution of molecules which have participated on mutual collisions but which at the beginning (that is, when X is still great compared to a) is very much smaller than the decrease in flow caused by the collisions. The total effect of the collisions is a decrease in flow in the zone X, = <» to X. ~ a. In the minimum zone X ^ a, the assumptions of Knudsen's calculation, particularly the assumptions of constant flow velocity over the channel cross section, appear to be fulfilled to some extent, hence T 4„ — Tj^n* The discussion of the circular-slit measurements is concluded with a few remarks about the pure effusion and the effect of the mirror reflec- tion. The effusion, that is, the contribution to the flow due to molecules which pass the slit without impinging on the wall, is easily computed. The result is discussed because it is a fine example of the importance of the cosine law. The flow in the annular slit is assumed to be from the inside toward the outside: I. Assumption: The molecules are emitted from the cylindrical surface 2JtR-, X a according to the cosine law, so that ^Ef ^ ^ = 2 iff \fie a2j NACA TM 1301 21 where J is the defined integral J = '1 dx P JO n% 1 -r X V^l - 1 ,1 . ^1 Rl TT arc sm — + 2 — - ^2 R2 Rg^ .^— s; II. Assuming, by way of contrast, that the molecules proceed from the plane circular areas 2rtR-, according to the cosine law Ri T II 1/^=1/^ l&a2 -^ ■-Ef R2^ - Ri^ The results are, like the assumptions, not identical. To indicate the numerical significance, the ratio "•Ef 2J "■Ef II rtR 1 was computed for several cases ^2 Rl M T I Ef Prism apparatus II Prism apparatus I, wide slit . . Prism apparatus I, narrow slit . I.598I+ 1.8897 1.0919 0.8123 1.1952 O.899I1 3.88 2.50 1.1+7 1.10 1.01 1.70 1.^3 1.17 1.03 1.00 22 NACA TM I3OI T I R It is apparent that the ratio _££ — —>1 when -2. -^1, which is T II ^1 Ef also apparent from the expression for J, and for the rest is immediately clear. It is difficult to decide which of the assumptions is preferable for the annular slits, or whether a third might not be better stillj for this reason, the effusion cannot be indicated with a high degree of accuracy. The foregoing calculations are applicable only in the absence of mutual collisions of molecules; the effusion decreases considerably with increasing pressure. An exact calculation of it is not of interest in view of the uncertainty of the first calculation and the small value of the effusion; hence, the decision to put the effusion at pressure p equal to Tgf-'-e ^ , where pX. = X,]_. The values of the effusion by assumption I, shown in table 1 below T_ „ \/M, amount to a few percent of the flow. The individual measurements T \/M with and without correction for the effusion are plotted in figure 13 • The other graphs contain only the uncorrected values, which were also used for defining the constants in formula (13) • Knauer and Stern have proved divergences from the cosine law for grazing incidence toward polished surfaces. Since precisely the molecules, moving grazingly toward the walls, contribute much to the flow, it is readily apparent that they produce a great increase in the flow when they are mirror reflected. The writer believed that up to a certain time interval, this reflection had great importance for the marked increase of flow in the zone from T ^ to T . But a more accurate min o calculation, indicated in the following, showed that with the small values of the mirror reflection, cited by Knauer and Stern, its effect is, at the most, a few percent of the free molecule flow. This is in good agreement with the fact that the ordinary calculation of the molecule flow (Clausing' s formula) explains the observed great values T com- pletely. NACA TM 1301 23 By calculations, not repeated here, it is found that when T denotes the increase in flow due to the mirror reflection, for small values of the reflecting power 0L_ rjE + 2Fa)da ' (ik)- where r.