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7 7 '.  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1357 AIR ADMIXTURE TO EXHAUST JETS* By E. Sanger 1. Introduction. The customary jet engines rockets,.turbojet engines, pulse jet engines show at actual flight speeds certain defects, like low efficiencies, failure in case of increased flight velocities, etc. Furthermore, all these power plants have, with the customary propeller piston power plants, the characteristic in common that they discharge considerable heat quantities with the exhaust gases without utilizing them. The ramjet engine has no static thrust, and, in the subsonic region, its thrust increases approximately with the square of the flight velocity; thus, one is forced, in many cases, to combinations with other jet engines for takeoff and slow flight. This situation gives rise to the theoretical investigation of how far such disadvantages can be fundamentally reduced by air admixture to the exhaust jet in special shrouds. The problem of thrust increase for jet engines by air admixture to the exhaust jet was introduced into aviation techniques by the suggestions of Melot (ref. i). Due to a too general interpretation of several theoretical investigations of A. Busemann (ref. 2), so far no practical use has been made of these suggestions. The following considerations show that, in the case of lowpressure mixing according to Ml4ot's suggestions, probably no thrust increase of technical significance will occur for the flight speeds of interest (however, the lowpressure mixture is highly promising for ground test setups and for special power plants of relatively slow sea and land vehicles). In contrast, application of the highpressure mixing in ramjet type shrouds, where surrounding air is admixed to the exhaust jet, appears advantageous throughout for some aeronautical power plants in the range of high subsonic and supersonic speeds. The relative increases become larger the higher the flight speed and the less satisfactory the thermic efficiency of the jet engine used. Luftzumischung zu Abgasstrahlen," IngenieurArchiv, vol. 18, 1950, pp. 310323. NACA TM 1357 Fundamentally, the four energy components of two gas jets mix in such a manner that, for constant total energy, the sum of the enthalpies increases at the expense of the sum of the kinetic energies, with the total entropy increasing as well. If the mixing of the jet takes place at a pressure exceeding the undisturbed external pressure, part of the enthalpy developed may be reconverted into kinetic energy (by the following expansion of the gas mixture) and thus be made useful. These conditions lead to combination of the ramjet engine with other power plants in the form of ramjet type shrouds and mixing chambers, with the purpose of utilizing part of the otherwise completely lost exhaust gas energies, especially the heat energies of the core jets for reheating of an additional ramjet engine. Since, in this kind of heating, kinetic energy is unadvoidably supplied beside the thermic energy, we call this principle, in short, impulse heating of the ramjet engine. For simplification of the following theoretical investigations, we assume in the calculation that the mixing of the jet always takes place at constant pressure; in practice, however, mixing at variable pressure is applicable as well, particularly in lowpressure chambers for pressure increase, in highpressure chambers for pressure drop. I. ConstantPressure Mixing of Gas Jets 2. Theory of constantpressure mixing. A gas jet leaves an opening of cross section F0 with the velocity v0 and the remaining state points p, pO, TO. It mixes with the free surrounding air which flows in the same direction with the velocity v2 and possesses the remaining state points p, p2, T2 (fig. 1). The following relations are valid for the mixing of a certain air mass m2 = P2F2V2 flowing per second through the cross section F2 with the corresponding exhaust gas mass m0 = pOFOvo: Continuity theorem: The sum of the masses flowing per second through the cross sections F2 and FO equals the mass flowing per second through the cross section Fx POFOVO + P2 F2v = PxFxVx NACA TM 1357 Momentum theorem: Due to the pressure being constant throughout, the total momentum remains the same in every cross section POFV20 + P2F2 = PxFxv2 (2) Energy theorem: The sum of the energies flowing per second through the cross sections F2 and FO equals the energy flowing per second through the cross section Fx with presupposition of a conversion of kinetic energy into enthalpy free from relaxation as in the calculation of the gas throttling 0Fo + gcpTo)+ 2F2v2( + gcpT2= xFxx + gcpTx) (3) Equation of state: The