A semi-empirical equation for the viscosity of air

UNCLASSIF DRES TN ’45 * N ______________________________________________ ___________ ENJD DATE E!IMEA 9 -79 I I AbC j I • O i~... I. 128 1125 ‘— I~ 2.2 ~ 1 _ _ _ _ _ _ _______ II 1 1 ’1 .25 L : ~~ ~ IIIII ~ ffl~j~.4 ~~ MICRO ’OI Y RI Ot III IUN I S I ( ‘I IA R I ~ NAIk ~N\t AU UI I ANII.\III’ :~ ~ _ .._ _. __JI i* ~ -~ - ----•----• - - -.— -— ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ “~~~~~~~~ . . . UNCLAS$IF1 ~~ (9 ~z. DI! E S1 ~ . • S S •.•• S S • • . S S S S S • •S • S • • • •.•. ....... • • • • • .• • •S • . • •. • . S • • • S . .. S S S S S• S S S S S S S S S S• S S • S S S ~~~~ ~~~~~ -~~~~~ - - - - ~~ - - - —- — - - -- - r- . -~ - - -~~~ -- -- ~~~~ - -~~~~~ - - - - -- UNLIMITED DISTRIBUTION --- ~~~- NO. 454 A SEMI-EMPIRICAL EQUATION FOR ThE VISCOSITY OF AIR (U) by ~ James J. Gottlieb David V. Rltzel DDC 1F~ O?H • 0 1 . . WUD 21K14 —~~ July 1979 ‘ !$WMIL: OSM . , ~~~~~ I M U J ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ WMNPIS u.s 1 . SM. l p.rrtSS.d . .~s.t ~~ ~~~~~~~~~ $. SI ~~~~~ SSS,y ~~~ ~~~~~ . — RAlSTON :MSWA _ ‘_ _ — - - -~~~~~— ~~—~~~~~~~~~~ - ~~- —--- ’ ~~S’~~~~ _______ ~~~~ UNCLASSIFIED UNLIMITED DISTRIBUTI ON DEFENCE RESEARCH ESTABLISHMENT SUFFIELD RALSTON ALBERTA _ (2 ) ~ SUFFIEL ~~~ECHNICAL NO~~~ OJ454 • / !MI_EMPIRICAL QUAhb0 I FOR THE JISCOSITY 0F/IR t(U) ~~.. ~~ ~ by L~d)~~~ ~~ / WARNING T I.. us. .1 SM, I.SSSI.S .II.. IS p .rmtft.d i.ibj .ci I . . 1 puspilsIsry s.d pstesl t bt. ~~ UNCLASSIFIED ‘. pt... - I — - -——-~~~~~~~~~~~~ ~ 2~~~~ ~~~ ~ - -----— --‘- --~ ~ -- .- -.-—~ - __~~~~~~~ —~—-— _ _--_ - ——_ _--_~ _- —— —~~~~ ~-- a ~~~ — -_ --_t - _ — —- — . _ -~---,r ~ ~~, - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I ~~ ~~~~~~~~~~ l -- .. ~~~~~~~~~~ - _ _ _ _ _ — _ _ _ _ • - UNCLASSIFIED DEFENCE RESEARCH ESTABLISHMENT SUFFIELD RALSTON ALBERTA SUFFIELD TECHNICAL NOTE NO. 454 A SEMI-EMPIRICAL EQUATION FOR THE VISCOSITY OF AIR (U) by James J. Gottlieb David V. Ritzel TRACT Sutherland ’s theoretical expression for the absolute or dynamic viscosity of a gas has been modified to represent experimental vi scosity data for air. The resulting semi-empirical equation accurately describes measured viscosities of air for temperatures ranging from the boiling point at 78 K to molecular dissociating conditions at 2500 K. Discrepancies between measur ed values and semi -empirica l results rarely exceed 2%. Other more limi ted expressions for the viscosity of air are also compared to measured viscosity data and discussed. (U) UNCLASSI FlED _ _ _ _ .I - ~~~~~~~ ---—- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - - — ~~~~~~~.- , , — -. — ~~~ ~ —S- ~~~~~~~~~~~ — --~ — —--.----- — ~~— — — --- - --~ ~~~~ -~~—-~ —- — ~~-— ----.-— ~~--- IABLE OF CONTENTS 1.0 INTRODUCTION . P ge ~ i . 2.0 VISCOSITY 2 • 3.0 PREVIOUS VISCOSITY EXPRESSIONS 4 4.0 A NEW SEMI-EMPIRICAL EXPRESSION FOR THE DYNAMIC VISCOSIrY OF AIR 5.0 CONCLUSIONS • • • • . . . . . 7 ... . 6.0 REFERENCES 11 TABLES FIGURES • D Di st Aiai1 9 d I°’ ~ 4 _ _ _ _ _ _ _ _ _ - — - - UNCLASSIFIED DEFENCE RESEARCH ESTABLISHMENT SUFFIELD RALSTON ALBERTA SUFFIELD TECHNICAL NOTE NO. 454 A SEMI-EMPIRICAL EQUATION FOR THE VISCOSITY OF AIR (U) by James J. Gottlieb David V. Ri tzel 1.0 INTRODUCTION - A means of specifying vi scosity as a function of temperature i s required when one analytically or numerically solves any fluid-flow problem involving viscous forces. Vi scosity nonnally enters the analysis through the Reynolds number, which is the ratio of inertial to vi scous forces. In turn , this number both characterizes the type of flow (laminar, transitional, or turbulent) and helps to select values of certain parameters such as the drag coefficient of an object or structure. For the solution of a fluid-flow problem, it Is preferable to represent viscosity data by a simple and convenient equation, which facil itates