-^ = the reflecting power when a is the grazing angle and a^ = the upper limit of the grazing angle for which a > ' « ' 1 •n\/p<- _'p2 F = i /2rt ^/Re f| + ^ arc cos ^J -^ i 2R^2^ For the reflecting power of a polished surface (speculum metal) compared to hydrogen, Knauer and Stern have given the following values-'-'^ a = 1 X 10-3 1.5 X 10-3 2 X 10-3 2.25 X 10-3 Radian ^aobs = °'°5 0-03 0.015 0.0075 ^aber = °'°50 0.033 O.OI6 0.0075 These results are closely approximated "by the expression r = 0.08i+ - 3^, as seen from the line r-j^gj.. Introducing this expression for r along with the constants Rn , Rq, and a for the prismatic apparatus II in formula (1^0 gives 1!-^\pM~< 0.075 for hydrogen. Reflec- tion measurements on other gases have never been published to the writer's knowledge. But an explanation of the reflections by wave mechanics indi- cates that the reflection of a given surface diminishes when the number of molecules of the reflected gas increases because the associated wave 9 The calculation is to be found in: Om Luftarters Stromnlng in snaevre Kanaler. Copenhagen I936, pp. 63-69. l°Ztschr. f. Phys. 53, 1929, p. 779. 214- NACA TM I30I length Z = ^ is inversely proportional to V \A and the condition for mirror reflection is Asin a < Z, where A is the mean height of the irregularities of the surface. Therefore, if r is assumed to be 1 ^a = I (0.084 - 3^) (15) for other gases. The values Tj- V^ in table 1 were computed from equations (l^i-) and (15) > they are, as will be recalled, the upper limits. It will be seen that whenever the reflection contributes to the flow, ule T \/ir= f, . cannot 1: for the quantity (T - T ) /I4T the rule T 'iW = ^/\\ cannot be rigorously fulfilled; it can hold only (, A.) The expression {ik) was obtained on the assumption that no mutual collisions occur. Maxwell computed the effect of the mirror reflection on the slip at higher pressures and found that f - \[l\IM 3. 2 - f (16)^ ^ V2 y M Ti f where f is the fraction of the incident molecule sent out diffused from the surface; the part 1 - f is mirror-reflected. Knauer's and Stern's values for the reflection are 1 - f = 38 X 10"° , f being practically equal to unity. With these values, the formulas (6) and (16) give 1 ' R_ V2 " R_ ^^R^ ^r; 1 1 TL. B. Loeb: The Kinetic Theory of Gases. 2nd edition, p. 288. NACA TM 1301 25 which was to be the contribution of the slip alone. The quantity B-|_ Vm" is shown in table 1; it is^ in all cases, smaller than the constant term B in equation (13) • Retaining the factor ■ in equation (I7) and equating B-, \/M to the term B in equation (13) gives the values for f which are shown in table 1 and are much smaller than unity, as well as being contradictory to the afore -mentioned small values of reflection. Since it is not likely that the reflecting power of the surface at higher pressure would be so much greater, it would seem that Maxwell's value of the slip coefficient in the manner attempted here should not be employed . We will now discuss measurements with the other gases. The measurements with the rectangular slit between ground glass plates (fig. 7) were made for the purpose of ascertaining the flow in a channel in which no mirror reflection occurs. Three test series with hydrogen were run off: I. Flow between the pure ground plates II. Flow between the same plates blackened with soot III. Slit as in II, but