HER? TMRSFER THRQUGH AN AIR CU‘RTAIN 00—: .h-s _' I (001—! TLnesls {or "no Degree of pk. D. WCE‘NAC SE‘T; J‘ “3333’ had Hetsroni 1963 \ Cg 0-169 ' "‘* “v? ‘r‘ p ‘ *1 llgfill/4l/LIAU/ll,/j/Cgllg@1fi/I//IEW LES—Yr] This is to certify that the thesis entitled HEAT TRANSFER THROUGH AN AIR CURTAIN presented by Gad Hetsroni has been accepted towards fulfillment of the requirements for Ph.D. degree in Aggiculggral Engineering Major professor Date%&". gfl/¢63 ”fir- WW, __., -—.——- :4 ‘21. ABSTRACT HEAT TRANSFER THROUGH AN AIR CURTAIN by Gad Hets roni The air curtain studied here consists of a stream of air discharged downwards from nozzles above a doorway towards a return grille in the floor. The curtain of air insulates the interior from outside temperature, dust, fumes, insects, etc. It also provides an unrestricted and attractive doorway. The fundamental heat transfer and flow characteristics of the air curtain were studied as dependent on the outlet air velocity and on the geometry of the curtain. A theoretical analysis was made, assuming that the air curtain is a two-dimensional jet. Semi-theoretical expressions for shear stress, stream function and eddy diffusivity were derived and plotted, based on an exponential type velocity profile, as suggested by Reichardt. A form of an error function was used for the temperature distribution in a two-dimensional jet. A mechanism of heat transfer through the air curtain was suggested, based on the process of the air entrained and spilled by the jet. Based on this mechanism a functional relation was derived describing the heat transfer through an air curtain. A two-dimensional air curtain was installed between two well- insulated chambers. The height of the curtain was varied from 5 to 7 ft. , the outlet velocity from 9 to 35 ft. per sec. and the thickness from 0. 115 to 0. 340 ft. Velocity, temperature and turbulence distributions were measured, together with the quantities of heat transferred through the air curtain. Gad Hets roni The exponential function for the velocity profile was confirmed experimentally. However, the coefficient in the exponent varied from the theoretical value for small aspect ratios and small outlet Reynolds numbers. The temperature profile based on the error function repre- sented the data reasonably well. The coefficient in the error function was determined experimentally. Hot wire anemometer studies revealed that the distribution of turbulence is asymmetrical around the centerline when a temperature gradient exists across the jet. The turbulence on the warm side of the jet was much higher than the turbulence on the cold side. The higher turbulence on the warm side caused a shift of the jet towards the cold side. The semi-theoretical correlation for heat transfer through the air curtain is given by b St = 0.0 0 A o 8 8 H where H is the height of the Opening, Zbo is the thickness of the air curtain at the outlet and Sto is the Stanton number at the outlet nozzle. The above correlation was found to redict the heat transfer within :1: 20 ’J H . b percent for the curtain parameter ’—-9— between the values 0.090 to 0.170. HEAT TRANSFER THROUGH AN AIR CURTAIN BY Gad Hetsroni A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCT OR OF PHILOSOPHY Department of Agricultural Engineering 1993 @MW,Q?z/flu ACKNOWLEDGEMENTS t. Y3 s, The author wishes to express his sincere appreciation to Dr. Carl W. Hall (Agricultural Engineering), whose continued interest and encouragement made this investigation rewarding and enjoyable. The guidance and unfailing assistance of Dr. A. M. Dhanak (Mechanical Engineering), who helped the author get some insight into the fields of heat transfer and fluid flow, are gratefully appreciated. Thankful acknowledgement is extended to the other members of the guidance committee: Dr. D. H. Dewey (Horticulture), Dr. J. S. Frame (Mathematics), Dr. H. R. Henry and Dr. E. M.- Laursen (Civil Engineering), for their helpful advice and suggestions. The author is indebted to Dr. A. W. Farrall, chairman of the Department of Agricultural Engineering, for making available the graduate assistantship that made the undertaking of the investigation possible. To the author's wife, Ruth, and daughter, Anath, whose unfailing love and devotion sustained the author throughout the challenges and frustrations of the graduate program; to them this manuscript is dedicated. a}: J}: a}: >:: 2:: 3:: 3:: >:< >:< >:< >3 )3 >3 3:: :3 >:: )3 ii TABLE OF CONTENTS LIST OF FIGURES . . . . . ......... . ......... NOMENCLATURE ............... . . ....... INTRODUCTION ..................... . . . . THEORY 2. 1 Basic equation and background ............ 2.2 Velocity . . . ..................... 2.3 Shear stress. . . . . . . ......... . ..... 2.4 Mass rate of flow ................... 2.5 Temperature . . . .................. 2.6 Heat transfer through an air curtain . . . . . . . . . EXPERIMENTAL 3. 1 Experimental setup .................. 3. 2 Instrumentation .................... 3. 3 Scope of tests and procedure .......... . . . RESULTS AND DISCUSSION 4.1Velocity..... ........... 4. 2 Turbulence ...................... 4. 3 Temperature . . . . ................ 4. 4 Heat transfer through an air curtain ......... CONCLUSIONS .............. . ........... APPENDIX ..... . ................ . ..... REFERENCES .......................... iii 12 14 17 19 20 24 26 34 38 50 59 66 72 74 85 FIGURE 10 11 12 13 14 15 LIST OF FIGURES Schematic diagram of an air curtain. ......... Traverse mechanism and probes . . ..... Instrumentation table ....... . ........ Calibration curve of the r.m. 5. meter versus fre- quency count ............ . . . . ..... . Dimensionless velocity profile ..... . . ..... Velocity and temperature distribution (Tests 30) . . . Velocity and temperature distribution (Tests 20) . . . Velocity, temperature and percent turbulence distri- bution (Test 41-2) ................ . . . Velocity, temperature and percent turbulence distri- blltion (TeSt 45‘6) o o o o ooooo o o ooooo o o 0 Velocity, temperature and percent turbulence distri- bution (Test 43-4) ..... . . . . . . . ....... The coefficient Cm versus the product of Reynolds number by the aspect ratio . . . . . . . ...... Semi-theoretical stream function . .......... Semi-theoretical shear distribution, related to outlet velocity... ........ Dimensionless apparent viscosity . . . . . . . . . . . Semi-theoretical shear distribution, related to local velocity ........... ..... .. iv 28 33 40 41 43 45 46 47 49 51 52 54 55 LIST OF FIGURES - Continued FIGURE Page 16 Dimensionless temperature profile .......... 6O 17 The coefficient CT versus yT ............. 62 18 The coefficient CT versus yT ............. 64 19 The coefficient CT versus yT ............. 65 20 f(a') versus a' ..................... 68 21 ¢ (H/bo) versus H/bo . . . . . . ...... . . . . . 69 22 Heat transfer through an air curtain ......... 71 APPENDIX A.1 A.2 A.3 LIST OF APPENDIC ES Equation of motion ........ . ........ Free convection through an opening in a vertical partition . . . . . ..... . . . ......... Sample calculation ................. vi Page 74 76 78 :11 afionzzw C. NOMENC LAT UR E width in the jet defined by equation (2. 6. 3), ft. a/JTCI'n H' half the thickness of the air curtain, ft. coefficient in the velocity distribution velocity spreading coefficient defined by equation (2. 1. 33) specific heat, Btu per 1b, 0F temperature spreading coefficient defined in equation (2. 5. 5) manometer head, in. manometer fluid acceleration of free fall, ft per 36;. sec. coefficient of overall heat transfer, Btu per hr, oF sq. ft. height of the door opening, ft. spill height, ft. thermal conductivity, Btu per hr, 0F, ft. numerical coefficient in equation (2.6. 17) momentum flux, lb per ft, sq. sec. sensible heat, Btu per hr. mass rate of flow, lb per sec. temperature, oF t-tc, Tmth-tC velocity parallel to x- axis (longitudinal direction of the jet), ft. per sec. velocity parallel to y - direction (transverse direction of the jet) ft. per sec. x, y, z rectangular coordinates O. B 7m thermal diffusivity, sq. ft. per hr dimensionless coefficient defined by equation (2. 5. 5) density of manometer fluid, 1bm per cu. ft. vii m <12>"f —b"f_ k 32'1“ t1‘37 "Mp—3.37 _ pCp 3)" (LL?) In addition to the equation of motion and the equation of energy, the equation of continuity must apply: 3_9_( u) + L0") = 0 (2.1.8) 3 x a Y The boundary conditions for the equation of motion are at y=0, v=0, T=o (2.1.9a) at y=00, u=0, T: o (2.1.9b) Multiplying equation (2. 1. 8) by if and adding to equation (2. 1. 3) results in i b(pfiz) + Mp???) = "LE 4,..— (2.1.10) 8* by bx by Integrating equation (2. 1. 