values of the quantities px, Tx are related to the known constant mixing pressure p = gpxRTx (4) In the jet core, the customary rectangular distribution of the state quantities over the cross section with the same mass flow as the actual distribution was assumed. The gas constants R and cp are assumed approximately equal for the two gases to be mixed. The turbulence dissi pation into enthalpy is assumed to be free from inertia as was said before. With these four equations, the four unknowns Fx, Px' Vx, Tx may be expressed by the known quantities F2, p02 v2, T2, FO' PO' vO, T0' For simplification of the notation, new symbols are introduced for the known quantities Mass sum p0Fov0 + P2F22 = M kgsec/m] (5) Momentum sum pOFO2 + P2F222 = J kg] (6) Energy sum pOFOvO v20 2 + gcpTo) + 2F2v2 2 + gcpT2) = E kgm/sec] (7) Temperature density p/gR = 0TO = P2T2 = PxTx = D kgsec2 o/m (8) NACA TM 1357 Therewith the four determining equations become pxFxvx = M PxF,2 = J vx( x PXFxVx + gCpTx = E PxTx = D From them, there follow the unknowns J vx = ME 1 J2 Fx = 2 x gcpDJ gcpM2D PX = ME 1 J2 2 ME 1 J2 Tx = 2 gc M2 (9) (10) (11) (12) and the mixture Mach number V;=\ 2 1 ax 1 2ME/J2 1 (12a) used later on. 3. Propulsive mixing efficiency. The ratio of the kinetic energy flowing off per second through Fx and the total kinetic energy which has flowed in is xFxV3x j2/M Tm =  =  p0F + P2F2 2E 2gcp pOFOOTO + PFZ2vT2 S m2v2o \2 1 + m2o22 \ "Of A 2 therefore, it is only a function of the two ratios m2 /m and (13) v2IVO NACA TM 1357 The propulsive mixing efficiency depends neither on the pressure p at which the mixing takes place, nor on the densities, temperatures, enthalpies, or Mach numbers of the mixing partners concerned or on the relaxation of the vortex conversion into heat. The kinetic energies disappearing in the mixing are converted via vortex motion into additional enthalpy as in the process of gas throttling. The mixing efficiency for m2lm0  0 and m2 mO c~ becomes of course equal, nm = 1, since, when one of the mixing partners disappears, a mixing, and therefore mixing losses, are no longer possible. The mixing efficiency becomes for v2 vO = 0, that is, mixing with surrounding air at rest, equal m = 1/(1 + m2 mo) = "mOrx thus for mixing ratios in practical use very small; the jet energy is converted almost entirely into vortices and heat. The mixing efficiency becomes, furthermore, with v2/v0 = 1 equal to Tl = 1, that is, when both gas jets have equal velocity of the same direction, the mixing occurs by diffusion without losses in kinetic energy, even when the temperatures of the two jets are different. In the range of arbitrary values of m21m0 and v2/vo, the mixing efficiencies show minima; these minima lie at values of m2/mO the higher, the smaller v2/vo; their course may be calculated to be m2/mO = v0 v2, that is, the losses become largest when both jet impulses are at first equal. The course can be seen in figure 2. If the gases to be admixed have a priori noticeable velocities in the direction of the gas jet, the mixing efficiencies are throughout consider able and the higher, the more gases are admixed inasmuch as m2/mo > v0 v2. Only for very small v2/v0 or m2/"O this rule is inverted, that is, the efficiency then deteriorates with increasing admixture until mOVO = m2v2 and rises again afterwards. Since, in case of mixing efficiences below unity, warming of the gas mixture occurs also when the temperature of both jets is equal, the supply of momentum always has a heating effect as well. NACA T 1357 For the mixture Mach number 2 2 i+ 2 a + o i+ 2 __ X_ 1_ xvl2 + X1 2 a12 (1 + m2V2 m 2 2 + i 1+ 2 a1 ( + 2m 0 1 + 2 av X1v2 X J1 1v+ 2 2 S2 aO 1 X(12b) Em "m22 +1 mOv2+ is valid. The mixture Mach number is therefore a function of the mass,. velocity, and Mach number ratios of the components. When the Mach number ratio of the components becomes unity, the ratio defined above of the air Mach number expression and of the mixture Mach number expres sion is equal to the mixing efficiency. 