easy computation of results. In the present work , it is shown that Sutherland ’s simple theoretical expression for the viscosity of a gas (Ref. 1) can be UNCLASSI Fl ED — I -. - UNCLASSIFIED /2 slightly modified to accurately represent viscosity data for air over the large temperature range of 78 to 2500 K. • It is worth mentioning that theoretical results for viscosity are not ver y useful In solving fluid-flow problems , because the equation or equations are generally much too complex to utilize effectively. For example , see the various expressions resulting from kinetic theory given in Reference 1. Furthermore , if any theoretical viscosity equation is sufficiently simple to use effectively In the computations , then it Is generally too inaccurate. Semi -empi rical and empi rical equations for air viscosity as a function of temperature are almost nonexistent in the scientifi c literature. Most researchers ei ther choose the tedious method of using tabulated viscosity values in computations or do not report their viscosity equations. Of the few expressions available In the literature (Refs. 2, 3 and 4), it has been found that most of them are either too inaccurate or too limi ted in their temperature range for some new problems of interest. For these reasons , the present report is valuable In providing a simple and accurate semi-empirical equation for air viscosity over a very wide temperature range. 2.0 VISCOSITY • - In all real flu id flows , momentum exchange and cohesion cause shear stresses between adjacent fluid layers in relative motion (Ref. 5). V iscosity is a fundamental property of a fluid , which defines the relationship between the shear stress and this relative motion . In a Newtonian fluid there is a linear relationship between the shear stress t and the velocity gradient du/dy normal to the flow, as expressed below. du 1 The coefficient ~ denotes the dynamic viscosi ty, which is also known as the coefficient of viscosity , absolu te viscosity or simply viscosity. From Equation 1 , the dynamic viscosity may be interpreted as defining the UNCLASSIFIED — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ , - - ._-—-• —_ .— -—~~-~ ~~~~ . ~~ ~~~~~~~~~~~~~ - ~~~~~~~ -------• ‘-. ~ UNCLASSIFIED - /3 ratio of the shear stress to the rate of shear deformation . This is li kened to the case in solid mechanics where the shear modulus defines the ratio of the shear stress to the magnitude of the shear deformation . In the case of liquids , for which the viscous mechanism is primarily cohesion , the dynami c viscosity diminishes with increasing average molecular-separation distance and thus rising temperature. For gases, however, the vi scous mechanism is principally molecular momentum exchange between adjacent fluid l ayers. Hence, a h i gher molecular k inetic energy or temperature results in a larger dynamic viscosity . Except in the case of an extremely hi gh pressure whic h brings molecules closer together to significantly increase the cohesion effect, the dynamic viscosity may be perature only. assumed to be a function of tem • There are several accepted ways of determining dynamic viscosity experimentally (Ref. 6). All devices essentially measure the stress or drag exerted by the laminar flow of the fluid. Coninon techniques include the measurement of torque exerted by coaxial cylinders in relative rotation when the small annular space is filled by the fluid , and the timing of fluid efflux through a hole in a vessel of known dimensions . Other methods use the damping of an ininersed pendulum , the drag on a moving sphere, or the measurement of lam inar flow in a duct. Viscosity data from such experiments with coninon fluids are readily available from most fluid—dynamics textbooks and handbooks (e.g., see Refs. 7 to 11). For convenient future reference , viscos ity data for air have been reproduced in Table I and are also displayed