10) between two variable limits a(x) and b(x) one obtains: b b IL(_3)d-T- T- uVI-fb—B—Ed (2111) an'pu y- b a p a aaxy .. Letting a = 0 and b = (I) and applying the first postulate, namely %}2‘ = 0,, together with the boundary conditions of equation (2. 1. 9) one gets 00 . ——f pu?‘ dy=0 (2.1.12) 0 where it was assumed that the order of integration and differentiation is immaterial. From equation (2. 1. 12) it is evident that: (n f pu dy= M= constant (2.1.13) where M is the momentum flux per unit width of the jet. Equation (2. l. 13) states the constancy of momentum flux at every cross section of a two-dimensional jet. The second postulate states that the velocity profiles in the zone of established flow are affine. In other words the rate of spread of the profiles must be linear, progressing away from the outlet nozzle, or u — = 2. 1. 1 u f (n) ( 4) where - i n—n (x) (2.1.15) Forming a ratio of momentum flux at some distance x from the outlet to the momentum flux at the outlet, assuming that the density is constant, 'and substituting equation (2. l. 14) one gets: (D .2 fxum fz(r))dn - 2 E =1: 0 _, —_- (Jig—)- in". (2.1.16) M0 nobo “0 b0 where (I) 2 Im- J f (man (2.1.17) The same ratio can be written for the zone of establishment, assuming that the functional relationship of equation (2.1. 14) holds true in this region, - b 2 a) 2 — '- 2 M of uody +b‘[ x u,n f (n) dn x0 .1. —=1: _ = 2.1.]. Mo 33b. b0 ( 8’ From which lm can be determined and substituted back into equation (2. l. 16) giving: um = 1‘9— (2.1.19) where x0 is the length of the irrotational core. This computation was suggested by Corrsin (1950) and Albertson (it a;1. (1950). The length of the irrotational core was also determined experimentally by the same authors. The length of the irrotational core for a three-dimensional jet was reported by Corrsin (1950) to be 7d to 8d (where d is the outlet diameter). Albertson St a_1. (1950) report that for a two-dimensional jet xo=10,4bo, and x0 = 6. 2d for the three-dimensional case. Since there exists at the present time no satisfactory theory of turbulent flow some assumptions pertaining to the nature of the flow have to be made. The theories providing for such assumption, are based on either statistical mechanics or on phenomenological theories, substantiated by experimentation. , The statistical approach was first used by Taylor (1935). Taylor based his theory on the assumption that the vorticity is conserved. However, no further progress could be made without introducing further simplifying assumptions. One such assumption is that the components of vorticity are transported unchanged by the turbulence. This is a drastic assumption and is clearly untrue for most cases, as was recog- nized by Taylor himself. Howarth (1938) carried forward Taylor's theory by introducing an apparent eddy viscosity, assuming that turbulence is isotropic and that the momentum is conserved during the mixing process. Taylor's theory clearly suffers from the necessity to make quite drastic assumptions, which were not confirmed experimentally. The limitations of Taylor's theory are pointed out by Liepmann (1947), Alexander (1953) and others. The phenomenological theories of turbulent shear flow date back to Saint-Venant (1843) who introduced the mixing coefficient 6. It was realized by Saint-Venant that 6 is not a constant throughout the flow and is dependent on the scale and intensity of turbulent fluctuations. Boussinesq (1877) suggested the coefficient em for momentum transfer by turbulence. The equation of motion thus becomes bu Bu _ B Bu u—;+v-8';-—b—;(€m-§';) (2.1.20) For a flow in a pipe 6m was supposed to be a function only of the Reynolds number, but was also found to be a complicated function of the location. Hinze (1948) felt that Boussinesq's hypothesis may apply to free jets. Hinze evaluated the apparent eddy viscosity and found 6m: 0.00196(x+a)um (2.1.21) where "a" is the distance from the outlet nozzle to an equivalent point source. Hinze also introduced a coefficient er for apparent concen- tration and temperature transfer by turbulence, such that the (energy equation is BI" 81" B N" + = — — 2. 1. 22 u-g-x V‘s— BY (6'. BY ) ( ) The ratio between 6‘, and 6m is inversely proportional to the distance from the nozzle x. He found this ratio to be a rather complicatedfunction of location and velocity, with a large number of empirical constants. The concept of mixing length was introduced by Prandtl (1925) and his school. Prandtl postulated that as the masses of fluid migrated laterally they carried with them the momentum concentration of their point of origin. A typical velocity fluctuation u' due to such migration 10 is characterized by 63 u' 0c 2. 1. 23 1 S7 ‘ > where 1 is the mixing length. If it is further assumed that v'ocu' (2.1. 24) then the shear is given by 63 BE T=p 2|—--—--—— (2123 1 BY by If it is now assumed that the mixing length is proportional to the distance from the outlet I=qx (212m then Prandtl's assumption leads to useful results because the stream . function for the flow can be obtained: — X. Y-xF(x) (Llifl The virtual kinematic viscosity (eddy viscosity), as resulting from Prandtl's hypothesis is e=Xfib§3 (212$ where X; denotes a dimensionless constant, to be determined experi- mentally. Prandtl's theory was extended by Howarth (1938) to include heat and mass transfer. Howarth suggested that the temperature distribution is similar to velocity distribution. However, Hinze (1948), Forstall (1950), 11 and others measured the temperature and velocity distributions in a free jet and found that they are not equal. The discrepancy was large enough to require a new theory. An empirical approach to the problem of free turbulence was suggested by Reichardt (1941). Reichardt noticed a similarity between the turbulent process and molecular phenomena of thermal conduction. Even though in their details the molecular and molar processes differ, it is highly probable that with the turbulent processes also, purely statistical distribution will be obtained. In fact, the measured velocity distribution in free jets exhibits a strong similarity to the Gauss error curve. The hypothesis that the distribution of turbulent momentum of the principle motion is capable of being represented by a partial differential equation of the type of the equation of thermal conductivity (the location parameter appearing in place of the time parameter), can be put: _2 -3 Z . __9u = 9 ‘1 (2.1.29) ex 3 y The solution to equation (2. 1. 29) is ._ C 2 u2= C1+ J exp-( 1) (2.1.30) b b if the "transmission quantity” is b db A— E a; (Z. 1. 31) where b=b(x) is a measure for the width of the mixing zone, and C1 and C7. are constants. Equation (2. 1.30) may be put in a similar form _. ._. 2 u= u exp-( -—-Z-—) (2.1.32) m 50’ where 0’: on)1 x (2.1.33) 12 which has been found by Albertson (1950), Reichardt (1951), Alexander (1953) and others to be in substantial agreement with experimental observation. In summarizing, there exists at the present time no satisfactory theory of turbulent flow. The concept of simple mixing length is not usable and does not agree with the experiments. The assumption of constant exchange coefficient by Prandtl and G'drtler does not agree with measured values, as pointed out by Liepmann (1947). The previous theories have been successful in predicting mean-velocity distribution 'which agrees well with experiments becauseany reasonable assumption regarding the dependence of u' on the mean flow parameters, results in a solution which fits the data reasonably well. This, however, is not at all a proof that the assumptions of the theory are correct. As Corrsin (1949) mentioned, it does not seem possible to differentiate among these theories on the basis of mean flow measurements made with a total- head-impact-tube. Since most parts of these studies deal with temporal mean velocities, the velocity distribution as suggested by Reichardt was chosen. This distribution involves the fewest objectionable assumptions and offers the easiest mathematical handling. Several researchers have used this approach and their results are very satisfactory. 2. 2 Velocity. Making use of Reichardt's theory the velocity distribution is as follows: = exp (- 112) (2.2.1) Eéi-‘llt‘ll where T! = —L— (2.2.2) '\1 2 me 13 Forming the ratio of momentum flux at any distance x from the outlet to the momentmn flux at the outlet, using equation (2. 2. 1) one gets (114 0) 2 f u dy --7- f exp (-n)dy M=1=° =um£ (223) Mo 65130 1.13130 . . After performing the integration and simplifying one gets in u0 30—... (2.2.4) x} 1T me Substitution of equation (2. 2.4) into equation (2. 2. 