4. Discussion of special cases. The discussion of the limiting cases of the mixing efficiency may be extended to the remaining properties of the mixed jet and leads to remarkable characteristics. The special case (frequently occurring in practice) v2/vO = 0, that is, admixture gas at rest gives because of the disappearing velocity component parallel to the axis of the admixed gas directly F2 = o, since continuity and momentum theorem can be satisfied simultaneously only when mixing occurs, thus PxFxvx 0 PoFovo. One may now consider Fx as an arbitrarily selectable independent variable and assume it to be pre scribed since to every arbitrary Fx there pertains a F2 = m. From the equations (1) to (4) there follows the quantity (undetermined at first) F2v2 1 X( 1 0 i + 1 Fx  FOvO 2 a 2 TO TOFO / S+ T+ 1 (14) \ 2 o o and therewith directly, from the equations (9) and (11) and (12), the unknown quantities vx, px, Tx. NACA TM 1357 A still more restricted special case not infrequent in practice is v2/VO = 0 and T2/TO = 1. Due to the throughout equal pressures p, one then has also p2/PO = 1, and the undetermined quantity F2v2 becomes x 1 +L 1) + 1) (14a) and hence furthermore the unknowns from equations (9), (11), and (12) F2v2 x 1 T + X 1 0 FOVO ox T VO 1 + F2 v/v0 0 a 1 + F1V 2 V Tx In the mixed jet temperature and density are, therefore, completely different from the values of the gases (all equal) before the mixing; the jet mixing has a heating effect as would be shown by a schlierenoptical observation. A third special case results with v2 = v0 M = Vo(PoF0 + P2F2) J = v(PoFo + PF) E = v0 FO( + gcpTO) + P2F2 + gpT2 D = pOTO = p2T2 and with the aid of equations (9) to (12) vx = V0, the flow velocity remains the same after the mixing Fx = FO + F2, the flow cross sections add x = PF + 2F2 the densities mix in proportion to the masses FO + F2 Tx = P0 + 2F2 the temperatures mix in proportion to the PF0 + P2F2 enthalpies NACA TM 1357 Similarly, all other special cases of the gas jet mixture may be derived from section 2. 5. The constantpressure mixing chamber. The free jet mixing treated in sections 2 and 3 takes place in the same manner in a closed mixing chamber, if the walls of the chamber have the shape of the streamlines of the admixed gas indicated in figure 1. In this case, the mixing ratio m2/m0 also may be arbitrarily limited. Due to the pressure in the admixed gas remaining constant, the flow velocity, temperature, and density of that gas remain constant in the mixing chamber before the mixture is achieved. The individual, usually decreasing crosssectional areas F of the mixing chamber may therefore easily be calculated from the decreasing quantity of the admixed gas. At the point on the mixing chamber axis where the mixing cross section is determined by SM()E 1 2 gcpDJ(,) the mixingchamber cross section is, according to the continuity theorem F = F() + F2(x) F2() (15) one uses therein the designations of figure 3 and the symbols have the same significance as in section 2. The independent parameter F2(,) may be selected arbitrarily between FO and F2(x) and results in each case in a mixingchamber cross section F. Whereas thus the mixingchamber cross sections may be calculated simply and unequivocally, the actual meridian form, that is, the coordination of these cross sections to given points along the axis of the device, is determinable only on the basis of empirical experiences regarding the actual opening angle of the mixed jet, the meridian form of the mixed jet, the velocity distribution in it, the rate of the turbulence dissipation, etc. NACA TM 1357 Without knowledge of the results of such tests, one may consider that the velocity distribution in a conical jet with a total opening angle of about 100 to 140 will be homogeneous. Another important research problem concerning the mixing chamber arises in the use of very hot core jets as are given off for instance from rockets and where the very strong thermal dissociation, which may contain in latent form more than half of the energy supplied to the core jet, is reduced in the mixing with surrounding air and'thereby will, in addition, very greatly heat the mixed jet. The question how far the mixing can be accelerated by special guide vanes in the mixed jet would have to be clarified separately. As mentioned before, it was further presupposed in the present consideration that the kinetic energy first converted into vortex energy in the jet mixing is further converted, still within the mixing chamber, practically completely into heat. This process, essentially caused by internal friction, is greatly accelerated by the large dif ferences in velocity existing, the high viscosity of the hot combustion gases, and the chemical reactions taking place simultaneously. Thus, it appears justified to calculate as in the customary gas throttling; experimental confirmation, however, is still lacking. As the examples completely calculated later on show, this assump tion is unfavorable for lowpressure mixing chambers, of slight influence for highpressure mixing chambers which are heated by power plants of small inner efficiency, and favorable for highpressure mixing chambers heated by power plants of high internal efficiency. II. Thrust Increase of Jet Engines by Admixture of Air to the Exhaust Jet 6. Theory of thrust increase. One has Core thrust: PO = p0Fov0 P2F202 Total