graphically in Figures la and lb. Since the ratio of dynamic viscosity ~ to fluid density p appears often in fluid—dynamics problems, the kinematic vi scosity v is often defined for convenience as follows: v= (2) ii/p. This definition is convenient for incompressible fluid fl ows because the kinematic viscosity , like the dynamic viscosi ty, is a function of temperature only. However, for a compressible flow the definition of kinematic vi scosity Is not very valuable. The kinematic viscosity is now UNCL A SSIFI ~~ Ik~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ . . - - -~~ —-- ~ ----.-- - - - - - - -- ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~~~~~~~~ UNCLASSIFIED /4 no longer a basic fluid property, but depends on both the gas temperature and the density. For an ideal gas having a pressure P, ten~ erature T, and gas constant R , we can express the kinematic viscosity as follows: v (T,P) p (T)/p(T,P) RTp (T)/P. (3) For compressible gases, there fore , it is most reasonable to deal with the dynamic viscosity and density as distinct fluid properties. 3.0 PREVIOUS VISCOSITY EXPRESSIONS One exam ple of an empirical expression for the kiiiematic viscosi ty of air , which appears in the literature (Ref. 2), is reproduced below. Ip /6 I!a b0 b1 b2 + PS /Pa \/P \O.7141 (b ~ 0 \~ + 6 s a’’ / ~ ~~ ~~ = = = J + b 11 + b 2T2) (4) 9.027754x10 5 ft2/sec -3.2934l3xl0 7 ft2/sec oR 8.741066x10 10 ft2/sec0R2 Because this expression was developed for a shock- or blast-wave air flow , the symbol s p0, a ’ p 5 , and P respectively denote sea-level atmospheric pre~ sure (14.696 psia), ambient pressure, peak absolute pressure just behind the shock-wave front, and absolute pressure further behind the shock front or in the shock wave. The polynomial expression in the curved brackets describes the kinematic viscosity of air as a function of temperature at sea-level pressure P0. The other factor in square brackets is a density correction to account for the density variation owing to a pressure change from P0. It should be noted , however, that the correct version of Equation 4 for the kinematic viscosity in a shock-wave flow would have an additional multipl ication factor in the square brackets, corresponding to the temperature ratio When this factor is incl uded in Equation 4, the correct expression can be simplified considerably to yield the fol~~wing result. v • P ~2 (bo + b 1T + b 2T2) (5) UNCLASSIFIED -- - _ _ - _ —- _ — --- - _ ____ _ _ ___ _ __. - - --- ,—-— -I ~ _ _ _ ~~~~~~~~~~ _ _ _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ _ ~~~~~~~ “ T _ ~~ _ _ _ _ _ _ _ “ ‘ ~~ ~~~~~~ UN CLASSIFIED --— _ _ _ ~~ - -----_ _ -- _ _ ~~ /5 It Is worth mentioning that Equation 5 is not restricted to problems Involving a shock-wave flow . It can be derived directly from first principles and previous definitions. Firstly, by definition let v equal p (T)/p, which also equals RTp (T)/P or (P0/P)RT~(T) /P0. Secondly, • • let RT~(T) /P0 in the latter expression be given by the polynomial b 0 + b 1T + b2T2 , as defined originally. Then Equation 5 follows directly from these resul ts. In order to conveniently compare the results of Equation 5 or the original expression (Eq. 4) to experimental dynamic-viscosity data given in Table I or Figure la , the shock wave can first be omitted = P), and then the resulting identical equations can be expressed ~a directly as dynamic viscosity , as gi ven below. P • — (b~/T + b 1 R ~~~ + b21) (6) The results of this equation are shown in Figure la , along with the measured dynamic viscosity data. It is readily apparent that the agreement is good only in a l imited temperature range of about 200 to 400 K. The kinematic viscosity of air, given by Equation 4, has been used quite extensively in calculating the Reynolds number for shock and blast work done at DRES during the last ten years. A large part of this work involved the