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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME RIREPORT ORIGINALLY ISSUED January 1945 as Advance Restricted Report L4L19 CHARTS OF PRESSURE, DENSITY, AND TEMPERA~TRE CHANGES AT AN ABRUPT INCREASE IN CROSSSECTIONAL AREA OF FIDW OF COMPRESSIBLE AIR By Upshur T. Joyner Langley Memorial Aeronautical Langley Field, Va. Laboratory WASHINGTON NACA WARTIME REPORTS are reprints of papers originally Issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. DOCUMENTS DEPARTMENT L13 I ' I.i:. Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundalior http://www.archive.org/details/chartsofpressure001ang NACA ARR No. TLL19 RESTRICTED NATIONAL ADVISORY COr c.ITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT CHARTS OF PRESSURE, DENSITY, AND TEMPERATURE CHANGES AT AN ABRUPT INCREASE IN CROSSSECTIONAL AREA OF FLOW OF COPMPRESSIBLE AIR By Upshur T. Joyner SUMMARY Equations have been derived for the change in the quantities that define the thermodynamic state of air  pressure, density, and temperature at an abrupt increase in crosssectional area of flow of compressible air. Results calculated from these equations are given in a table and are nilotted as curves showing the variation of the calculated quantities with the area expansion ratio in ter.i of the initial Mach number as parameter. Only te subsonic region of flow is considered. INTRODUCTION The wellknown BordaCarnot expression for change in pressure when an incompressible fluid passes an abrupt area expansion has long been used fcr estimating the pressure changes in comnressible air flow at an abrupt expansion. Thismethod is simple but not exact. The expressions for pressure, density, and temperature ratios given herein are for subsonic flow and are in precise agreement with the exact expression for the velocity ratio in a compressible flow at an abrupt Area expansion developed in reference 1. In the present report, Mach number, or the ratio of air flow velocity to existing sound velocity, is used as a parameter; whereas in reference 1 the parameter was the ratio of existing airflow velocity to the velocity that the air would possess if accelerated isentropically until its velocity was equal to the then existing local RESTRICTED 2 NACA ARR 1o. L4L19 sound velocity. This difference in parameters must be considered when equations fro.n the two papers are compared. By use of the same three fundamental equations used herein, a somewhat similar equation showing pressure and density changes across a shock loss in a pipe of uniform cross section was developed by Hugoniot and is given in reference 2. The expressions obtained herein for pressure, density, and temperature changes in a compressible flow at an abrupt expansion are too involved to be of practical use. T'\e present computations have therefore been made and are presented in tabular and graphical form. SY",:DOLS A crosssectional area of flow, square feet a velocity of sound, feet per second f area ratio (A1/A2) M ..i ch number (V/a) p static pressure, pounds per square foot V velocity of flow, feet per second Y ratio of specific heat at constant pressure to specific heat at constant volume, dimensionless R universal gas constant, Btu per slug per OF T absolute temperature, L160 + OF p density of air, slug per cubic foot Subscripts: 1 before abrupt expansion 2 after abrupt expansion NACA ARR NTo. ITL19 5 ANA LYSIS Prom the fundamental equations .or the conservation of energy, of continuity, an. for che conservation of momentum, equations are obtained tnat give the variation of the pressure, density, and temperature ratios with the area expansion ratio f in terms of the initial Mach number as parameter. Figure 1 shows the condition of flow assumed for the present calculations. The static pressure at a (fig. 1) is taken to be the same as at b for subsonic flow, as has been proved exneriirentally by Nuaselt (reference 5). uniform velocity, distribution before and after the expansion is assi1.t'n.. The ratio of specific heats y is taken as 1.L0G5. The fundamental equations are the equation for the conservation of energy: V 2 _1 _)22 Y 2 V12 +1 122 + Y P2 + + (1) 2 Y 1 Pl 2 Y 1 P2 the equation of continuity, PA1V1 = p2A2 (2) and the equation for the conscrx ition of momentum P2222 pAlvl12 = A 9 " Pl) (5) Prom equation (1) Y 1 2 2 Y 1722 2 2 1 a1 2 Vl 1 P2 or 2 M + 1 = 2 ( + 1+ (4) YACA ARR No. I4L19 From equation (5) P2= 1 + yf 2 V ) P 1 + = 1 + yfM2 12 P Prom equation (2) V2 P2 = P2 VI P2 A2 P2 When equations (5) and (6) are substituted in equation (4), y 1 + 1 =Y 2 l1 2 i2 2 I Y2+ 1 = 2 f2 + 1 2 2 eP2^ P2 + _L 2 2 2 'p 2 P2 2 Pl\2 2 + 1 P122 + 12 _M2 + 2 2) i 2 P2/ 2 2o (7) When equation (7) is solved for pl/p2 and the resulting equation is inverted, P2 Pl f21l2(y + 1) 1 + yf.1l2 2yfMl12(1 f) + 1 2f2 .12 + f211 (8) In order to obtain an expression for the pressure ratio, equation (8) is substituted in equation (5) and the following