thrust: P = P4F4 v pFv2 p4F 4v pF2 Factor of thrust distribution:  PO PFov p2F20v2 0F0 0 2F20 10 NACA TM 1357 Of the characteristic parameters listed in figure 4, 14 are unknown F, v2' P2' T2 F3, p3, v3, P3' T3 F4', p4, v4' P4 T4 The parameter p2 is assumed to be known since it is, by design assump tions, a priori arbitrarily selectable within certain limits. Of flow equations, there are available: Zone F F2: Continuity and energy theorem, adiabatic equation and gas equation Zone F2 F3: See section 2; for the F2 there one has to put here F2 F20; use is made of equations (9), (10), (11), (12), and of the relation for constantpressure mixing p3 = P2 Zone F3 F: Continuity and energy theorem, adiabatic equation and gas equation, and the pressure reduction condition P4 = p Thus one has, for the 14 unknowns, an equal number of determining equations. The unknown quantities p4, v4, F4, v2 appearing in the thrust factor P/PO are to be calculated from the prescribed quanti ties p2, p, v, p, T, FO, PO, Vo, pO, TO, F20, F2' From the flow equations of the first zone there follow the relations 1 P2 (2) (16) 1 T2 = P2 (17) pgR\P2/ NACA TM 1357 V2= VZ+2gcpTrl(j) (18) F = F2 + gpT (19) where the symbols defined as parameters for equations (5) to (8) become SP v,2 where the symbols, defined as parameters for equations (5) to (8) become Mass sum: 1 M = P0FOF0 + p(F2 F20) () kgsec/m (5a) Momentum sum: Energy sum: E = p0Fovo2 + p(F2 20) ) 2 + 2gCpT + gcTo) + 1 X1" (F F20) v2 + 2gcpT + gpT) kgm/sec] Temperature density: D =2 kgsec2 o/m4 gR 6 [ (6a) (7a) (8a) 12 NACA TM 1357 and finally the directly used unknowns themselves P4 = gcp M2D15 X gsec2/m (20) 2 M 1 2 X1 ME XJ T4 = 2 p oK] (21) gcpM2 P2 1 2 X1 4 = + 2 1 X m/sec (22) M2 M 22 \P 1 F4 2 (P) [2 (3 gcpD F 2(M 1 P Therewith, one may at last write the desired factor of thrust distribution p F v2 pFv2 P 2 2 O Po0FO P2F20v2 X( 1 X1 2gcX P2 1 2 oFvov 4S L2 S P2X 2 22 + P 1 2 2 2 + 2gcpT 22\ IP\X pFO 2 p 20v 1 +  1 ) (24f *The NACA reviewer has pointed out that the quantity (p2/p)l/X in the denominator of this equation was erroneously inverted in the German text. NACA TM 1357 7. Discussion of the thrust increase. The thrust factor P/PO does not immediately signify the thrust increase of a jet engine by addition of the shroud under otherwise equal circumstances. Rather it describes, in the first place, the distribution of the thrust between the core jet and the shroud for the respective state of flight p, v. However, for this state of flight, the core jet without shroud, might have a thrust essentially different from the thrust it would have with a shroud. In practice, it will happen not infrequently that the corejet thrust without shroud is considerably smaller or even zero so that the actual thrust increase for a certain state of flight by addition of the shroud can be much larger than P/PO, even infinite. This case occurs for instance for pulse jet tubes and high flight velocities. On the other hand, the thrust of core jets might be reduced by addition of a shroud, as in rockets, although this effect will often remain negligibly small. In the last special case P/PO then actually signifies the thrust increase of only the core jet by addition of the shroud. The physical technical significance of equation (24) will become even clearer by discussion of a few special cases. (a) Special case v = 0, that is, static thrust. Equation (24) is specialized to the form V2g X 1 P0FO, 2 2 a0 )FoV 1 OFov 2o PF20T /2)X P2 TQ_/ P 2 P(F2 F20) gcpT px 1+X  P2 12 P _; 2 2 +l+j + 1 + 1 T0 _ 2p(F2 F20o) T(P Z1 P2 p (24a)* p = F0v i *NACA reviewer s correction: The erroneous term in the denominator of the German text was changed to the correct form PoFoo 2 a0 P PO 1X + p  1X (P2) X P  # 14 NACA TM 1357 With p2/P > 1 the solution becomes imaginary; this case, as is immediately clear, is physically not realizable. With p2/p = 1 one obtains P/PO = 1; this case is the same as in the jet mixing with free surrounding air and an increase in thrust does not occur. With p2/p < 1, real solutions with P/PO > 1 are possible as long as the sum of the three terms under the large square root sign remains positive. Since the expression 1X~ L (p/p) X is always negative, the second term will always be negative and the first, too, becomes negative when the enthalpy of the core jet is large compared to its kinetic energy, thus, the Mach number vo/a0 is small. One understands immediately, by means of the following consideration, that, for the static case, even for P2 < p, cases may exist where the flow is physically not realizable. From the elementary gas dynamic relation X X p 1 Xl 2 1 P2 2gcpT2) 2 there follows that the pressure ratio for the