experimental evaluation of the unsteady- drag coefficient for different sized cylinders in bl ast-wave and shock-tube flows (Refs. 2 and 12 to 21). For example, from the measured free-flight motion of a cylinder in a blast wave, the drag force can be determined by using Newton’s second law of motion. Then the drag coefficient is simply the drag force divided by the product of the cylinder area and the dynamic pressure of the blast-wave flow. This drag coefficient is valid only for similar flow conditions characterized by both the flow Mach number and the Reynolds number which depends inversely on the kinematic viscosity. Another significant part of the DRES work invol ved the prediction of drag response from a blast-wave flow for shipboard antenna masts, a fiberglass whip UNCLA SSI F lED -z~~~ ~~~~~~ : ~ — —“-~~—,-~~ 4- ____________________________________________________________________________ — — —— _ _ _ _ UNCLASSIFIED . /6 antenna and a UHF polemast antenna (Refs. 22 to 26). Based on the Mach and Reynolds numbers of the blast-wave flow, a corresponding drag coefficient for each structural element can be selected in the analysis to predict the drag response of the structure . Similar work on structural response to the drag loading of a bl ast wave , which uses drag coefficients experimentally evaluated for cyl inders at DRES , has been done for DRES by Mechanics Research Incorporated (Refs. 27 and 28). In all of the work mentioned in the previous paragraph, the blast waves under consideration were sufficiently weak that the peak temperature behind the shock front never exceeded about 400 K. Hence, In past calculations using the kinematic expression given by Equation 4, the temperature was not outside the range of appl icability for the factor in curved brackets (200 to 400 K). However, since the correct kinematic-viscosity expression of Equation 5 should have been used, because it contains the additional multipl ication factor Ta/T a the previously mentioned work is affected. The error depends on the blast-wave strength. The discrepancy in kinematic viscosity from using Equation 4 instead of Equation 5 containing the additional factor Ta/T can be as high as 2, 4, 10, 18, 25, 32 and 40% for of 1 , 2, 5, 10, 15 , 20 and 25 psi respeak shock overpressures ~~ pectively. Furthermore, the error in kinematic viscosity from using Equation 4 instead of experimental viscosity data (Table I), which are not represented accurately at high temperatures, can be as high as 1 , 3, 10, 21 , 32, 43 and 54% for peak overpressures of 1-, 2, 5, 10, 15, 20 and 25 psi respectively. Such di screpancies would result in errors of similar magnitude for the Reynolds number which is inversely proportional to kinematic viscosity. However, the resulting discrepancy In the drag coefficient, selected on the basis of Reynolds number, would normally be smaller , except when the Reynolds number passes through the critical value defining the transition regime between subcritical and supercritical flow. - A second example of an empirical expression which appears in the literature (Ref. 3), this time for the dynamic viscosity of air in a shock wa ve , is reproduced below. UNCLASSIFIED A— _ _ _ _ _ _ _ _ _ _ ~~ - •- - —~~~~~~~~~~~ p ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ UNCLASSIFIED p • O.2lxlO 8 pS0.21’. /7 0.715 I (7) 0.247xl 06 P5 T ~ This equation does no give meaningful results, even if reasonable assumptions are made as to the units for the two constants and the existence of typographical errors. J A final example of a dynamic-viscosity expression , which is appl icable over a wide temperature range, can be found in Reference 4. This expression is reproduced below. a T1-5/(b + 1) a 2 27Ox lO 8 slugs /ft sec b =1 98.7 °R (8) = ORO.s Results of this equation are shown in Figure la. It can readily be seen that this equation represents the experimental dynamic-viscosity data with little error from 78 K to about 1200 K. Such a good representation is noteworthy. 