the air traps of the inlet tubes were cooled with liquid air The principal data of these three test series are correlated in table 2 J the individual results for series II and III are represented in figures I6 and 17 . The slit width a could only be determined by means of the laminar flow data. At high pressures ,a + 2^3^ a3b a^^bp P = T^TT P + 12LT1 ^ 12LT1 ^ 2Lti or T = A-^-p + 'B-^, A-L and B-]_ being constants within a certain pressure zone. Figure 17 indicates the linear relationship between T and p at high pressure. It is seen that the slope A^ can be fairly accurately defined. The value of a-^, computed from the equation A-|_ = ^ , is the average value a3j but the surface irregularities are of the same order of magnitude as a (about 0.01 mm). 26 NACA TM 1301 The minimum lies at the pressure where X, ~ a. Knudsen's formula, applied to the rectangular slit, gives l+i/^l/ie a^b ^Kn 3!' nF M L The values Tj^ |^ M computed from it are given in table 2. The agreement between Tjj^^^^^ and Tj^ is less good than for the flow in the circular slitsj Tjj^in. ^^ about 20 percent greater than Tj^. The reason for this is, in part, that the slit width is not well-defined because of the irregularities and, in part, that the flow for the same reason differs from the flow between smooth surfaces. To compute the flow Tq at the lowest pressures the writer attempted to use Clausing 's formula ^Cl 2 V jt V M V2 ^ a a^b the formula holds for b » L » a, which is not entirely complied with because b is only 1.4 times L. The calculated values Tq-^ are about 25 percent greater than the observed values T^j this discrepancy is due to the nonagreement of the geometric assumptions. The most significant result of these measurements is that they show that the decrease of T in the zone from X, = 00 to X ~ a is not produced by the mirror reflection, but rather by the collisions of the molecules among one another. The measurements with the cylindrical slit (fig. 8) and with the gas filter (fig. 9) were made for the purpose of checking the rule T V^ = ff^ \ in absolutely constant channels. Hydrogen and carbon dioxide were used. The results for the cylindrical slit are represented in figure I8. At high pressures, the measurements with the two gases are in complete agreement, as Indicated by the fact that when the slope A-, in the equation T = A-]_p + B-|_ is graphically defined and a is computed from the equation A-|_ = jri , a = I6.I1-1 for CO2 and a = 15-9^ for Hg, if the previous values t\2q are utilized. The slit width a being very small compared to the radius R and the length L, the cylindrical slit can be regarded as a rectangular slit of width 2rtr in the calcu- lation, and this produces the preceding formula for A^^. NACA TM 1301 27 The position of the minimum is, as for the other slits, determined by the fact that ^i^ ~ ^' The application of Knudsen's formula yields which, on inserting the numerical values, gives Tj^ fW = 5-50, in close agreement with the value Tjjj^^^^ 1/M~ (compare fig. I8). In the range X ~ 20^ to \ 21 1000|i, t here i s a small discrepancy between T /W for the two gases so that T \/M/„ •. is always greater than T l/M/rm a for "the same value of -2.. The difference is about k percent, at the most. A plausible explanation is that the true effec- tive path length \^ in the range X, = 00 to X ~ a is not the same Xl as the path length computed from the equation X = — . This is apparent from the remark that X-|_ and hence X is determined by the internal friction in such a manner that only the collisions, which occasion great direction changes, are regarded as cut-offs of free path- lengths . But in a