flow either in the static or dynamic case depends only on the Mach number. If there becomes for instance v/a <2 v2/a2, that is, if, in the mixing process, the enthalpy increases more than the kinetic energy, thus decreasing the Mach number, the higher external pressure can no longer be attained at all after the constantpressure mixing. This case will occur in practice quite fre quently in the mixing of hot slow combustion gas jets with cold air, particularly when, in addition, reverse dissociation occurs. Generally, the static thrust increase will be zero, thus P = PO, when X 2 1 /2 2 P 2ME J P P  or v2 gcpT2 2 P2 \ 2ME 2 ME 2 NACA TM 1357 15 that is, one will have to work sometimes with very high mixing chamber inflow velocities v2 in order to attain high mixing efficiencies if the static thrust value is important. The v2 optimum for static thrust may be immediately determined for any given p2/p from the equation for P/PO0 With the aid of equation (12a), one finds generally that, for an efficiency BD of the enddiffuser different from unity the undisturbed external pressure p is again attainable when 2 a2 VO I X12 vo X 1 v2 2 ao 1 v2r X I 1 W\o0+ m2  + 1 mg 2 2 a2 X  2 2V a2 (12c) (b) Special case v0 = 0, that is, momentum less heat supply, with the static thrust becoming immediately zero, as known from the standard ramjet engine. The effect of the fuel injection as a core jet will be mostly negligible in this case; that is, the core thrust PO becomes zero. For the pure ramjet thrust, there then follows with equation (5a): equation (6a): M = MO + pF2(2 2 + 2gCpT 1 J = pF2 )f2 + 2gcpT '~f 2 2 m2V2 2 2 1 j + 2 1 2 X 1 V2 0O~ 2) NACA TM 1357 1 equation (7a):E = MOgH + pF2 P21X \p/ + gcpT) when M0 is the mass of the added fuel and H its thermal value in kgm/kg, from equation (24) X1 + J21\ X rP2) With the known assumptions F2/F.o, MO0, this formula may be transformed into the simple approximation formula for the thrust of the ram jet given before P = pFv2 1) It differs, in addition, by the constant values of the specific heat from the exact calculation of the ramjet engine according to ZWBUM 3509 (NACA TM 1106). (c) Special case T = TO without further peculiarities. (d) *Special case (F2 F20) = 0. With M = p0FOvO, J = PO'A , E = POFOVO + gcpTO) X1 P= 1 P2)X *ACA reviewer's footnote: The symbol F20 was misprinted in the German as F2. NACA TM 1357  becomes PO IFO 2g 2 T ] 2gc T X POF + L. F + 2 P POO P 2 X1 ( 1 pOFOv2 PF2o V2 1 + 2 1 ~ X (PX Thus after eliminating the admixed air, in general, a thrust increase remains, which follows from the formulation, since the thrust PO of the core jet had been referred to the moderate flow velocity within the shroud and additional useful pressure drops originate due to the higher flight velocity. Only with p = pO one finally has P/PO = 1, for instance, in case of rockets as core jets, when with F20 = 0 also p/p2 = 1. (e) Dependence of the P/PO on F2. In equation (24), in the expression X1~ X1 X X V2ME 1 (1 + J2P ( the terms quadratic in F2 become equal to the second numerator term pFv2, and thus disappear for very large F2 so that P/PO then becomes, for very small v/vO, approximately proportional to F). It therefore becomes proportional to the square root of the admixed mass whereas otherwise P/PO, independently of F2 and the admixed mass, tends toward a fixed limiting value. This behavior is known from the elementary theory of the Melot device as well as of the ramjet device. Generally, one may, by augmenting F2, increase the thrust not without limit but only up to a fixed limiting value. Only at the flight speed zero it increases theoretically with /f without limit. NACA TM 1357 (f) Dependence of the P/PO on p/p2. For p/p2 = 1 thrust increase does not occur in any case; equation (24) yields P/PO = 1. For P/P2 < 1 there originate throughout real values for P/PO which are larger than unity, whatever may be the flight speed v, ratios of masses, velocities, or temperatures. Under these circumstances, there exists therefore no range of flight speed or of the other operational conditions where the shroud would lose its thrustimproving effect. For P/P2 > 1 the situation changes completely. Here, cases with P/Po 1 as well as cases with P/P0 = 0 are possible. The term chiefly responsible for the thrust 2 X1 22 ME p X P4F4v4 =M + M2 1 contains under the square root sign two energy terms: the total kinetic energy J2/MP present at the end of the mixing chamber F3 and the (ME J2) addition kinetic energy 2 originating during the process M2 of expansion in the discharge nozzle between F3 and F4 from the total enthalpy present in F3 when p/P2 < 1, or kinetic energy recon verted into enthalpy when p/p2 > 1; thus, compression