4.0 A NEW SEMI-EMPIRICA L EXPRESSION FOR THE DYNAMIC VISCOSITY OF AIR Many of the rigorous theoretical formulations for dynamic viscosity, which can be found in Reference 1 , are too complicated to use readily in either analytical or numerical calculations. However, a theoretical model originally proposed by Sutherland (see Ref. 1) results in the rather simple expression given below for the dynamic viscosity of a gas. u s u rI i+s/ i olrT l o s I Ll+ S/T — T~ The symbols ~~~~ T~ and S respectively denote reference values of the dynamic viscosity and absolute temperature, and the so-called Sutherland ’s constant. The theoretically derived value of Sutherland ’s constant S is 113 K or 203 °R for air (Ref. 1). It must be pointed out that Sutherland s theo~ retical expression cannot be used directly in numerical or analytical calUNCLA SSIFIED - k -- - - — - - ——----- - - - _ ~~~~— —-—------ - _ ---~~~~~ —~~~~~~~ -~~~~—-_ _ - -- ~ -~~~~ - - ~~~~~~~~~~ ~~~ UNCLASSIFIED /8 culatlons , because the reference value of the viscosity i~~ at temperature is unknown or unspecified theoretically. When experimertal data are used to specify ~~ then the resulting expression of identical form is no l onger theoretical but is semi—empirical. Sutherland ’s theoretical expression can easily be rearranged to yield the following result, a~,T’ 5/(S + 1), (10) whe re a 0 is a constant based on reference values of dynami c viscosity u 0 and temperature T0. This form is identical to that of Equation 8, which was reproduced from Reference 4. In Equation 8, however, the constant b is 198.7 °R , whi ch is slightly lower than Sutherland’s value of 203 °R. If Sutherland ’s constant S is to retain its value of 203 °R or 113 K in Equation 10, then the experimentally determined value of a0 will also differ slig htly from the similarly determined constant a in Equation 8. = In the present work, Sutherland ’s value of the constant S has been retained. The other constant a0 in Equation 10 was selected such that Sutherland’ s formula fitted the experimental viscosity data best in the temperature range of 100 to 800 K. Results for this case are sum- marized below. a0T ’5 /(T + S) l.47x10 6 kg/rn s K 0 5 (11) = a0 S = = or 7.35x10 7 ib m /ft sec °R0 5 113 K or 203 °R The results of this semi-empirical expression for dynami c viscosity do not differ substantially from those of Equation 8. Hence, if the results were plotted in Figure la , they woul d lie essentially on the solid line labelled Equation 8. Equation 11 , l i ke Equat i on 8, therefore represents the measured dynamic viscosity data for air with little error over the temperature range of 78 K to about 1200 K. A simple e qu a t i o n for the dynamic v i s c o s i ty of air is occa- UNCLASSIFIED -— -—---- - — — - --— -- -- _ • __ _ _ _ ___ !;___ •______ - r 1 ~~~~~~~ ,4 , _ I r. , ) lf ~~~~~~ t q IS . n_ ~~ . . - ~~~~~~~~~~~~~~~~~~ .__— ~~~~~~~~~~~~~~ -- - -~ -.-~-~ — — — — ______________ - - - - — UNCLASSIFIED /9 sionally required to accurately represent experimental data at temperatures higher than 1200 K. For this case, Equation 11 was simply extended by adding a suitable multiplication factor. This factor was selected by the authors because it conveniently reduced the differences between the results of Equation 11 and experimental data . The resulting semi empirical expression is given below. • i’.~r[1.0 + l.53x1O_ ’.(l + a a0 = l.47x10 6 kg/rn S K0-5 1 - 2] 1) (12) or 7.35x10 7 ib m /ft sec °R0 5 S = 113 K or 203 °R Results -of this expression are shown in Figure lb , along wi th measured dynamic-viscosity data for air. It can readily be seen that this equation represents the measured data not only very accurately, but also over twice the temperature range covered by Equation 8. Discrepancies between measured values and semi-empirical results rarely exceed 2% in the temperature range of 78 to 2500 K. It should be noted that this new semi—empirical equation for the dynamic viscosity as a function of temperature is quite easy to use in analytical and numerical calculations. 