equation results: p2 1 + YfM12 + y\2yf..:!2(1 f) + 1 2f2:'.i2 + f11 I = (9) Pl y + 1 (5) (6) NACA ARR Io. LLL19 5 By differentiating p2/p' wIth respect to f, it can be shown that the maximum staticoressure recovery for any value of 1 is obtained when Y +J 2 12 + 1 f = (10) 2(y + 1) N12 The locus of maximum pressure ratio is shown in figure 2. In equations (3), (9), and (10), the sign of the radical has been chosen so that the results obtained are in the region where the assumptions are valid. The temperature ratio is obtained by us. of the general gas law and the computed values of pressure and density ratio as pl P1 P2 = rT2 T2 E2 /  = p/ (11) ? P2, /P Figures 5 and 4 show the variat'.on of density ratio and temperature ratio with area exoa'nsion ratio. In order to make the results shown in figures 2 to 4 usable in cases for which only the conditions after the expansion are known, the value of h!2 in terms of MN and f is given in figure 5. If N2 and f are known, Y1,l can be determined from this figure. The relation clotted in figure 5 is developed as follows: 6 IACA ARI No. L4L19 / (v2/a2)2 V22 YP2/P2 V2 P 1 /P 2 1 V 2l/ P By use of equation (6), I P2 P2 Pl PI Of fM M2 = 2 f2 RESULTS AND DISCUSSION The calculated values of pressure ratio, density ratio, and temperature ratio are given in table I. The values have been computed to 8 decimal places because of the form of the equations,in which small differences in large quantities are involved. In the region where M and f were both small, that is, 0.1 or 0.2, it was necessary to carry some of the calculations to 12 decimal .laces in :or.er to obtain smooth curves for the quantities calculated. NACA ARR No. L4L19 As in the case of incompressible flow, the calcu lated changes occur gradually after the abrupt increase in crosssectional area of flow, and the calculated and measured results are in best agreement at a distance the order of 6 to 10 diameters of the large cross section downstream from the abrupt area increase. The comparison of pressure ratios for compressible and incompressible flow is shown in figure 2, in which a longdash line gives the pressure ratio calculated on the basis of incompressible flow for the same initial conditions that are assumed for compressible flow at an initial Mach number of unity. It is evident that the effect of compressibility is vanishingly small for values of the area ratio of expansion below about 0.25. The shortdash line in figure 2 shows the pressure ratio to be obtained with isentropic expansion and an initial Mach number of unity. The experimental points from reference 3 shown in figure 2 were obtained from the only experiments known to the author in which pressure ratio has been measured at an abrunt expansion with compressible gas flow at high Mach number. These data were obtained for an area expansion ratio of 0.246, however, for which the difference between compressible and incompressible flow is insignificant. These exnerimental results agree well with the calculated results but are by no means conclusive. Agreement of experimental with calculated values at an area expansion ratio of 0.7 or 0.3 would be conclusive evidence of the difference in pressure ratio obtained with compressible flow from that calculated by the EordaCarnot formula for incompressible flow. An experimental investigation of the changes in pressure, density, and temperature at an abrupt increase in crosssectional area with compressible flow would serve to determine corrections for the effect of nonuniform velocity distribution and friction on the idealized results obtained from the present calculations. Langley Memorial Aeronautical Laboratory 'Tational Advisory Committee for Aeronautics langley Field, Va. 8 A CA ARR No. .L L419 1. Busemann, A.: Oasdynamik. Har.db. d. Experi'entalphys., Bd. IV, 1. Teil, Akad. Verlagsi.esellschaft m. b. H. (Leipzig), 1951, pp. 405407. 2. Ackeret, J.: Gasdyrnamik. Handb. d. Phys., 3d. VII, Kap. 5, Julius Springer (erlin), 1927, P. 525. 3. 'lurselt, W.: Der Druck im Ringquerschnitt von Rohren mit plitzlicher Lrw:iterung beim Durchfluss von Luft mit hoher Geschwiidigl:eit. Forschung a. d. Geb. d. Ingenieurwesens, Ausg. B, Bd. 11, Heft 5, Sept.Oct. 19l10, pp. 250255. NACA APR No. .: I I' .. .. I co hrO Nrlj O1O m hr"0 r'ao c,' o . .i o3rj .u in .ta' 0NN N4 .. , 0 o\C .l 0 oN'N r 4 ,.i 3,o o cr nj mo. a ,. 0w ."co " N 4 ' O m N, ,a " I N 0 N N0 0'1 0 \" l< 0 ,o 0 M 0 N CP 6<8 4 rr 0 4 CO f, Ol rD .4 Lr ''o"t0 o nT 0000 0 9 C H i, a r 0 _010 0" i 0 0,.r DPcOWo 0aN F 0 SIBr cir rflcc jNN 0 N40000. ."do N \ PO CN'.D U aJ, 0 c NC 0) OiC t1,lfl f I frO  hr u5 r 3n 3 o 0 30 N N 0' "s.ol cyO, q^Fu. F N .40 NCN"0%r i' .LT 0 04 rIJOrqu r4 : r ~ 'iir1fli0.3tN0ti NO nN in4 N ?Oin .4 r It%.l4O.inrdili rj N :" r co a 0 0 .4  4r 4 4 oo WMO 0 C CZ ar00 0.0 N 0000000 "j  r4 r" CN r !l.L4% JrCC N '0 'IOX rj J GIW\vr W % r4% P^ M un N %'ac 01 N N "OOL 2C ' '* 000000 I C! 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