flow exists as is here to be considered. When p/p2 is larger than unity but still compression flow so little kinetic energy ME J2 2 1 M2 so small that in the X1 p  p\2/ is consumed that there still remains (PF1v 2 pFv2) (P0Fv2 p2Fv2) > 1 we obtain again a positive thrust increase P/PO > 1. NACA TM 1357 When p/p2 has exactly the magnitude that p +J gcpT 1 X1 X x1 S(P X] + 2(P) X 2l/ 2gcPT )X = 2E + j2 pF2v2 + Vr21 ( SoFov PFo2 2gcT v2 ( X 2 \ 0 OY0 PF20 + 2 P P the thrust increase disappears, that is, P/PO = 1. If p/P2 further increases, the left side of the above equation becomes smaller than the right side; due to the shroud, there appears immediately a loss in thrust, that is, P/PO < 1. If P/P2 is augmented still further, for instance, until V X 1 X 1 (P2 X P2 P2p/ v2p/ is valid, the thrust of the core jet is exactly used up by the processes in the shroud; there remains P/PO = 0 and the total thrust of the arrangement becomes zero. All these cases are definitely realizable design, and have probably been actually realized as proved by the numerous failures of related tests. This holds true also for the case of still larger p/p2, for instance for the case that exactly all kinetic energy J2/M2 present in F3 is used up during the recompression to the external pressure which is achieved when J2 X1 P 1 + M2 F2 2(ME 1 J2) NACA TM 1357 In spite of all the energy absorption, the entire power plant behaves the same as a pitot tube, has therefore on the whole, only drag. From these considerations, there results that the jet mixing must take place, if possible, at pressures exceeding the static pressure of the outer air so that expansion to external pressure occurs after the mixing, whereby a compression to external pressure is not necessary. That is, fundamentally, melot type lowpressure mixing nozzles are less favorable or quite useless, and ramjet type highpressure mixing nozzles more favorable. Since lowpressure shrouds therefore promise advantages only for the static case and at very low speeds of motion, we shall deal with them here only by comparison or for ground setups. (g) Dependence of the P/PO on the inner efficiency of the core jet. The ratio between the kinetic jet energy and the total jet energy of the different jet engines, characterized by the jet Mach number varies to an extreme extent. For rockets, this ratio approaches sometimes 50 percent, for turbo jets it lies around 10 percent, and for pulse jet tubes it drops to a few percent. For the basic application of momentum heating to the exhaust gases of these power plants, it is therefore worth knowing how far the attainable thrust increase P/PO depends on these inner effi ciencies. One may write equation (24) for this purpose in the form XX1 X1 1 J 1 (P+ pF2v + 1  P22 F2 )2 X v ) P O X1 1 1 2gcT p9) X (P2 X J pF2v2 + 2gc~T  In many cases which are important in practice, the ratio P/Po is determined predominantly by those terms which contain M, E, and J, that is, it will increase with NACA TM 1557 Large values of ME/J2, that is, large ratios between total and kinetic energy or small jet Mach numbers according to equation (12a), however, signify nothing else but low thermal efficiencies. For otherwise equal conditions, one may therefore expect higher thrust improvements with core jets of low inner efficiency, that is, small jet Mach number (for instance, pulse jet tubes) than for instance with turbojets or rockets. Since the thrust of the shroud is determined by the thermal energy loss, and the core thrust by the "useful" kinetic energies, the above statement is quite plausible. The influence of the thermal efficiency frequently is greater than the influence, large in itself, of the mixingchamber pressure. In practice, the values of ME/J2 lie approximately between 10 and 40 whereas the values of X1 1 (p/p2)X (at the high subsonic speeds treated here first) vary between 0.05 and 0.1, and the second term under the square root sign always remains close to unity. Thus, the differences in the thrust factors may become larger due to the selection of different core jets than due to different mixing chamber pressures. III. Examples 8. Air admixing shroud heated by rocket exhaust gases. Let a rocket be prescribed with the jet characteristics PO = 0.02 kgsec2/m4, V0 = 2000 m/sec, FO = 0.04 m2, TO = 17500 K; hence P0FOv0 = 1.6 kgsec/m, pOFv2 = 3200 kg, p0FO0v/2 + gepTo) = 6,720,000 kgm/sec (= 90000 hp). Let the dissociation heat still bound at the mouth of the rocket be 3,620,000 kgm/sec, corresponding to a lower mixturethermal value of the rocket propellants of 1540 kcal/kg. The jet Mach number is VO/aO = 2.63, the expansion ratio of the rocket Pi/PO = 23. Favorable arrangements