1) yields which gives the temporal mean velocity at any point as a function of the outlet velocity, of the location, of the geometry and of a numerical con- stant Cm. The constant Cm was determined by Albertson e_t a_1. (1950) for a two- dimensional jet issued into an infinite medium. Albertson gives cm a: 0.109 (2.2.6) In order to determine the transverse velocity the equation of continuity will be integrated with respect to y, Y B— — u v=-f —— d (2.2.?) 0 5" y Differentiation of equation (2. 2. 5) with respect to x and substitution into equation (2. 2. 7) yields, after a change of variables 14 3: chmfi -Z— In[ [-exP( nz) - 41) eXP(- nz)]dn 2072' J— 0 where A=J -—2—139——- (2. 2. 8) 01—me the first part of the integral is recognized as the error function of n, and the second part can be integrated by parts. Finally 3 = N]? Cm All, [n exp(-nz) - i2?- erf (77)] (2.2.9) In addition the stream function can be determined from equation (2.2.5). The stream function \y is defined by __e__\i/ MI "by : v=' 5;; (2.2.10) Substituting equation (2. 2. 5) into equation (2. 2. 10) and integrating, one obtains Y 77_ W: Iudy=~12 mef quexp(-nz)dn 0 0 01' ‘15,] flbocmx :0 erf (n) (2.2.11) which clearly satisfies equations (2. 2. 10). In Appendix A it is shown that equations (2. 2. l) and (2. 2. 9) together with expression for the shear T satisfy the equation of motion. 2. 3 Shear Stress. For a two-dimensional turbulent flow, the boundary layer approxi- mation of the equation of motion can be written as 15 where the bar over the symbols indicates temporal mean and the prime indicates a deviation from that mean. The second term on the right hand side is the derivative of the shear stress T , 1: I T: (1%; -pu'v' (2.3.2) However, in highly turbulent flow the first term on the right hand side of equation (2. 3. 2) is usually much smaller than the second term, and can therefore be neglected, hence T=-p:'_v-' (2.3.3) The shear, stress can be determined theoretically from equation (2. 3. 1) by simple integration with respect to y. ._._oo.._m> m GI N\_D\> m._ 2. N _ 0.. ad 0.0 To «.0 .12 _ _ _ _ _ _ E + +9 gm 0 a. u . . GB + Q + + 1N0 + z o + E o m a 49+ 8. QUIE- q a f. a mo. Amy... D m w > 1.9.0 % m. o «6.... ED 3... mas .o... I. m 00.. + 1... . Q g 01 or 00 G 6.. . 9+ a. u a m on homes +,. + a a 5N EUFG Q n+0 _ G + E lm.o 6m emue . ma 6,... . .8 a mu ._.mwk + o... n. a a on + ..1 on o o + .. 41 Ho - I905 f1/soc. E . I. F, ' i, cm-mos c, -0.154 1) *‘0‘ Tm- 20.40 r .8 F .6)- .4 » .2 r x-2.o n. .341 » .2) x-so n. I . . TEST 31 (VELOCITY) a “\ o—--—o TEST :2 (VELOCITY) s ‘34“. o—---o TEST 32 (TEMP) .4 .2 .CIlcl ."l‘" ‘u'o - 30.22 Mm. cm-mos c, - 0.154 TM- 145- F ' TEST 33 (VELOCITY) o—--o TEST 34 (VELOCITY) o-—-O TEST 34 (TEMP) o\‘.§n‘_ ‘0—‘—‘———+ .o 1.0 m.) V FIGURE 6. tile! 0 ,«ll 4 blind I X I 2.0 fl. 6., sasss Tum cm-mos c, -0.154 Tm- 17.4- F m. V Robbin T X I 3.0 H. .1 ITEST 37 (VELOCITY) o—--OTEST as (VELOCITY) o—--oTE$T 38 (TEMP) Go ' 32.!0 fl. luc. 1.1 U,’ ?_ ' cm- 0.109 A 0,-0.154 Tm-MJ‘F .8» .6» AL .2» x-zon. 6“» .6» .4» .2» x-3.on. 1 .a» 0% . ITEST 35 (VELOCITY) a» "v, O—--OTEST 36 (VELOCITY) ' “3 0—--0TE3T 35 (TEMP) AP \1 .2» ' 11.5.0111 . . 1 1 J 1 ‘1‘ .fi’42I2463 10 (n) y X VELOCITY AND TEMPERATURE PROFILES. 2b,: 0.115 FT. 42 It should be noted that when no temperature gradient existed across the air curtain the jet was inclined a few degrees towards the left side (the warm side). When a temperature gradient did exist across the air curtain the jet was shifted towards the right side (cold side) and was inclined a few degrees to the right. The amount of shift from the original inclination to the left, depends mainly on the air velocity. The higher the velocity the less the jet is susceptible to shifting. The reason for such a shift was not well understood. At first it was believed that the air in the warm chamber expanded due to the heating and the larger pressure in the warm chamber caused the shift of the air curtain. However, in measurements made no static pressure difference could be detected across the jet. - It was then thought that at higher temperature the constant Cm varies, causing an asymmetry in the velocity profile, resulting in deflection of the jet from its original location. However there was no indication in the literature to substantiate this explanation and it was abandoned. . Turbulence measurements yielded some information that may be a clue to the explanation of the shift phenomena. < This will be further discussed in section 4. 2. In Figure 7 the data from the next series of tests are presented. The outlet thickness of the jet was 2bo = 0. 280 ft. , which gives an aspect ratio of 15.8. The points given in the figure are data points and the curves repre- sent equation (2. 2. 5), where the constant Cm was determined for each test separately. For high outlet velocity (Tl-o = 21. 76 ft. /sec.) the constant was Cm = 0.109. For the lowest velocity tested (1:0 = 9.68 ft./sec.) the data was correlated well by equation (2. 2. 5) with Cm = O. 220. The air curtain at 5 ft. from the outlet nozzle, in test 30 (:0 = 9. 68 ft./sec.) when temperature gradient existed ‘across it, is in a zone of transition. The effect of transverse pressure and buoyancy force is almost large enough to cause the curtain to break. It may be safely 43 TI ‘ {kg 8 ' 31' "’2. ; ° Meson/ac. 5- 2:3.) '0 25,-0.2» n ‘ I .. .z x -. wow . \‘I 07-0330 . ~9 .2 r- x- 4.0.; 5 ts. 7.9m? . 5;: ._ H I .8 .S mum m " ' 25-01230 :1 1 1E4; c.-0uoa ; --‘~~‘ Cf'Qm 3 . .S A 2 -—-- TEST 2: (VELOOITY) "—' TEST 29 (W1 . o—--o TEST 24 (vaoa'rv) ‘1 o—-—o TEsTao (vaccmn um... 7531' 24 new, °--° TEST 30 (TEMP) [.01- .o- -~--~~k. ‘1- \‘~\:“ .4 1. X.” ET. ' ‘5‘.‘ 1 “Moo 0 0 o ~~“‘~‘ 1 m 1 I 1 1 m4;; 1 A. __;y :7 Inssnzfizasnmw x ......... . “mm...““m,”,,,,,,,,,,,,,,,m WW [1141111111 ............................... [lg-2|." nine. 21190200 91 0,-1240 cm». ' °"°"°’ 2b.-0zao n °"°3°° c.-0.170 1'.-15.9°F c,- 0200 1'..- 26.6'F \?~‘~ I-—I TESTETWELDGTY) 0-“0 TEST 2. (MY) “n“ TEST 2. mm H TEST 25 (MI!) o---o TEST 26 (VELOCITY) °---° TEST 26 (TM) A '— faith I 5033:. r1 1 §A :n .0 10 m.) ’ HOME 7. VELOCITY All) TEMPERATURE PROFILES. 211-0180 FT- 44 predicted that if the momentum of the air curtain at the outlet would be further reduced, the transverse forces will cause a transverse flow of air through the air curtain. In this case the air curtain actually ceases to exist and loses its effectiveness. The phenomena of the shift of the air curtain from its original position, can be noticed in this series of tests also. Again the amount of shift depends on the outlet velocity and probably on the temperature difference across the air curtain. The higher the outlet velocity, the less the jet is shifted toward the cold side. The higher the temperature difference, the more the jet is shifted. . In Figures 8 through 10 velocity profiles are presented for an air curtain issuedat a thickness of 2bo = 0. 340 ft. The aspect ratio in this case was 13.4. The points in the figures present data obtained from velocity measurement. The solid line represents equation (2. 2. 5). Again the constant» Cm was found to vary from the theoretical value of 0.109, and was determined for each test separately. The tendency observed previously appears here too, namely, the lower the outlet velocity the larger the discrepancy between the theoretical Cm): . 109) and the observed Cm. In addition, to correlate the data from this series of tests by equation (2. 2. 5), it was necessary to divide the equation by the constant C2. This constant varied with the outlet velocity. Thus theair curtain cannot be considered a truly two dimentional jet. The coefficient C2 increased proportionally to the distance x from the outlet. Two feet from the outlet the coefficient was usually close to unity. At three feet it increased to 1.15, then to 1.19 at four feet and 1. 21 at five feet from the outlet nozzle. The shift of the jet due to the temperature difference is noticed in this set of experiments also. The amount of shift is found to be inversely proportional to the temperature difference across the air curtain. ’10-! r1“ 1%.] 2 an a 11.. - 13.4. 45 .%m .CIICI d-I . I g H TEST 41 Roo- 13.0” cm - 0.135 017.01 '83-” C. 'E." LO-D l—‘ VELOCITY MSTIIIUTM .1. o---o TEMPERATURE DISTIIIUTION 0””0 TUIUULEICE DISTIIIUTION CEO I I no ,0- ’A\h 1 ‘ -4.‘ / 25.2“”! V ‘0..- _ u -m- M“? I Run“ I I C. .'.37 1 L 1 1 1 1 1 1 1 4 1 1 #1,! T C s 4 3 t I f n 2 3 4 5 I 4. must a. mocm. TEIoEIuTuIE 1|) maniac: names. 2b,-o.14o rT. 46 1‘1 T u' : $-O ' m) . (:1? u. T. 9 !- Lqém (t ‘I U 1 I TEST 40 TEST 45 t. 16000 WHO 11 16 600 I . .0. . C. I 0131 Cm - 0131 c, - 0.220 a»: T,‘ - 14,20! S~6 01-0 ‘~o 2 r2 Nuns \c. -1.00 1 0 11 1 m #16. I 1o~10 IT'S Kurt: c, -1.04 ‘ fl 0 2 ~z \. \ V l $lb° l 11 )— 10 1o 1.0»10 K o a a .\ 0 . a»e \ \ o c ' $.43.” ._ 10‘ \ . ' ‘ I > C. '0]? 1 6 G M 2 2 1. . l \ 2 2 1 1 1 1 1 L 1 1 L n? T V DO '0‘” 10»10 L ._. VELOCITY ousmwunou s-o ' °\. \ ._. 