5.0 CONCLUSIONS Different expressions for the dynamic and kinematic viscosity of air have been taken from the literature and assessed in this work. It was found that most of these expressions were too inaccurate or too limi ted in thei r appl icable temperature range. However, expressions based on Sutherland ’s theoretical model of the dynamic viscosity of a gas were found to be generally successful in representing measured vi scosity data from the boiling point of 78 K to about 1200 K. Furthermore, it was shown that Sutherland ’s basic equation could be easily modified to give a semiempirical expression (Eq. 12) for the dynami c viscosity of air which represents the measured data very accurately from 78 to 2500 K. Discre- UNCLASSIFIED _ _ _ __________________________________________________________ -- ‘I ’ .- - - - - I UNCLASSIFIED / 10 pancles between measured values and semi-empirical resul ts rarely exceed 2%. The new semi-empirical expression for the dynamic viscosity of air Is recoimiended for future viscosity calculations at DRES . UNCLASSIFIED _ _ _ _ _ - ~~~~~~~~~~ ~~~ - -~~~~~ - -— - - - — - ~~~~~ — - --- - , ~ —.-- - - -—. - ----- -- — ~ ~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~ ~~ —~~~ - - UNCLASSIFIED /ll 6.0 REFERENCES 1. S. Chapman T.G. Cowling 2. S B . Heu sen The Mathematical Theory of Non-Uniform Gases . Third Edition , Cambridge (1970). Drag Measurement on Cylinders by the Free Flight Method Operation PRAIRIE FLAT (U). Suffield Technical Note ~~ 249 (Jan. 1969). UNCLASSIFIED. - 3. W .E. Gilbert Private Correspondence. 4. R.J. Janus Graphs for Use in Obtaining Desired Mach and Reynolds Numbers in a Shock Tube (U). Ballistics Research Laboratories Memorandum Report No. 1132 (March 1958). • Elementary Fluid Mechanics. Fourth Edition , 5. J.K. Vennard 6. C.B. Anderson Mechan ics of Fluids. Chapter 3 in the Standard Handbook for Mechanical Engineers. 1. Baumeister and L.S. Marks (Eds.), Seventh Edition . 7. H. Schlichting Boundary-Layer Theory. Sixth Edition , McGraw-Hill Book Company (1968). 8. R.C. Weast (Ed.) “Thermodynamic and Transport Properties of Air ” and the “Viscosity of Gases.” CRC Handbook of Chemistry and Physics , Fifty—fi fth Edition , CRC Press (1974). 9. F. Kreith Principles of Heat Transfer. Thi rd Edition , In text Press (1973). Wiley and Sons (1965). 10. J.P. Holman Heat Transfer. Fourth Edition , McGraw-Hill Book Company (1976). 11. R.L. Daugherty J.B. Franzini Flu id Mechanics with Engineering Applications. Seventh Edition , McGraw-Hill Book Company (1977). 12. S.B. Heu sen Correlation of Drag Measurements in Operation PRAIRIE FLAT with Known Steady Flow Values (U). Suffield Memorandum No. 12/69 (April 1969). UNCLASSIFIED. 13. S.B. Meilsen R. Naylor Aerodynamic Drag Measurements and Flow Studies on a Circular Cyl inder in a Sh~ck Tube (U). Suffield Memorandum No. 7/69 (May 1969). UNCLASSIFIED. UNCLASSIFIED _ - _ _ - UNCLASSIFIED /12 6.0 REFERENCES (Con ’t) 14. S.B. Mellsen Drag on Free Fl ight Cylinders In a Blast Wave . The Shock and Vibration Bulletin , Bulletin 40 (Part 2), pp. 83 99 (Dec. 1969). - 15. S.B. Heu sen Measurement of Drag on Cyl inders by the Free Flight Method Event DIAL PACK (U). Suffleld Technical Paper No. 382 (Dec. 1971). UNCLASSIFIED. - 16. S.B. HeUsen Drag Measurements in Event DIAL PACK. The Shock and Vibration Bulletin , Bulletin 42 (Part 4), pp. 157 173 (Jan. 1972). - 17. S.B. Hellsen Drag on Cylinders in a 20 to 25 millisecond Blast Wave (U). Suffield Memorandum No. 114/71 (July 1972). UNCLASSIFIED. IS. R. Naylor Unsteady Drag from Free-Field Blast Waves (U). Suffield Memorandum No. 42/71 (Jan. 1973). S.B. HeUsen 19. S.B. Heu sen UNCLASSIFIED . Measurement of Drag on Cyl inders by the Free Flight Method Event MIXED COMPANY (U). Suffleld Technical Paper No. 419 (Mar. 1914). UNCLASSIFIED. - - 20. A .W.M. Gibb D.A. Hill Free-Flight Measurement of the Drag Forces on Cylinders Event DICE THROW . Proceedings of the DICE THROW Symposium, June 21-23, 1977. Defence Nuclear Agency Report DNA-4377P-2, Vol . 