slit- in molecule flow the collisions initiating only small direction changes are also significant because (as stated previously) a great part of the transport in a slit is due to molecules, which fly approximately parallel to the slit walls, and this contribution to the flow is greatly diminished even by small divergences. Therefore, the effective path length Xq must be less than X for the same value of the pressure in the low-pressure zone. Knauer and Stern, Mais and Rabi, and others have demonstrated that the same holds true for the effective path length which is determined by scattering of molecular rays. Mais adduces that the effective center distances for collisions between potassium atoms and hydrogen or carbon dioxide molecules, etc., are aj_2 = 7.2 and I3.3 A for deviations < U.5' " Oip = 5.7 and 9-D A for deviations < 1.5 '12 •^12 ~ 5«0 snd 5-9 A for great deviations o The effective mean path length X. ~ i_ decreases therefore much more in CO2 than in Bj^. "1/ -"-^W. H. Mais, Phys. Rev, U5, p. 773, 193^- 28 NACA TM 1301 On assuming that 1\/W= f , the length A - B, cut off on a horizontal line in figure I8 between the (T VM, i J curves, is equal V ^1/ to the log]_Q of the ratio of the effective path lengths. Hence, it follows that ^gH^ Is everywhere greater than X.gC0 ■^°-'^ ^^^ same value of -E.. The ratio ^ is around 2, when log3_o — = -2.5. ^1 ^eCOp ^1 This appears to be in good agreement with Mais's results. The pure molecule flow in a cylindrical slit has been computed by Clausingl3. Application of his formula gives Tq-|_ fM" = 12.3. The measured value of T yM at the lowest pressures was about 9> "the dif- ference is due, in part, to the fact that the mercury vapors were not frozen out and, in part, to the fact that the measurements were not extended to siofficiently low pressures. Figure I9 represents the test data with the glass filter as flow channel. The corrections for the inlet tube resistance were rather great (about 50 percent at the lowest pressures) since the resistance in the filter was quite small. When X > about 10^, T|/Tl is practically constant. The variation of T l/TT = f / -. is exactly the same for both gases throughout the entire pressure zone. According to Weber's invest- igations of such filters 1^, the flow at low pressures should be regarded as pure effusion, which agrees with the fact that no minimtmi exists. This is apparent by an application of Knudsen's theory of effusion through an opening in a thin plate-'-^ . Npi = A(v' - v") A = area of opening, N , = number of molecules per second through opening, V and v" are collision numbers per sec cm^ at both sides of the plate, where the pressures are p' and p". When each pore in the porous body can be regarded as a series of chambers, between which openings of the magnitude A-^, A2 . . . A^ are presented, the notations of figure 20 give N. = Ai(v' - v'l) = A^Cv'i - v'2) . . . ^i(Vn-l " v") ■^Over den Verblijftijd van Moleculen . . . Amsterdam 1932, formulas 151 and I5U, pp. 108-109. l^S. Weber, Teknisk Tidsskrift I9IT, Nr. 37. ^^Ann. D. Phys. 28, p. 999, I909. NACA TM 1301 29 from which follows Ni Ni Ni Al A2 • ■^n or Ni = ^=-Y ( V - V") If the porous plate consists of a number (i) of such pores, the total effusion is (18) Hence, it follows that N is proportional to v' - v'% or that T is constant at low pressures. Khudsen's theory is applicable to pressures where X, is great compared to the diameter of the opening A. At higher pressures the ratio ^ „ increases with p. Applied to equation I8, V' - V it indicates that N must increase also with increasing pressure in good agreement with the measurements. Translated by