0—»-0 TEquuTuaE DISTRIBUTION ' e\ ‘ , o-mo TURBULENCE 01511110011014 5“ \’ G -0 \ \ “0.2,"0 E. 5562’” c, -120 ‘ ‘ t / o be FIGURE 9. VELOCITY. TERPER‘TURE AID TURBULEIICE PROFILES. 2b,,- O.34O FT. 47 i T 1 RT; ’5‘” 1 1. >10 11111—11. ngT u 111,- 22,000 In, - 12.000 0. - 0.121 I». c, - 0220 Y. - 10 .07 SP6 ,P .1». 1-2 5:11.73 0, -1.03 1 1- 1 ;k I) I 10-10 10)!) .e. 1H 1H 01-. I’O'T‘O ,0’ 2*: *(1161 C. "J? . 45° 10m h. -—- VELOCITY 01311110111101. O-‘-. TE'PEIATUIE DUSTIIBUTION o---o mount: ounmunou 0»: FIGLIE TO. VELOCITY. TEPERtTIIRE MID TURBULEIICE PROFILES. 2h.-O.34O FT. 48 In summation it may be said that equation (2. 2. 5) does correlate the data well and represents the velocity distribution in a two-dimensional air curtain. The constant Cm, as found in the literature (for a two- dimensional jet issued into an infinite medium), applies here only for high - velocity and for high aspect ratios (Tests 31 through 38). For smaller aspect ratio and lower velocity the value of Cm does vary from 0. 109. There were not enough data taken to determine the exact dependence of Cm and C2 on either the aspect ratio or the outlet velocity. These two coefficients are probably temperature dependent. There is a considerable difference in velocity distribution of isothermal and heated jets both in the axial and transverse directions. This was pointed out by Cleeves (1947) and is substantiated here. Some idea of the change of the coefficient Cm with the aspect ratio and outlet Reynolds number can be gained from the following Table and from Figure 11. Table 2. The coefficient Cm for the various tests. M L 1.— Test No. Aspect ratio (AR) uo(ft/sec) Reo Re011:(AR):1110-3 Cm 35-36 38.4 32.10 11,006 425 .109 33-34 38.4 30.22 10, 380 398 . 109 37-38 38.4 25.65 8,800 338 .109 31-32 38.4 19.05 6,659 256 .109 27-28 15.8 21.76 17,869 2.82 .109 23-24 15.8 15.52 12,596 199 .155 25-26 15.8 12.40 10,426 165 .170 29-30 15.8 9.68 8,067 128 .220 43-44 13.4 21.35 22,000 295 .121 45-46 13.4 16.56 16, 600 222 .131 41-42 13.4 13.57 13,488 181 .135 49 5001- I -o- 2110 = 0.115 ft. 1. 1:!) . —o— 2110 = 0.280 11. 400.. 1;) -A— 21110 = 0.340 ft. I 0 l I I I .20 .22 '5' m Figure 11. The coefficient Cm versus the product of the aspect ratio (AR) by the outlet Reynolds number (Re), on a semi-logarithmic paper. ' It is quite obvious that Cm is inversely proportional to both the outlet Reynolds number and the aspect ratio. It is inversely proportional to the product of the Reynolds. number by‘the aspect ratio. From the data it appears that when this product is higher than 300,000, the jet can be considered as truly two-dimensional. ~When this product decreases, the discrepancy between the air curtain and a true two- dimensional jet increases. , In Figure 12 the theoretical stream function was plotted._ For this purpose equation (2. 2. 11) was made dimensionless as follows: 50 )1! _ f‘x— Hobo _ «From 70—0- erf(r)) (4.1.6) A theoretical value for the coefficient was used in the plot, namely, cm = 0.109. The lines in Figure 12 represent the stream function at equally spaced values. The figure is given only up to a distance x = 5 ft. from the outlet nozzle. The function was not extended to the zone where the spill occurs since the spill mechanism is not well understood and the stream function from the previous equation most certainly does not apply to this region. However the figure demonstrates the entrainment and _ -1 1 it is quite obvious that umoc x2- and Q (X1 x+z-. 4. 2 Turbulence. The theoretical shear stress is plotted in Figure 13 in a dimension- less form, using equation (2. 3. 7), namely 1 l T "2 = [~12— x/bo ”PI-n?" erf1 (4.2.1) 10 no where T) : __y__ y/bo (4.2.2) ’\( 2 me NIZ Cm x/bo From the figure it is quite apparent that the shear stress has two maxima, and is symmetrical around the centerline. This plot represents a semi-theoretical distribution, since it is based on Reichardt's theory, which correlates experimental data adequately. The plot is theoretical in that it is based on calculated values rather than experimental measurements. In Figure 14 the coefficient of apparent viscosity em is plotted in a dimensionless form, according to equation (2. 3. ll), l5‘ —\ 1 20‘- -r 25~ ) 1 , 1. 1 Figure 12. Semi-theoretical stream function in a dimensionless form (C1.n = 0.109). 52 1)% . [éfi‘exénrrfmfl'fi X , , -l/.76 Figure 13. LA. Semi-theoretical shear distribution, related to the outlet velocity, at various locations in the flow (Cm = 0. 109; b0 3 0.170 ft.)- 53 A..— Em 1 ij3 NET—x erf(n) = R 8° 81., n (4. 2. 3) where n = ——Y—-—— (4.2.4) ~12 me The plot was made for four distances x/bo with Cm = . 109 and for aReynolds number at the outlet corresponding to that of Test 44, namely Reo = 22,000 Figure 14 can be considered as a semi-theoretical plot, based on the exponential velocity profile. This figure may be used for determining the heat transfer coefficient, if Reynolds' analogy or some other relation- ships are assumed. It should .be noted that the ratio Em/v reaches a maximum value at the centerline of the jet and has a general shape similar to that of the velocity distribution. The assumption that 6m >> v is further confirmed for the particular values chosen here. Another form of the shear stress distribution is shown in Figure 15. The shear stress T is now related to the local velocity Ti, rather than to the outlet velocityTi-o that was used in Figure 13. . For this purpose equation (2. 3. 7) was modified as follows 1 — 1’ Em 7- i‘ .2 - I 7' 2 WW?) erf(n)] (4.2.5) p u cm = 0.109 (4.2.6) where It should be noted that curves obtained have quite a different shape than those in-Figure 13. The curves have one minimum at the centerline of the jet and increase without bound toward the two edges. Since the shear stress is given by = -p u'v' (4.2.7) I500 1000 500.. 54 I500 I0002 500 /\ 249,517.65 A/.\ ‘ X/b. =2353 1 Figure 14. 4. I X/bo =29.41 .,__ 1% 2 4 6 6 5%. The ratio of apparent viscosity to the kinematic viscosity at various locations in the flow. (Re 2 22-000: C... = 0-100 and Figure 15. .Semi-theoretical shear sdistribution, related to the local velocity (dotted line) and the velocityldistribution (solid line), (Cm = O. 109, ha = 0.170 ft.). 56 and since it was assumed that u' is proportional to v', it may be concluded that Figure 15 represents the turbulence, when a different scale is used in the ordinate. The assumption that v' is proportional to u' and thatTiTv-l is proportional to u' is quite a drastic one, and was neither investigated nor confirmed in this investigation. . However, as pointed out in section 2. 3, it is frequently used and it gives some insight into the turbulence process, even though it may not be quantitatively exact. With this in mind -Figure 15 may be compared with Figures 8 through 10. ~ In these figures the results from turbulence measurements are plotted in a dimensionless form, together with velocity profiles. All data were taken with an air curtain issued at a thickness of Zbo = O. 340 ft. The outlet velocity was varied from 13. 57 to 21. 35 ft. per sec. In Figure 8 data are plotted for an air curtain issued at 13. 57 ft. per sec- (Tests 41 and 42). There are eight turbulence profiles repre- senting four distances from the outlet nozzle. -On the right hand side in Figure 8 are the velocity and percent turbulence profile when there was no temperature gradient across the jet. On the left hand side are velocity, percent turbulence and temperature profiles when such a gradient did exist. The dotted line connects data points from turbulence measure- ments, conducted with the hot wire anemometer. - The points represent percent turbulence or (WAT) x 100. The percent turbulence at the outlet was less than one percent in most cases. At a distance 2 ft. from the outlet the percent turbulence was very high when a temperature gradient existed across the air curtain. This may be due to the fact that at a distance 2 ft. from the nozzle the jet is still in the zone of transition where the flow is unstable and percent turbulence may be high. . Further from the outlet nozzle, where the flow becomes fully developed, the percent turbulence decreases, and keeps decreasing with x. For the air curtain with no temperature gradient the turbulence is approximately symmetrical around the centerline. There is always a 57 minimum at the center point, and a gradual increase of percent turbulence progressing away from the centerline. It should be noted that the very high percent turbulence at the edges of the jet is due to a low velocity at these points, rather than high fluctuations of the velocity. The air curtain with a temperature gradient across it presents quite a different situation. Here the turbulence profile is asymmetric, where the turbulence on the warm side has, in most cases, a peak much higher than at a corresponding point on the cold side. For example at a distance of 3 ft. from the outlet nozzle the turbulence on the warm side is about 12.4 percent at a distance y = 0.60 ft. from the centerline, com- pared to 5. 