2 (July 1977). 21. A.W .M. Gibb D.A. Hi ll Free-Flight Measurement of the Drag Forces on Cylinders in Event DICE THROW (U). Suffield Technical Paper No. 453 (Feb. 1979). UNCLASSIFIED . 22. B.R. Long The Analysis of Shipboard Lattice Antenna Masts under Air Blast and Underwater Shock Loading: Part III Final Report (U). Suffield Technical Paper No. 431 (June 1975). UNCLASSIFIED . - - 23. G.V. Price C.6.Coffey Blast Response of 35-ft Fiberglass Whip Antenna Event DICE THROW . Proceedings of the DICE THROW Symposium,June 21-23 , 1977. Defence Nuclear Agency - Report DNA-4377P-2, Vol . 2 (July 1977). 24. C.G. Coffey G.V. Price Blast Response of UHF Polemast Antenna Event DICE THROW. Proceedings of the DICE THROW Symposium, June 21-23, 1977. Defence Nuclear Agency Report ONA-4377P-2, Vol . 2 (July 1977). - UNCLASSIFIED ~~~~~~~~~~~~~~~~~~~~~~~~~~ - - —— .——--— ,———-— — -* ~~~~~~~~~~~ - —-.—— ———--————— —— -—— — ~~ ——— —— — - UNCLASSIFIED H 25. C.G. Coffey G.V. Price Blast Response of UHF Polemast Antenna Event DICE THROW (U). Suffield Technical Paper No. 449. (Nov. 1977). UNCLASSIFIED. 26. G.V. Price Numerical Simulation of the Air Blast Response of Tapered Cantilever Beams (U). Suffield Technical Paper No. 447 (Nov. 1977). UNCLASSIFIED. 27. R. Geminder Analytical and Empiri cal Study of Shipboard Antenna R.W. Hicks - Masts Subjected to a Blast Environment: Vols. I , II and III. Mechanics Research Incorporated Report, MRI-C2230-TR-l (Nov. 1969). 28. R.W. Hicks R. Geminder Calculated Strains in Antenna Mast Located at 19 psi Subjected to the “DIAL PACK” Blast Environment. Mechanics Research Incorporated Report, MRI-2422 (Dec. 1970). I UNCLASSIFIED - - -. — /13 6.0 REFERENCES (Con ’t) - —- ___ _ _ _ _ _ - -- - -~~ —-- -: ~~~~~ -------- - - - - - - - -~~~~ ~~~ - - --- UNCLASSIFIED TABLE EXPERIM ENTAL SOURCE TEMPERATURE (K) 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 “~~~~ 850 900 950 1000 1100 1200 1300 14 00 1500 1600 1700 1800 1900 I VALUES OF THE DYNAMIC VISCOSITY OF AIR DYNAMIC VISCOSITY _S (10 kg/rn SOURCE 5) 6.924 10.283 13.289 14.88 18.46 20.75 22.86 24.85 26.71 28.48 30.18 31.77 33.32 34.81 36.25 37.65 38.99 40.23 41.52 ‘t~ . . ~~~ 46.9 54.0 56.3 58.5 60.7 62.9 223 14.6 323 373 473 573 19.6 21.8 25.9 29.6 79 90 169 5.51 6.27 11.30 242 291 313 327 347 502 607 630 682 739 754 810 838 15.39 17.08 18.27 19.04 19.58 21.02 26.38 31.23 31.75 34.13 35.01 35.83 36.86 37.50 911 40.14 1023 42 .63 44.19 273 204 273 44.4 49.3 51.7 (K) DYNAMIC VISCOSITY (1O ~ kg/m s) TEMPERATURE ~~‘ ~~~ 0 ~~~ 893 2000 65.0 2200 2300 69.3 71.4 1083 1196 2500 75.7 1407 2400 73.5 UNCLASSIFIED . ~~~ 1307 17.1 13.33 39.16 46.43 49.06 52.06 - -- -‘ ~ ~~~~~~~ • - -- . -~~ — - -- UNCLASSIFIED TABLE I (Continued) EXPERIMENTAL VALUES OF THE DYNAMIC VISCOSITY OF AIR SOURCE TEMPERATURE (K) 300 400 ~~~~~~~~~ 500 600 — 700 ~~ U 800 900 1000 i£itin IIJ’J i— -v ~ 1200 1300 1400 1500 DYNAM IC VISCOSITY (10 ’ SOURCE kg/rn s) 18.4 22.7 41.9 47.0 . ‘I- (10~ kg/rn s) 14.9 15.6 253 255 261 266 272 16.1 16.2 16.5 16.8 17.1 283 289 293 294 17.6 17.9 18.1 18.3 303 18.6 305 311 18.7 19.0 17.1 C W C ~) 256 273 311 367 422 . • (K) 273 alA ~ V T .. 1 49.4 51.7 54.0 VISCOSITY 233 244 26.5 29.9 33.1 36.2 39.1 DYNAMIC TEMPERATURE 533 589 644 700 756 811 1089 1367 1644 1922 16.52 17.34 19.12 21.43 >, - , ~ 23.96 28.13 29.76 31.80 33.5 35.1 36.8 44.6 51.3 54.9 57.6 313 322 19.0 19.5 344 353 355 366 20.2 20.9 20.8 21.5 394 473 23.3 25.8 UNCLASSIFIED i L ~~~~~~ ~~~~~~~ - -- _ _ _ - - ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - - .