I NATIONAL ADVISORY COMMITTEE 
 i FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1274 
 
 BEHAVIOR OF FAST MOVING FLOW OF COMPRESSIBLE GAS IN 
 
 CYLINDRICAL PIPE IN PRESENCE OF COOLING 
 
 By G. A. Varshavsky 
 
 Translation 
 
 "K Voprosu o Povedenii Bystrodvizhushchegosya Potoka Szhimaemogo 
 Gaza V Pryamoi Tsilindricheskoi Trube pri Nalichii Okhlazhdenia. " 
 Zhurnal teknicheskoi fiziki (U.S.S.R.). Vol. XVI. no. 4. 1946. 
 
 Washington 
 
 September 1951 
 
 bOCUMENTS DEPARTMENT 
 
1^^^'^^' ,.f 
 
 NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS 
 
 TECHNICAL MEMOEAKDUM 1274 
 
 BEHAVIOR OF FAST MOVING FLOW OF COMPRESSIBLE GAS IN 
 CYLINDRICAL PIPE IN PRESENCE OF COOLING 
 B,y G. A. Varshavsky 
 
 INTRODUCTION 
 
 The investigation of the distribution of energy in the f lov of 
 a compressible gas that moves without resistance in a straight 
 cylindrical pipe and gives off heat indicates that the existence of 
 a "thermal" Laval nozzle in the supersonic region is possible 
 (reference 1) (that is, Bg^ > 1, the cooling of the flow under 
 these conditions results in a rise in the Bairstow (Mach) number). 
 This interesting result is actually strongly distorted by the effect 
 of the resistance. For the simple case of gas cooling (heat con- 
 duction at the wall), the well-known relation between the resistance 
 and the heat transfer makes the existence of a "thermal" Laval 
 nozzle improbable. If only the heat transfer by contact is taken 
 into account, the existence of a "thermal" nozzle is impossible; 
 however, if radiation from the products of combustion -is also 
 considered, the "thermal" nozzle is possible only in a narrow range 
 of high temperatures and for large dimensions of the nozzle (pipe 
 diameter) . 
 
 1. SOLUTION OF EQUATIONS 
 
 Assumptions are based on the analysis of the simultaneous 
 solution of the following equations: the momentum equation (in 
 which the friction forces are taken into account by the usual 
 "hydraulic" resistance coefficient) 
 
 dw 1 dp >, w / , \ 
 
 ^ d? = - p dl -^ 2D (^^ 
 
 the equation of continuity 
 
 wp = constant (2) 
 
 "K Voprosu o Povedenii Bystrodvizhushchegosya Potoka Szhimaemogo 
 Gaza V Pryamoi Tsilindricheskoi Trube pri Nalichii Okhlazhdenia. " 
 Zhurnal teknicheskoi fiziki (U.S.S.R.). Vol. XVI, no. 4, 1946, 
 pp. 413 -416. 
 
NACA TM 1274 
 
 and the heat -transfer equation 
 
 'jj= a|T(l +^a2J -t^ j dS (3) 
 
 where v (m/sec) is the velocity of the flow at a certain section of 
 the pipe; p (kg/m^) ia the static pressixre at the same section; 
 T (^) is the temperature of the gas; p ( (kg) (sec2)/m'* ) is the 
 density; D (m) is the diameter of the pipe; t, is the coefficient 
 of resistance; G (kg/sec) is the weight of the gas per second; 
 Cp (cal/(kg)(°C)) is the specific heat of the gas; a (cal/(m2) (hr) (°C)) 
 is the heat transfer coefficient; Ba = v/c is the Bairstow numher 
 (^ch number); c (m/sec) is the local velocity of sound; t--, (°K) 
 
 is the temperature of the wall; and dS is an element of the pipe 
 area that corresponds to an element of the length dx. 
 
 The solution of this system for the general case was given by 
 the author and M. D. Weisman in 1934 (reference 2) and leads to 
 complicated expressions not capable of a clear qualitative analysis. 
 In investigating the problem of the formation of a "thermal" Laval 
 nozzle, the system was reduced by the author to nondimenaional 
 variables and was solved under the assumption tQ = (that is, 
 
 at a wall temperature negligibly small in comparison with the 
 stagnation temperature of the flowing gas). 
 
 The conclusions as to the possibility of formation of a 
 "thermal" Laval nozzle for tQ = will naturally be the more 
 favorable in this sense because for certain wall temperatures 
 comparable with the gas temperature, the intensity of cooling 
 will be less than in the case considered by the author. 
 
 The transformed system of equations is written as follows: 
 
 e de ^JH ^' ('^ 
 
 2 
 Pil_ = constant (2' ) 
 
 
 
 a(e[M¥)3] 
 
 i _ 
 
 i-( 
 
 ■^-^-3 
 
 4 Md^ (3' ) 
 
 NACA comment: The symbol t^^ in equation (3) appears subse- 
 quently as tg. Brackets and braces in equations (3), (1')^ 
 
 and (3') do not appear In the original version but are con- 
 sidered desirable In the Interest of clarity. 
 
WACA TM 1274 
 
 where = ^/'^n ^^ ^^^ nondimensional temperature (Tf^ is the ini- 
 tial gas temperature); it = p/Pq is the nondimensional pressvire 
 (Pq is the initial pressure); M = a/wgc p is the Margulis 
 criterion ((g)(m)/3ec2) is the gravitational acceleration); 
 
 P = Ba ; and | = x/D is the nondimensional length. 
 