8 percent at corresponding y-location on the cold side. InFigure 9 data from Tests 45 and 46 are presented. The outlet velocity is 16. 56 ft. per sec. and the percent turbulence at the outlet is always less than 0. 82 percent. The tendencies observed previously are present also in this figure, i. e. , the higher percent turbulence on the warm side and the gradual decrease of turbulence progressing away from the outlet nozzle. The peaks at the warm side, however, are not as pronounced as in the Figure 8 . This may be due to 'a smaller temperature difference across the jet (14. 20F compared with 25. 60F in previous test) or it may be due to the higher outlet velocity. In Figure 10 the data from tests 43 and 44 are presented. The outlet velocity is 21. 35 ft. per sec. and the percent turbulence at the outlet is always less than 0. 65 percent. The observation that the peak in the turbulence profile is higher in the warm side than in the cold side, is certainly noticeable here too. Also, the percent turbulence decreases as the jet progresses away from the outlet nozzle. In summation, it may be said that the percent turbulence decreases as the air curtain progresses away from the outlet nozzle. In addition, there exists an asymmetry in the turbulence profile. . For a jet issued at uniform temperature into a medium at the same temperature the 58 turbulence profile is symmetrical around the centerline. However, for a jet having a temperature gradient across it, the turbulence at the warm side is considerably higher than the turbulence on the cold side. This‘asymmetry is proportional to the temperature difference across the jet, and may also be inversely proportional to the velocity at the outlet. The exact effect of temperature and outlet Reynolds number were not determined in the present investigation. The higher turbulence on the warmer side causes several phenomena. .Since the turbulence is higher it may be safely assumed that the shear stress is higher on the warm side. This causes an asymmetry in the forces acting on the jet, which in turn causes the jet to be deflected from its original position. The phenomenon of the jet deflection toward the cold side, due to the temperature gradient, was observed and discussed before. This deflection is caused by the higher percent turbulence, even though the exact mechanism cannot be stated explicitly. It may be stated that the higher turbulence on the warm side is due to increase of the apparent viscosity 6m proportionally to the temperature, which is similar to the effect of temperature on the kinematic viscosity. However, the coefficient 6m represents a physical process which is not well understood and the above statement does not help in clarifying the phenomenon and its causes. ‘It can be safely assum'ed that the turbulence is increased due to an increase in the buoyancy forces. This assumption is somewhat sub- stantiated by the fact that the degree of asymmetry is proportional to the temperature difference across the jet. The higher temperatures on the warm side cause warm air to rise at the edge of the jet. This process of counterflow induces turbulence at the edge of the jet, which then pro- gresses into the main flow. The effect of temperature on the apparent viscosity and the percent turbulence in a free-turbulent shear flow deserves further attention and study. 59 4 . 3 Temperature . The temperature profiles, as obtained from measurements, are presented in Figures 6 through 10. The temperatures are presented in a dimensionless ratio T/T—m where T := t — tc and Tm = tW - tC. The temperatures of the cold and the warm sides were established at a distance of 2 ft. from the outlet nozzle. The traverse mechanism with the thermocouple was moved towards the warm side until the temperature reached an asymptotic value. The same procedure was followed in the cold side. Since at distances larger than x = 2 ft. , the traverse mechanism could not be moved far enough to reach an asymptot, the temperature difference at x = 2 ft. was considered the maximum temperature difference Tm In Figure 16 the data from temperature measurements are compared with the form of the error function which was assumedfor the temperature distribution. For this comparison equation (2. 5.4) was modified as follows. By defining b%_as the distance yT at which the temperature is three quarters (or one quarter) of the maximum temperature difference existing across the air curtain, Tm, or at y=:i:b%_, T=;}-TmandT=g-Tm (4.3.1) it follows that 3' b1 71" =%—=%—[1+erf( T )1 (4.3.2) m N] 2 C x T or b i- ): 1 therefore b or «Fa—c X: i— (4.3.3) 60 f. _ m 43- Test 24 r 1A- Test 26 )r ‘0' Test 28 )- C] .99- D D O 098 A 0 DA .95’ o a '- 0 .0 A l—=2i[l+orf(8)] 90? 00A A ‘6'" o O Q ‘ I- . ‘ ' 99 3:". A 080- (900W 0 D ' 00 A" "l I D A A 070' "L‘ 7 DA DAD A D 1"“ 9'}: 'AA A A '60 of‘é: “ Figure 16. . Dimensionless temperature profile plotted on an arithmetic probability paper. Substitute equation (4. 3. 3) into equation (4. 3. 2) to get: 3- [1+ erf (34,521)) (4.3.4) This function does not contain any arbitrary constants. '1‘"- 1:- Temperature ratios from three tests, each at three distances from the outlet nozzle (x = 2 ft. , 3 ft. and 5 ft. ), for outlet thickness of Zbo = 0.280 ft. and height h = 83%- in. are presented as points inFigure 1‘6. rThe solid line 61 represents equation (4. 3. 4). Since the figure is drawn on an arithmetic probability paper, this is a straight line. Equation (4. 3.4) is seen to <_:_orre1ate the data reasonably well. Close to the center (at YT = O; i = i— ) the correlation is excellent, since the curve was made toffit the data, at this region - or yT = O was defined at the point where Tm = 71;. Further away from the center the correlation is less satisfactory. This is due to the fact that small errors in measurement, due to turbulence and inaccuracies, are multiplied as one progresses away from the centerline. The solid line in Figure 16 was also used as a mean to determine the theoretical [3 and then the theoretical CT. From the temperature measurements in each location the dimensionless temperature T/Tm was formed. The value of [3 was then obtained from the abcissa in Figure 16, corresponding to the ordinate T/Tm. Since the value of x, as well as the value yT Were known, a theoretical value of CT was then calculated from the formula YT C:— T VZ—xfi (4. 3. 5) The values of the coefficient CT thus obtained are presented in Figures 17 through 19. In Figure 17 the variation of the coefficient C with respect to yT is shown, for the different velocities and distances: from the outlet. This figure presents the results of tests 32 through 38, where the thick- ness of the air curtain at the outlet was 0. 115 ft. and the aspect ratio was 38.4. The scatter of the data is not too wide and may be due to experimental errors and effects of turbulence. An average value of CT = . 154 is suggested as representing the data of this series of tests. It should be noted that for the same series of tests Cm = . 109 was confirmed. Therefore, for a two-dimensional jet issued into an infinite medium a value of 62 x .3 m: .o n 33 a.» mugs—26 2: 343., so 332:8”. 2: mo 8E .2 Rama .9 _. .0 kin... O O O o O O O ._.)" $ka _ m. _ AW _ .V W _ W W .m |Ow a Boo _ and? ' '\‘ (In. ' 'N. I nI n. I ofiuuoc 50.3 .m 0 011.10 oauuoc Eoum .m ell-'9 vfiuuoc 83.3 .N .685: 3.3 n OB 708%: 2 .mn n fuofl .a mm .8 7255: 3.2 .e .e . owv LT mm «was (OI on $3. I»: 3 “mos Iol Nm “mos 63 C =0.154 (4.3.6) T or CT 6=_.__C = @- (4.3.7) In suggests itself. This value is in agreement with the value given by Reichardt, as indicated in section 2. 5. Figure 18 shows the variation of C versus yT for another series of tests. In this series the aspect ratio :as 15. 8, with the outlet width 2bo = 0. 280 ft. The scatter of the data is wider than in the previous figure. A reasonable average for the coefficient is CT = 0. 200. . It should be noted that for test 28 a value Cm = 0. 109 was found for the velocity distribution, and a value of C = 0. 170 approximates the coefficient for T the temperature profile. This test therefore comes closer to the sug- gested theoretical value of C than any other test in this series. T’ In Figure 19 the results are presented for tests 42 through 46. The scatter is seen to be rather wide and a value of CT = 0. 