of thrust increasing airadmixing shrouds at flight speeds 0 and 800 km/h are to be indicated. For v = 0 km/h naturally only a lowpressure mixing chamber comes into question. Since drag of the shroud itself here matters little and NACA TM 1557 the static thrust is known to increase without limit with the admixed masses, an optimum magnitude for this case cannot be stated as is also shown by the differentiation of equation (24) with respect to F2. If one chooses arbitrarily a diameter of 2m for the F2 cross section, and the state of the outer air to correspond to the standard atmosphere, and does not at first assume a value for the mixing chamber pressure P2 which is likewise arbitrarily selectable, there follow for F20 = 0 and v = 0, the values for M, J, and E as functions of p/p2 from the equations (5a) to (7a), and the corresponding thrust coefficient from equation (24). It is known that P/PO = 1 for p2/p = 1, likewise for small values of p2/p. Hence, there must necessarily exist an optimum value for p2/p with respect to thrust increase which may be determined by differentiation of equation (24) with respect to P2/P to be about 0.7 as also shown by the representation of this equation in figure 5 (with out reverse dissociation). P/PO there becomes approximately 2.5, that is, the total thrust is increased by the shrouding from 3.2 to 7.5 tons. If the rocket nozzle is adjusted to the new pressure ratios, P/Po becomes 33, the exhaust velocity increases to 2080 m/s, and there follows a small additional core jet thrust of 0.15 tons with which the entire calculation would have to be repeated. If the end diffuser of the shroud has an efficiency different from unity, one attains only a slightly higher mixing chamber pressure P2 which, according to fig ure 5, has only a small effect on P/PO because of the flat thrust maximum. Corresponding to the pressure drop of 0.7 p there is an inflow velocity of 238 m/s and an inflow Mach number of 0.72. The mixing efficiency becomes about 55 percent and the mixture Mach number, according to equation (12a), remains sufficiently far above the inflow Mach number to insure the reincrease to the external pressure. With the aid of equations (16) to (23), the remaining desired quantities may be calculated. The dimensions of this shroud are represented in figure 6 and are reminiscent of windtunnel proportions. Possibilities of its application for very slow aircraft, launching devices, or water and land vehicles seem therefore dubious. On the other hand, these lowpressure shrouds are interesting for ground test setups, for instance, for investigation of rocket engines operating at low nozzle opening pressures or of entire rocket devices with high approach flow velocities and for subsonic and supersonic wind tunnels of very large test cross section with rocket propulsion, particularly by highpressure lowtemperature (for instance, water vaper) rockets. NACA TM 1557 In the calculation, so far, the possibility was disregarded that the considerable dissociation of the rocket jet does not stop during the relatively lengthy mixing procedure, in spite of the very consider able cooling of the combustion gases, but does reassociate so that the dissociation energy mentioned at the beginning additionally heats the mixture. M and J remain, of course, unchanged, whereas E increases correspondingly; hence, the thrust coefficient also varies according to equation (24) in the manner represented in figure 5 (with reverse dissociation). The optimum of the mixing chamber pressure now lies at p2/P = 0.9 and reduces the thrust quotient optimum there to P/PO = 1.6; thus, the total thrust then increases from 3.2 tons to only 5.1 tons. The further heat supply by reverse dissociation and possibly afterburning of the combustion gases has therefore, in the lowpressure mixing chambers, a very deteriorating effect on the thrust improvement. In this case, minimizing the vortex conversion into heat would therefore be advantageous. At a flight speed above zero, one finds, for the shroud, a region with two optimum mixing chamber pressures, one below and one above the external pressure of the air at rest. The first, with increasing v, soon becomes insignificant so that one is concerned only with the high pressure mixing chambers at all flight speeds of practical interest. The optimum thrust increases become largest in the static case or for very low flight speeds (region of good Melot effect), pass through a region of very moderate values at medium subsonic speeds, and finally, approaching sonic velocity (region of good Lorin effect), increase again to higher values which, however, remain far behind the high initial values. Only in the supersonic