 By elementary tremsf ormations , the system (l' ) , (2'), and 
 (3 ' ) is reduced to the equation 
 
 dp 
 
 [(C-4M)p - fj kp [l +(^)p] 
 
 1-p 
 
 (4) 
 
 i 1 
 
 that for constant values of C and M may be integrated . By 
 
 simple transformations, corresponding expressions for 6 and it 
 
 can also be written. The relations (4'), (4''), and (4'*') are 
 
 the result of the integration 
 
 i - 
 
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 The constant of the values ^ and M and the existence of the 
 normal relations between them for the case of large values of 
 Bairstow (Mach) number has been confirmed several times in both 
 Russian and other foreign literature (references 3 and 4). 
 
 "Hereinafter \ = (k-l)/2, 
 
NACA TM 1274 
 
 2. ANALYSIS OF SOLUTION 
 
 (a) Radiation neglected. The hydrodynamic theory of heat 
 exchange gives the relation hetween ^ and M 
 
 5 = 8M 
 
 (5) 
 
 By auhatituting equation (5) in equation (4), the following expression 
 IS obtained : 
 
 dp _ Cp(iAp)(i-kp) 
 de " 20-1) 
 
 (6) 
 
 A study of equation (6) shows the existence of three regions 
 of variation of 3 : 
 
 1. p > 1. In this case, p drops along the pipe when 
 
 approaching p = 1. 
 
 2. l/k < 3 < 1. Here p increases when approaching p = 1. 
 
 3. p < l/k. In this region 3 drops and approaches the normal 
 flow of an incompressihle gas in the presence of cooling. The 
 behavior of P, 0, and jt for particular cases is shown in 
 figures 1, 2, and 3. 
 
 (h) Radiation considered. In the case of the presence 
 of triatomic products of combustion (water vapor and carbon dioxide) 
 in the flowing gas, a certain quantity of heat will be given to the 
 walls by radiation. This condition leads to an increase in a and 
 a breaMo'vm of relation (5). Determination of the increase in a 
 that is required so that a "thermal" nozzle may exist in region 1 
 is made possible by equation (6). It is thus necessary that 
 
 M > 
 
 fl) 
 
 (a) 
 
 The application of relation (6) facilitates finding the heat stress 
 of the endothermal reaction and makes the existence of a thermal 
 nozzle fundamentally possible. 
 
NACA TM 1274 
 
 that is 
 
 M> -L- (7) 
 
 685 ^ 
 
 An increase in the value of a hy 17 percent corresponds to equa- 
 tion (7) as compared with the value given "by the hydrodynajnlc 
 theory of heat interchange . 
 
 The computations made for a nozzle with a 200-millimeter 
 diameter emd hy using the air products of gasoline combustion with 
 an excess coefficient equal to 1 indicate that for Pq = 1 at 
 atmosphere, the coefficient of heat transfer Increases by 
 10 percent because of the radiation. In working with a larger 
 air -fuel ratio or oxygen-fuel ratio, compositions for which a 
 increases by more than 17 percent are possible. This increase 
 in a will however occur over a small part of the pipe. After 
 a certain lowering in the temperature and a corresponding decrease 
 in the radiation, a drop in Ba along the pipe begins (as in the 
 usual case) . 
 
 3. CONCLUSION 
 
 1. The hydrodynamic theory of heat exchange applied to the 
 investigation of the possibility of the formation of a -"thermal" 
 Laval nozzle on cooling the gas by heat conduction at the wall 
 leads to a negative result; that is, the formation of such a 
 nozzle is impossible. 
 
 2. When radiation is considered for the case of the flow of 
 gasoline combustion products In an air-rich mixture, a certain part 
 of the pipe in the region of high temperatures may work as a thermal 
 nozzle. 
 
 After a certain lowering of the gas temperature, however, the 
 pipe will operate normally (with a drop in the Bairstow (Mach) 
 number along the pipe) . The effect of an increase in Mach number 
 in this case holds only for: (a) relatively large diameters of the 
 pipe and (b) products of combustion obtained in an air-rich mixtxrre. 
 The practical application of the thermal nozzle even under these 
 conditions is in the author's opinion impossible. 
 
 Translated by S. Reiss, 
 
 National Advisory Committee for Aeronautics. 
 
NACA TM 1274 
 
 REFERENCES 
 
 1. Vulis, L. A.: On the Transformation of Energy in a Flow; 'etc. 
 
 Otchet Nil. 
 
 2. Varsbavsley, G. A., and Weisman, M. D. : Bull. NIVK, 1, 1S34 . 
 
 3. Gukhman, A. A., Varshavskj^, G. A., and Others: Jour. Tech. 
 
 Phys., vol. 4, 1934, p. 10. 
 
 4. Jung, ln(7\TaT: Warmeubergang und Reibungswiderstand hei 
 
 Gasatromung in Rohren hei hohen Geschwindlgkeiten. 
 Forschungshef t 3S0, erganzung 3u Forschung auf dem Geblete 
 des Ingenieurvesens, Bd. 7, Ausg. B, Sept./okt. 1836. 
 
MCA TM 1274 
 
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NACA TM 1274 
 
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UNIVERSITY OF FLORIDA 
 
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