220 approxi- mates the results. This series of tests was conducted with outlet thick- ness of O. 340 ft. and aspect ratio 13.4. The deviation from the theo- retical value is much larger than in the previous figures. The values cT = 0.154 with 2b(, = 0.115 ft. CT = 0.200 with 2h0 = 0.280 ft. .cT = o. 220 with Zbo = 0.340 ft. were used for calculation of the temperature profiles according to equation (2. 2. 1). The solid lines in Figures 6 through 10 represent equation (2. 2. 1) with the coefficient CT as indicated. It is seen that these lines correlate the data reasonably well. In general the temperature profile becomes flatter, progressing away from the outlet nozzle. 64 .A .um owN .o n 33 Hr» oonmumflp was mumso> BU “aofloflmooo 65 mo “3m .wH ondmfim . km‘. 0 o o it) w o s l . _ _ a _ a _ 300 u a. .I 0 4‘ H 1 0 0 "P I O (outm... I. Odd-It On. >, \ .u- u.\ I I u 1%» "H‘." \ s: omaiq. n 1r...» Lb-» .n < I .34... l B/fibMKbl I {all I cannon 59G .m Oillno wines E03 .m 9.|-l9 oanuos Eoum .N A.oom\.um moi "one Iql om “ooh. A635: 2.3. new )OI. 32.3. 6.035: 3.2 new. JPI £33. 33::th use Isl 3:3. 65 .A .aw ovm .o n .58 HS oocmumfip 05 w~+wu0> BU “Cofioflwooo 93 mo “cam” 42 onnwfih I‘ — _ q fl _ _ UHOU I d d I H I d n' dlm‘d.|:m’d are, , 8.3.5.4 u t ' em- Iflfill 0nd ".4“ I l ’8) .WWIH . O 5. :4 .T.... I: ‘1‘. : d I @v./: q o my... a. tart/mt. .. £4 .. t . e\ i - u .Z. I I- 4 . I I/fluaxfl \ . 2 / :u . n 4 d u. c 0 Id /U m..// (4| IQ! [lax u g x % \ . fl I d. I II . // x \ . I I I /.- \D/ \ _ I I .\...£%..¢.t¢.wm§ 6., .. . t... 4 . -m. l 6‘ I. ll \ ’ \P, d, . \ U .wrfl . _ , z ,D. \ ./. \ . I I. I - \QIIIQ z/ ,9/ R \ D, d /.d.. \ I/ . \ / Z r I: x / m\ d..\ n d . I Q \..z Iv Taoism 8.: u owe Aduflfi 4.3 n ow. A635: 82 u cw. ofiunoa 80.3 .m .¢ .m UHNNOQ SOHH @fiNNO—H EOHH Saxon 50.3 .N .0. 3. the ”IQ: «iv 30H. ID! Nv «och. 66 In most cases the temperature profile and the velocity profile are assymmetric, namely, Ti = am at y = 0 and T = i—Tm at VT = 0 but y a! yr. In most cases the temperature profile tended to be shifted more to the warm side. However there were quite a few exceptions and no general tendency can be given here. The effect of the temperature gradient on the apparent viscosity and the turbulence was discussed previously. It may be concluded that since the apparent viscosity 6m is proportional to the temperature gradient the effect of temperature on the heat transfer is twofold, i. e. , since .. <1: q ‘5de and sinc e emoc AT it is obvious that q will increase as some power of the temperature gradient, and this power is likely to be closer to two than to unity. The effect of the temperature gradient on the heat transfer properties of an air curtain was not studied in detail in the present investigation, due to time and equipment limitations. This phase of the problem certainly deserves much more work, theoretical as well as experimental. 4.4 Heat Transfer Through An Air Curtain. The results from the measurements of quantities of heat and temperatures are presented in Table 1. The results are presented in dimensionless quantities, where Re is a parameter describing the H bo outlet velocity and the geometry and Nu/Pr describes the overall heat transfer coefficient. To correlate the data, reference is made to section 2. 3 where the semi-theoretical expression describing the heat transfer through an air curtain was derived, based on the suggested entrainment- spill mechanism. 67 The equation there obtained was Nu ' H _ = — 4.4.1 Pr K f(a )Re ’ b0 ( ) - where the function f(a') was defined by the equation: (C; a) f(a') 2 NET— f exp(-nz) [1 + erf( -%—)] dn (4.4.2) at and where K is a constant, to be determined empirically. The magnitude of a' can be evaluated as follows. From equation (2. 6. 3) and from the expression for the mass rate of flow at the outlet nozzle one gets a 230b,: oo = 2) Edy (4.4.3) 0 Upon substituting equation (2. 2. 5) and changing variables equation (4.4. 3) becomes a! 'J 2 A CmH' f exp (-n7‘)dn 2 b0 (4.4.4) 0 which can be further simplified to read: b0 ' erf(a') = fol? (4.4.5) CmH' Upon substitution of the appropriate values for b0 and H' one gets the dimensionless width a'. .For the present investigation a' was always smaller than 0. 5. Therefore equation (4.4. 2) was integrated numerically on a digital computer for~a' ranging from zero to 0. 5. The theoretical values were substituted for the coefficients, namely C = 0.109 and 6: 1.414. m The plot of f(a') versus a' is shown in Figure 20. The equation of the straight line passing through the points is the following: f(a') = 0.3058 - 0.2718a' (4.4.6) 68 fla‘I=.3058—.27I8a‘ I :3 .l .2 .3 .4 .5 Figure 20. The function f(a') versus the dimensionless width a', as obtained-from numerical inte- gration on a digital computer. Equation (4.4. 6) can now be substituted back into equation (4.4. 1) to get 153-1- = K (0.3058 - 0.2718a')Re ’31— (4.4.7) Pr b0 Since affinity of the velocity profiles was assumed, the dimensionless width a' is a function only of the geometry, namely, of the ratio H/bo. . Equation (4.4. 7) can then be modified and the functional relationship determined, i. e. , H . Nu _ ii 0.3058 K ’73-; - RePr — q> (b0) (4.4.8) 69 .8- .6- l- .4- .r- - 02" 485 H = 11' (who) .0321“) r l L l g l L I J l 1 ._ 20 40 60 m I00 200 H) Figure 21. The function 4) (§-) versus the ratio H/bo, on logarithmic papero. -In Figure 21 this function is plotted on a log-log paper for five values of the ratio H/bo. For this purpose experimental data were used to calcu- late the ordinates, for the various values of H/bo encountered in the investigation. The equation of the straight line passing through the points is 4) (37,-) -.-. 0.0321 (b0) or ’0.500 (5%) °-' 0.0300(1)?) (4.4.9) Upon substitution of this function <|> (ii) into equation (4.4. 8) one 0 gets the semi-theoretical expression: Nu ’H — = . K _ . .1 Pr 0 2758 Re b0 (4 4 0) 70 The experimental data are presented in Figure 22. These data are correlated by a straight line with a slope of 0.0808, as determined by the statistical method of least squares. These data therefore indicate that the coefficient K is: K ‘3! 0.341 (4.4.11) The heat transfer through the air curtain within the experiment range of the curtain parameters Re Ji- of 51, 000 to 141, 000 is be expressed by the final semi-theoretical correlation Nu H _ = 0 0 ._.- O o l Pr 0 08 8 Re I ha (4 4 2) Equation (4.4. 12) is accurate within :1: 20 percent. A The dimensionless heat transfer for an air curtain issued at zero velocity is calculated in Appendix A. 2. The value obtained is presented in Figure 22 on the ordinate. The suggested heat transfer mechanism due to the entrainment-spill process holds true only for the parameter Re £1- larger than approximately 50, 000. For smaller values of this paramgter the air curtain ceases to exist and the heat transfer is due mainly to free convection. The line describing the heat transfer in this region is probably line (b) in Figure 22. However, this region has not been studied at the present investigation. Equation (4.4. 12) can be modified as follows: Sto = 0.0808 J91} (4.4.13) where Sto is the Stanton number at the outlet. This equation is also . - b accurate within :i: 20 percent for the curtain parameter 1% between the values 0.090 to 0. 170. 71 .5925 .30 cm awsosaa Mommas: «no: mo unmfioflmooo mmoficowmcmfifip 0:» mo soflgonuoo fiducogwnomxm .NN 30E .5 mm-I N_g +_2_>__T p(uT -+V —— 0X b_-Y- (A. This equation has to be satisfied by the assumed velocity profile and by expression derived for 3 and T . From equation (2. 2. 5) .1: = 0 A exp (-n"‘) (A. fig 2b fl y where A: -——-9-—— ; r) = —---—— (A. «I 17 me NI me From equation (2. 2. 9) _ _. 1r v = «I 2 Cm A uo[n exp(-nz) - 4 erf(n)] (A. From the first postulate b = 0 (A. x Now .1) .2) .3) .4) .5) or 33' A— '3‘; :-. 7:9- exP(-nz)(4fiz - 1) (A.1.6) and — ._. _ 3.3 _ 6‘1 __a_fl __ N]? 11 Ar) Y ”Fr? by " ' Cm: eXP (~02) (A.1.7) 74 75 From equation (2. 3. 7) Z T = — 39139 exp (ml) erf (n) (A.1.8) NIT): and _ 5T .. u b 1 .3? - 5:17:21 '7 erf (n) - up (412) n 16XP(-nz) (A.1-9) Subsitution of equations (A. 1. Z), (A. 1.4), (A. 1. 5), (A. 1. 6), (A. 1. 7) and (A. l. 9) into equation (A. l. 1) yields :0 A exp(-nz)[ ‘3‘}:— exp(-n2) (4172- 1)] + Ail-ON]? Cm[n exp(-nz)- 4w erf(n)] ~__ _2 [—€::n ex135772)] 3 $332 [n erf(n) W: eXP(-nz)] eXP(-nz) " (A. 1.10) simplifying: AZ A3 -2— exP(-nz)(4nz-l) - -2- [4nZeXP(-nz)- ~11? n erf(nH ? b 2 = ——L—c x [ n erf(n) - exp (‘7? )1 m 17 Substituting for A from equation (A. 1. 3) one gets: 2 ? ('- 11,: —L_x.[ -exp(-nz) + «(1r n erf(n)] = L?[ 1r 7’) erf(n) ' exp (“1721] 5’ 1T Cm M 1'me (A. 1.11) This last expression is obviously an equality. Besides proving the correctness of the mathematical manipulations, equation‘(A. 1. 11) shows that the expression assumed for velocity distribution, equation (2. 2. 