J. Vanier National Advisory Committee for Aeronautical 30 NACA TM 1301 Table I Flow in the Annular Slit 1 Fig. 10 11 12 13 2 R]_ cm 1.1952 0.899^ 0.8123 0.8123 0.8123 3 R2 cm 1.8897 1.0919 1.598i^ 1.598U 1.598^+ 5 aopt^^ Gas 9.23 3.53 H2 8.17 H2 8.13 H2 8.15 Air ' H2 O2 CO2 A ' 6 7 8 9 10 11 12 13 Tminl/^ To 1/M TKnl/^ A B C D ^str^^ 2.00 2.01 1.92 1.98 c^.O 3.9 3.8 3.8 , 0.70 1.6 0.67 0.19^ 0.560 0.393 1.8 3.38 1.09 2.U 1.03 0.73 0.86 0.60 1.5 7.98 1.04 2.6 1.02 0.72 0.78 0.655 1.5 7.9^^ 7.98 1.95 1.55 1.62 0.86 2.0 9.02 lii ^opt - ^st^^^ 0.21 0.15 0.19 0.19 0.17 15 R2 + Rl R2 - Rl hM 10.1+ 3.07 3.07 3.07 16 Tcif^ 5.28 1.72 2.95 2.92 17 b/c 1.89 l.i^3 \M 1.19 18 19 20 0.32 A 0.12 0.02 o.4o 0.15 0.07 0.61 0.15 0.07 .0.10 0.03 0.02 0.02 "Bi/M" l.U 0.60" 21 f 0.83 0.83 0.83 0.87. NACA TM 1301 31 Table II Flov of Hydrogen in Rectangular Slits Si, L cm ze of Sli b cm t an Obseiwed Tmin/M T^I/m Calculated I II III 0.971 0.963 0.963 1.320 1.323 1.323 19.2 18.1 18.2 000 NO MD NO ro ro -(=- 1.6 l.k 1.5 0.80 0.75 0.76 2.li^ 1.89 1.92 32 NACA TM 1301 Figure 1.- Pressure -gage assembly. NACA TM 1301 33 Figure 2.- Prismatic apparatus I. 3^ NACA TM 1301 Figure 3.- Prismatic apparatus n. NACA TM 1301 35 Figure 4.- Theory of Herschel's interference fringes. 36 NACA TM 1301 Figure 5.- Herschel fringes photographed in transmitted (top) and reflected light. Figure 6.- The hypotenusal surface of prism P-^ of the first prismatic apparatus showing measured distances in units n . MCA TM 1301 37 m L<« . • .• * ' r^. . • « ♦ - ♦ , • * ::%%^ » • . . » « « • ♦ • , ' < ♦ Figure 7.- Rectangular slit between ground glass plates. 38 NACA TM 1301 Figure 8.- Flow channel with cylindrical slit around C. ^T ^ A^ ^ Figure 9.- Filter plate. NACA TM 1301 39 ■^^^, or K^ en ,^ ^^ r)f£e\5=v='^— S^ = CO -b 1 — 2" < <^ ^ Kg and K2 — ^K^. The results are completely identical within measuring accuracy. NACA TM 1301 1^1 ^opt. - 8.17n. Figure 12.- Flow of hydrogen in prismatic apparatus 11 The graph shows the uncorrected values of T \Jm ( .). 3-s well as those corrected for inlet tube resistance ( © ) along with the measurements of the quantity ordinate scale. Trj \/m" at 100 times smaller k2 MCA TM 1301 o -5 -4 -3 -2 -I OLogP' Figure 13.- Flow of hydrogen in prismatic apparatus n. a^-^ = 8,13 n. The air locks (or traps) of the inlet tubes are cooled with liquid air. The effect of the effusion is indicated. The upper point of the two points linked together is not corrected, the lower one is the value of T \pM. corrected for the effusion. NACA TM 1301 ^3 /4 /2 10 6 4 T-Vm y / a= 9.23IU y y y -/ y z % \ y • CO^ 1 i X 1 0^ A 1 1 p. X' 1 1 o 4- 7 AV -; Figure 14.- Flow in the wide annular slit at high pressure. > . -/p X///////////, '///////////A 1 __ __ ^ 1 V//////////, y//////////A Figure 15.- Cut placed normal to the plane of the annular slit; the axis of rotation lies in the drawing plane. hh NACA m 1301 2.0- T^^Ym h^ 1.5 1.0 0.5^ 6> ° (fi II -J L= 0.963 cm b^/.32 cm L I II 10" 10' Tn^yivT ^ 10 11 /A/ •4 -3 -2 -/ '-"^t Figure 16. 30 P 40 Figures 16 and 17.- Flow of hydrogen in the rectangular slit. O Indicates liquid-cooled air traps • Indicates no cooling MCA TM 1301 45 Figure 18.- Flow of hydrogen and of carbon dioxide in the cylindrical slit; r = 1.00 cm, L - 0.487 cm, a = Ar = 16. Ou. k6 NACA TM 1301 /50 WO 7"Vm~ •/-/. oCO, SO • • • •• 09 <»• • J /O f^ -4 -3 -2 -I Figure 19.- Flow of hydrogen and of carbon dioxide through glass filter; the top row of dots and circles indicates the values corrected for inlet resistance. v' I v; v^l V3 n Figure 20.- Each pore equals a series of chambers. 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