- - - ~ ~~~~~-- ~~ b. ------ ~~~- - - — — - --- - - - -— ~ UNCLASSIFIED so I I I I I I —- - -:__— ~~~~~~~ ~ - - ---- ~~~~ -- - ~~ — ~~~~~~~~~~ - -- SIN NO. 454 1 1 I I I I - 0: ~~~ 1L 4 O- F: ~~~~~~~~~~~~~~~~~~~~~ - ” 1/ H 20 : 10 p 7 — EXPERIMENTA L. DATA ~ S 0 S + / I — 0 - X 1 1 2O0 1 SCHUCHTING (Ref. CRC HANDBOOK OF - 7) CHEMISTRY AND PHYSICS (Ref. 8) KR(ITH ( Ref. 9) HOLMAN (R.f . IO) DAUGHERTY AND FRANZ INI (Ref. II) POFERL AND SVEHLA (CRC HANDBOOK OF CHEMISTRY AND PHYSICS , Ref. 8 ) I I 400 1 800 1 I I 800 I I 000 I 200 ABSOLUTE TEMPERATURE ~ - I - I 400 T (K) FIG. Ia. COMPARISON OF EMPIRICAL AND EXPERIMENTAL RESULTS FOR THE DYNAMIC VISCOSITY OF AIR (78 TO 1500 K). UNCLASSIFIED - - —~~~~~ -- --—— - — - - -- —--~ - -- --- _ ._ ~~~~~~~~~ : : ‘ j ~~~ ~~~~ ~ - - -— — — —— - ————— ~~~~~~~ ~~ - ——— — — —— — —-—- - — — —— ~ - — UNCLASSIFIED . 90 I I I I I 1 1 111 STh NO. 454 1 1 I I I I 1 I 1 1 I - I ‘ ~~ ~~ + ‘ [i.o 153 + o ~ ( )‘] _-_ ~ - \ \ \ ;z I . 4 7 x 1 C 5 S k g / r n s K° — > U ~~~ S’ II3 K U , UI 0 60- - / S S~~~o — 5 — , 0• 50— — ./~~ 3O ~~ 20 f — t $0 o • 0 S + [ — ~ —r 0 I — EXPERIMENTAL DATA — III 400 I 600 SCHLICHTING (Ref. ? ) CRC HANDBOOK OF CHEMISTRY AND PHYSICS (Ref. B) KREI TH (Ref.9) HOLMAN (R.f. IO) DAUGHERTY AND F RANZINI (Ref I I) POFERL AND SV EHLA (CRC HANDBOOK OF CHEMISTRY AND PHYSICS , Ref. B) — — — 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 900 $000 1200 1400 1600 $800 2000 2200 2400 ABSOLUTE TEMPERATURE t I (K) FIG lb. COMPARISON OF EMPIRICAL AND EXPERIMENTAL RESULTS FOR THE DYNAMIC VISCOSITY OF AIR (78 TO 2500 K). UNCLASSIFIED - - -- - — -“ --— - — .- --—-- . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ___._ ~__ _ _ ;—,-- _ UNCLASSIFIED ~~~~~~~~~~~~ OSCUMINT CONTROL DATA — N $ 0 ._-_LW u nwS be .*Is. I~~s.*~ sIssNtL~ I u si Wile. lsil SI end IndenI i SI~sn ~~ enulill ~~~imsifl N ~ ~~~~~ ~~ ~~ JL11 I. ORISINAT*IS ACTIVITY CLAMIPICATION ~~~. DOd DEFENCE RESEARCH ESTABLISHMENT SUFFIELD I ~ 4. ~~ GROUP A SEMI-EMPIRICAL EQUATION FOR THE VISCOSITY OF AIR (U) O RIPTIVI NOTIS IType si ,ypurt AUTNOR 5I ~ ft.ass en~~s. SIne ~~~~~~~~ Technical Note n s. i,ilddIs ANWI ~~ Gottlieb , James J. and Ritzel , David V. OCUMINT DAT! ~~~ . 7.. TOTAL NO. OP PAGIS July 1979 PROJeCT OR GRANT NO. ~~~~. 28 ORIGINATOR S DOCUMINT NUMUR($I SUFFIELD WUD 21Kl 4 TE CHNICAL NOTE NO. 454 Sb OTNIR 000UMINT NO.5)(Amy sWisi W~ s desumund ~~ CONTRACT NO. 1 7b. NO. OP USFS 19 — nun*st. * SISTRISUTIO N SIATIMINT UNLIMITED DISTRIBUTION II. SUNI.SMSNTARV NUlS 2. IPONSO5)NG ACTIVITY ~ 13. MSVRACT Sutherland s s theoretical expression for the absolute or dynamic viscosity of a gas has been modified to represent experimenta l viscosity data for air. The resulting semi-empirical equation accurately describes measured viscosities of air for temperatures ranging from the boiling poi nt at 78 K to molecular dissociating conditions at 2500 K. Discrepancies between measured values and semi —empirical resu l ts rarely exceed 2%. Other more limited expressions for the viscosit y of air co mpared to measured viscosity data and discusse d. are also (U) . ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~~~~~~~~~~~~~~ -— ~~~~~~~~~ - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ~ - --~ ~~~~ -.- - - ~~~y be .- —--— UNCLASSIFIED ISV WOROS Dynamic Viscosity of Ai r Kinematic Viscosity of Air WiSTRU~~~~~~ I . ORIGINATING ACTIVITY : Enter iN. neme end S UUS of the ~~ iw.n.paqiflfl iuu,n the document. DOCUMENT SECURITY CLASSIFICATION: Sn* die overall ~N. security clesu.hcasson of the document Includkl ~~uaisI ~~~tiiil5 terms whenever MEliculsis. GROUP: Enlur security rscliuuiflcutlon goup numbs,. The three ~~uuUs see defined in Aunendis ‘Vol thu DUB Security Repjls ione. 3. OOCUMINT TITLE: Enter INs ...pia (e document title In alt , p.ial totters. Titles in elI cause iheuld be wideusifled. 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