1) does satisfy the equation of motion, providing the shear stress T is given by equation (2. 3. 7). This is not in itself a proof of the correctness of Reichardt's theory and hypothesis, but is a prerequisite for any assmned velocity profile or shear profile. 76 A. 2 'Free Convection Through An Opening In A Vertical Partition Therprocess of natural convection through an opening in a vertical partition has been studied very little. The system under consideration is one in which two sealed chambers, at different temperatures, are separated from one another by a vertical partition having a rectangular opening of height H and width w . 7 Due to temperature difference a density difference will exist between the two chambers. The more dense cold air will therefore be spilled out, and warm air will be introduced into the cold chamber. An example of a case like this will be a cold enclosure, with a doorway that is not protected by an air curtain, or a doorway protected by an hypothetical air curtain issued from a slot over the doorway at zero velocity. Assume the absolute pressure (p0) at the elevation of the opening centerline is everywhere equal, since the density differences are small. Brown (1962) used this assumption and confirmed it experimentally. - Let the density in thecold chamber be pc and that of the warm chamber be pw. The pressure p at a level 2 below centerline will be pc =po+pcgz (A.2.1) and in the warm chamber pw=po+pwgz (11.2.2) The pressure difference at that level is Pc - pw = (pC - pw).g z (A.2.3) Assuming that the air in both chambers is ideal fluid and that Bernoulli's equation applies, i. e. , the velocity of flow from one chamber to the other at elevation z is m v: J2 “SW” ‘ (A.2.4) p where _ + . p = M (A.2.5) 77 The mass rate of flow (0') from one chamber to the other will be H obtained by integrating equation (A. 2.4) from z = 0 to z = —Z-: _ - 3 Q!:Cp —§—Jg ——-M3 P HT (A.2.6) where C is the coefficient of discharge, similar to the coefficient used in sluice-gates. The magnitude of C usually varies between 0.6 to 0. 9. - The quantity of heat carried from one chamber to the other with the mass rate of flow (2' can be approximated by q :Q'Cp Tm ‘(A.2.7) where T,n = tw - tC (A.z.8) Define a coefficient (ho) for the over-all heat transfer between the chambers under the se‘ conditions: ho=‘w—I§-T- (A.Z.9) where the subscript zero stands for zero velocity of the air curtain. ~In dimensionless form equation (A. 2. 9) with equation(A. 2. 7) can be written as Nu:BE_:%Jg£%—%__my_).}i3 EBB—— k =-3C-:—'\/Gr Pr or Nu C r—-- where the Grashof number is . 3 Gr = ———z—%Af H 78 Equation (A. 2. 10) cannot be exact because the viscosity and thermal diffusivity have been omitted. On the other hand, in heat transfer texts the Nusselt number is usually given as Nu = A (0103 (Pr)b (11.2.11) where A, a and b are numerical constants. The exponent on the Pradtl number is usually close to unity and the exponent on the Grashof number is between 1/2 and 1. Brown (1962) studied the problem of natural convection through rectangular openings in a vertical partition and suggested the equation: 131—9- : 0.044 (Gr)°-59 (A.2.12) Pr for the problem. Now let the following condition exist for an air curtain issued at zero velocity: 1C 65°F pc = 0.07570 Ap 83°F pw= 0.07323 3 0. 00247 (1bm per cu.ft.) 0; 07446 (lbm per cu.ft.) 1:w v = 1.649x10“ (sq.ft. per sec.) for average temperature? = 74°F H = 83.125 (in.) g = 32.174 (ft. per sq. sec.) Then the Grashof number can be evaluated as: Gr = 1.305 x 101° and with this value of Grashof number equation (A. 2. 12) yields: = 41. 2 x 103 This figure is plotted in Figure 22 on the ordinate, where Re = 0. A. 3 Sample Calculations. Test 44 will be used to demonstrate the calculations performed. - In Table A.3.-1 results from measurements at a distance x = 3. 0 ft. from the outlet nozzle are shown. 79 The columns in the table are as follows: 1 . Location - indicates the distance y from the jet's centerline in feet, as measured by means of the point gage. 2. Total head - indicates the reading (d) from the manometer (connected to the total head impact tube) in inches of manometer fluid. 3. Velocity head - the square root of the difference between the manometer reading (from the previous column), and the manometer reading at zero velocity (do = 1. 3430 in.). 4. , Velocity ratio - the ratio of the velocity at any location (II) to the velocity at the outlet (Tl-.0). Since the density difference is neglected this ratio is also given by the ratio of 5171-30 at any location to m at the outlet. 5. Potentiometer reading, in millivolts, as obtained from the potentiometer, connected to the thermocouple on the traverse mechanism. 6. Temperature - the reading from the previous column was converted to temperature, in 0F, by means of conversion tables. 7. Temperature ratio - the ratio of the temperature at any location (-t-) subtracted from the temperature of the cold-side (Ye), to the maximum temperature difference across the air curtain (Tm = 14. 80F) as measured at the distance x = 2. 0 ft. from the outlet. 8. [3 - a dimensionless number , defined by equation (2. S. 5), was read from Figure 16, corresponding to the temperature ratio from the previous column. 9.- The distance, YT' from the centerline of the temperature profile (yT = 0 at T/T-m': §-). This distance was measured after the points from column 7 were plotted on graph paper; 10. The coefficient CT was calculated from the equation _. YT fix 8 where the a re riate and were substituted from revious columns. PP P YT P CT ll. 12. 13. 14. 15. 80 Meter balance, I. The values of the hot wire anemometer's balanced circuit are given. The readings are four times milliamperes and are a measure of velocity, if a calibration curve is used. The read- ing at zero velocity was 10: 344/4 m. a. for this particular test. Noise level, Mn- The count from the electronic counter, with an Open gate for 100 seconds, was converted to r.m. s. millivolts by means of Figure 4, to obtain the noise level. The root mean square voltage Mn+v’ when the hot wire anemometer was balanced on' "warm, " namely, with current flowing through the hot wire. The root mean square voltage, M when the square wave was n+v+s’ applied. The root mean square was measured by counting the frequency on the counter and converting to r.m. s. by means of Figure 4. The percent turbulence was calculated from the previous columns by means of the formula: z z “j u _ 100!4x3.05 [ I 1 {M n+v Mn - ‘ —17__T—_' Z 2 u 1+ 3;? I - 1‘ 1V1n+v+s - Mn+v where N = 1.4 is the resistance ratio and B = 154.0 is the bridge null. oo.¢ m.¢®~ b.0vH >©.H Ann 1 1 1 1 1 N. m>.m o.¢NN o.NwH H>.H mwm Fwd. ooo.~ wwo. NNO. mm¢m.~ o. Fm.v 0.00N m.®mm ¢>.H awn MNH. mmo.~ ONH. mwo. om¢m.a m. Ho.¢ m.Nom M.H>N v>.H mom aha. obo.a mmm. wwo. oomM.H v. ©©.N m.>~m m.>©N N©.H 50¢ MNH. ooo.~ 5mm. owfi. NNQM.H m. ww.N m.vom 0.5mm mm.H Haw MFA. owo.~ N©¢. Nwfi. ochm.~ N. AN.N N.m>N 0.0mm >@.H oaw and. ONO.A wmm. com. mmmm.~ A. m. 3.8 m.oe~ o.~m~ 3.2 m2. SN. oood omm. NS. can; o.o mo.~ H.¢©N m.mHN o>.H @N¢ ova. mwm.o mwm. mam. mwwm.H H. vo.d o.>hN o.wNN om.H omw mmH. m¢o.o hmm. mom. omwm.H N. o¢.N m.>¢N w.m- Hm.H MN¢_ mofi. ¢N®.o whw. Nwfi. 005m.a m. ON.M m.mwN ~.mmN oo.~ Haw ooa. m0©.o mum. wva. h¢©M.H w. m~.m O.HwN m.N¢N oo.~ Now amm. mow.o HON. NOH. mmmm.~ m. mo.m w.¢mN w.wo~ ®©.H mom 0mm. mbw.o 00H. ooo. movm.~ o. wN.m m.h¢N o.>ma Hw.H owm omm. comic Hod. omo. MV¢M.H N. NN.¢ m.th m.hmH ow.~ whm mam. omw.o omo. ONO. vmwm.~ w. fio.m m.@h~ w.wHH ©©.H mbm ©HN. Hmw.o 1 1 1 o. 1 1 1 1 1 cam. ONw.o 1 1 1 o.~ mom .o o .NS 0 .vo ave .H m3. 1 3.0 .o 1 0mm . 0oz»; uodsO . mm man map may outflow. -f 5.8 5.8 5.8 8.8.3. 3 5.8 I 8% 3:5 .9 mocmfimm H0 used cm was: was: :03 . rm: m+>+¢2 >+G2 G2 Hod—02 .uOnH lm >uwoofim> HMHOH. {mood .um o .m n x .wv pooh. no“ mcoflmgodmu was madmamndmmoz .H .m Jo. gash. 82 In Table A. 3. 2 the results of measurements in quantities of heat and temperatures are given. The columns of this table are as follows: 10. 11. The date and time for each test. The duration of each test, in hours. . The amount of energy required to maintain the warm chamber at constant temperature in Btu per hr. .The heat losses through the walls, qe. The losses were determined by the calibration of the warm chamber to be 89. 2 Btu per hr. , 0F. The temperature difference between the warm chamber and the outside was multiplied by this figure to obtain the heat losses in Btu per hr. .- The net quantity of heat transferred through the air curtain, q, is given in Btu per hr. This quantity was found by subtracting qe from q1,i.e., q = C1! "' Qe- . The average temperature at the warm side, tw' All the hourly record- ings from the thermocouples in the warm side were averaged for the test period. The average temperature of the cold side, to, obtained by similar procedure. - The temperature gradient across the air curtain is defined by All} '-'- tW - tC' . The outside temperature, to, an average of hourly recordings of two . . o thermocouples located out51de of the warm chamber, in F. The temperature difference between the warm chamber and the outside, i.e. , At = t - t . z w o The overall heat transfer coefficient through the air curtain, in Btu per hr. , oF, sq. ft. -The quantity of heat q is divided by the area of the doorway (A = szé-x83i—/l44 = 30. 3 sq. ft.), and by the temperature difference existing across the air curtain, Atl. 83 mo mo mammogram? mo Ho>3 m £33 £95 miss? 05. Amflouau umofi ohms Mo .8 com 3m mm.~ 0 ~48 u to mo ”3.9083 0&8 .nofimnnflmo new as one? .280qu memo» Go>mm £09,523 and? on... wan—snags >0. posflgnoump who? miss? 93 flaws??? mommofi «mos 9:“. "Mumaom 2.5m mic 22 22 see 2.: :mo each 3.22 30.2 mouoo .2 .ooo - 8:3 .2 dog :18 v.3 v.3 22 0.2. 28 Boo 2%.... 2.3.: 25.3 ooumm .2 .ooo - 8.8 .2 Jon. 3.2 ego 2.3. 2: 5:. 2mm Nose 2.2m 2.2.2 oom.o oouoo .2 .85 - 28 .o .ooo $2 «.mm You 2: mg: 28 ~86 2.3.2. 83.: 25.: omumm .o dog -32 .o .ooo 2.2 23 ~13 2: Ni. wdw goo 826 £20.: oomd 858 .o .ooo .. omuoo .o .85 8.3. 22. 23 22 ode ~13 mmwg. $2.2. 292 38.2 82 .m .ooo .. emumm .s .ooo «mom mi. 23 22 0.3 now 28;. Base. 2%.: 28.2 omamm .e .ooo - £2 .2. .89 .3 .Um ho .2 .36 o» :4 co 3» U 0w 5 .mp3 053. was 030 3 3m Ahov snowshomfiofi Ann\.3mv “mom 05.3. .3» amour MOM mucogmnsmdoz ouaumnomfiuob was “mom .N .m .4 03mm. 84 In addition a dimensionless number describing the heat transferred through the air curtain was calculated, namely .153 - EH Pr pCpV Where the overall heat transfer coefficienth is the average of the values presented in column 12. The properties of the air were obtained from the International Critical Tables, for the temperature of the air at the outlet (ta). For this test ta = 73.6 OF therefore p = .0745 lb per cu. ft. , v = 1.647 x 10" sq. ft. per sec. and CF = O. 24 Btu per 1b, 0F. The velocity of the air at the outlet, for this test, was calculated from the equation Ho 4% %l (d'do) Where 7m = 50. 55 lbm per cu. ft. is the density of the manometer fluid, obtained from the International Critical Tables for the appropriate temperature. . Substituting the prOper values A — 50.55 u0-2.315JW X0.356 or _ uo = 21.35 ft. per sec. and the outlet Reynolds number Re: 3}, b0 ___ 21.35 x 0.170 v 1.647x10“ Re = 22,000 I H 83.125 Re :6: — ZZ'OOOJIZXO.170 - 140,710 Finally REFERENCES Abramovitch, G. 1939. The theory of a free jet of a compressible gas. ‘NACA TN 1058. Albertson,-M.«L.; Dai, Y. B.; Jensen, R. A. and Rouse, H. 1950. Diffusion of submerged jets. Trans. ASCE 115: 639-697. Alexander,.L. G. _e_t a_1. 1953. Transport of momentum, mass and heat in turbulent jet. Univ. of 111. Eng. Exp. Stat. Bull. No. 413. American Society of Mechanical Engineers Research Publication 1937. Fluid Meters, Their Theory and Application. 4th edition. Bickley, W. G. 1937. The plane jet. .Phil. Mag. 23:727-731. Birkhoff, G. 1957. Jets, Wakes and Cavities. Academic Press Inc., N. Y. Bjorkrnan, R. V. 1961. . Air curtain improves plant heating. Air Eng. January. Boussinesq, J. 1877. . Essai sur la theorie des eaux courantes. - Memoires presentes par divers savants a l'Academie des Sciences 23. Brown, W. G. and Solvason, K. R. 1962. Natural convection through rectangular openings in partitions-1 Int. ~ J. Heat Mass Transfer, 5:859-868. 85 86 Cadiergues, R. 1957. Warm air curtains. Heating and Ventilating and Journal of Air Conditioning, . 30:423. Callaghan, E. E. St a_1. 1949. Investigation of flow coefficient of circular, square and elliptical orifices at high pressure ratios. .NACA TN 1947. Chia-Chiao Lin 1947. Velocity and temperature distributions in turbulent jets. The Science Repts. of National Tsing Hua Univ. 4(5):419-450. Chou, P. Y. 1947. The laminar mixing motion of two incompressible gases. ‘ Chinese‘Journal of Physics, China, Vol. 7, p. 96. Cleeves, V. 1947. Isothermal and nonisothermal air-jet investigation. Chem... Eng.- Progress, 43(3):123-134, . March. Corrsin, S. 1949. Extended applications of the hot wire anemometer. NACA TN 1864. Corrsin, S. 1949. Diffusion of submerged jets: Discussion. -A.S.C.E. 75:901. _Corrsin,. S. .et a1. 1950. FTir-t-her experiments in the flow and heat transfer in a heated turbulent air jet. - NACA Rept. 998. Corrsin,.S. and Kistler, A. L. 1954. - The free stream boundaries of turbulent flows. NACA TN 3133; January. Eckert, E.~R. G. and Drake, R. M. 1959. - Heat and Mass Transfer. McGraw Hill Book Co. , N. Y. 87 Elrod, H. G. 1954. . Computation charts and theory for rectangular and circular jets. Trans. ASHVE 60:431-444. Ferrari, C. 1935. The transport of vorticity through fluids in turbulent motion. .NACA TM 799. Flow. Corporation. 1958. ~Model HWBZ hot wire anemometer theory and instruction. Flow Corp. Bulletin No. 37B. Forstall, W. and Shapiro, A. H. 1950. ~Momentum and mass transfer in coaxial gas jets. ASME Jr. of Appl. Mech. 72:399-408. Forthmann, E. 1936. Turbulent jet expansion. NACA TM 789. Gc'Srtler, H. 1942. Berechnung von Aufgaben den freien Turbulenz auf Grund eines neuen N'aiherungsansatzes. .ZAMM 22(5):244-254. Gygax, - E- E. 1956. Air curtain seals door opening. Heating, Piping and Air Conditioning, 28(1):146-147. Gygax,. E. - E. 1957. Air curtain entrances grow wider. Heating, Piping and Air Conditioning 29(10):124—126. Helander,- L. e_t a_1. 1954. . Characteristics of downward jets of heated air from a vertical discharge unit heater. Trans. ASHVE 60:359-884. Hetsroni, G. 1961. 1 Heat transfer through an air curtain. Unpublished thesis for M.S. degree, Michigan State Univ. ,- East Lansing, Michigan. 88 Hinze, J.- 0. 3t _a_1. 1948. Transfer of heat and matter in the turbulent mixing zone of an axially symmetrical jet. Appl. Sci. Res. A-l, 435-461. Howarth, . L. 1938. Distribution in plane and axially symmetrical homogeneous jets. Proc. Cambridge Phil. Soc. 34:185. Hukill, W. V. and Smith, E. 1946. Cold storage for apples and pears. USDA Cir. No. 740. Karman, T. von and Howarth, L. 1938. - On the statistical theory of isotropic turbulence. Proc. Roy. Soc. ,1 London, ser..A 164 (914):192-215. Koestel, A. 1954. Computing temperatures and velocities in vertical jets of hot or cold air. Trans.lASHVE 60:385-410. Kuethe, A.M. 1935. Investigations of the turbulent mixing regions formed by jets. Journ. Appl. Mech. 2, A87-95. Kurek, E. J. 1962. All about air entrances. Air Eng. 4(6):38-42 June, 62; 39-40 July, 62. Liepmann, H. W. and Laufer, J. 1947. Investigation of free turbulent mixing. NACA TN 1257. Lin, . C. C. 1947. Velocity and temperature distributions in turbulent jets. The Science Reports of National Tsing Hua Univ. series A, vol. 4, October. - Loitsianskii, L. G. 1944. . Integral methods in the theory of the boundary layer. NACA TM No. 1070. 89 Michael, W. R. 1960. - Luftschleiertiiren fur Kuhlraume. Die K'a'lte 13(12):679-682. Milne-Thomson, L. M. 1955. Theoretical Hydrodynamics. The Macmillan CO. , N. Y. Norton, W. 1959. 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Measurements of entrainment by axisymmetrical turbulent jet. Jr. of Fluid Mechanics 11(1):21-32 Aug. Rouse, H. 1959. Advanced Mechanics of Fluids. J. Wiley and Sons, N. Y. Ruggeri, R. S. (31: a_1. 1950. .Penetration of air jets issuing from circular, square and elliptical orifices directed perpendicular to an air stream. .NACA TN 2019. Schlichting, H. 1933. - Laminare Strahlausbreitung. ZAMM 13:260. Schlichting, H. 1960. Boundary Layer Theory. McGraw Hill Book Co.; N. Y. Schmidt, W. 1941. Turbulente Ausbreitung eines Stromes erhitzter'Luft. ZAMM 21:265-278. .Seban, ~ R. A. et a1. 1962. Ve-I-oc-ity and temperature profiles in turbulent boundary layers with tengential injection. Jr. of H. T. Trans. ASME 84(1):45-54,. February. Sleight, P. 1961. .. Curtains of air. Compressed Air Mag. 66(10):15-l7, October. Squire, H. B. 1948. Reconsideration of the theory of free turbulence. - Phil.-Mag. 39:1-20;- Jan. Steiner, E. 1958. Device for producing a room-closing air curtain. U. S. Patent No. 2, 863,373. 91 Szablewski, W. 1950. The diffusion of a hot air jet in air in motion. .NACA TM 1288. Taylor, G. I. 1935. Statistical theory of turbulence. Proc. Roy. Soc.,- London, ser. A 151 (873). Taylor, G. I. 1937. Transport of vorticity and heat through fluids in turbulent motion. Proc.‘Roy. Soc. 135A, 685. Tollmien, W. 1926. Calculation of turbulent expansion processes. ZAMM 6:468-478. Williams,.R. M. 1958. Refrigeration losses through open doors and conveyor passes. Unpublished thesis for M. S. degree,~ Michigan State Univ. , East Lansing, Michigan. “11111111111“