region does the highpressure shroud become equivalent to the lowpressure shroud in the static case. In flight, at v = 800 km/h, the air drag W of the shroud must be taken into consideration in determining the optimum conditions. One can again express the thrust increase (P W)/PO as a function of the quantities to be determined, F2, and p2/p, and find their optimum values. For example, for the conditions represented in figure 7 it is found to be P2/P = 1.2, and a thrust increase of about 20 percent is found which corresponds to a thrust coefficient of the shroud of approxi mately c, = 0.2. Thus with the assumptions made here (full reconver sion of the dissociation and turbulence into heat) one finds this thrust increase is 30 percent of the maximum thrust possible from a prescribed ramjet shroud. NACA TM 1357 Whereas, in the lowpressure mixing, one could observe expansions of the entrance and exit cross section, the highpressure mixing chamber showed the known narrowing of these two cross sections. On the whole, one will conclude from the moderate thrust increases of this example that the momentum heating of the ramjet engine by rocket exhaust gases probably will have technical significance only for special conditions. 9. Ramjet tube shroud heated by exhaust gases of turbojet or pulsejet tubes. Let the necessary characteristic parameters of the entering and leaving gas jet of a jet engine of moderate jet Mach num ber at the flight speed of 200 km/h be prescribed. The arrangement of a ramjet shroud with maximum thrust is desired; the flow velocity at inlet and outlet of the power plant is to remain unchanged, and the flight speed is to be v = 800 km/h. One will choose an arrangement according to figure 4 where the core jet operates within the shroud in the adiabatically decelerated air in a medium of the same surrounding velocity as in the initial state (200 km/h) but with increased values of pressure, density, and temperature. Due to these changes alone, the thrust increases by 15 percent, the fuel consumption by 17 percent. With consideration of the drag of the shroud, one may again represent, with the aid of equations (Sa) to (7a) and (24), the thrust coefficient as a function of the cross section area F2 of the shroud, and one obtains a flat maximum for instance in the neighborhood of F2/FO = 2.3 of (P W)/PO = 1.2. The total thrust increases due to the shrouding from the standard value at 200 km/h to a value by 20 per cent higher at 800 km/h without noticeable change in the fuel consump tion, thus, with a multiple of the total efficiency. The practical importance of this arrangement lies not so much in the moderate thrust increase in itself as in the fact that the increased thrust may still be expected for a flight speed at which the operation of a simple pulse jet tube is altogether questionable. The shroud simulates at 800 km/h flight speed the conditions of 200 km/h which are more favorable for the core jet, particularly the lower diffuser entrance velocity. The internalpressure level of the core jet increased by almost the whole freestream impact pressure and therewith provided the compensation of the high additional opening pressures on the air control valve flaps, and improved the air supply from the rear for the frontal ram effect of the air. Whereas the efficiency of the jet tube mentioned as an example amounted to about 1.5 percent for v = 200 km/h, it can be increased by the shrouding, for v = 800 km/h, to 8 percent so that it comes quite close to the known efficiencies of simple turbojets and the range of flying missiles thus equipped may become very considerable. NACA TM 1357 Figure 8 shows the approximate thrust variation against the flight velocity of a pulse jet tube without shroud, with the previously men tioned free shroud, and, finally, with a shroud attached to an existing airplane fuselage or wing in such a manner that no additional drags are caused by it. The shrouding causes the thrust variation of the pulse jet tube to become similar to that of a turbojet. This circumstance which is very favorable to the pulse jet tube shows simultaneously that the effect of the shrouding on a turbojet remains by far smaller since the standard equipment of the latter anticipates a large part of the effects utilized by the shrouding. 10. Air admixing shroud heated by exhaust gases of piston power plants. A 000hp piston power plant yields for 800 km/h flight speed about 950 kg propeller thrust and approximately 3.8 kg/sec exhaust gases of 6000 C temperature. The freestream impact pressure near the ground is q = 3080 kg/m2. If one wanted to mix this quantity of exhaust, gas with an air quantity as large as possible, for instance 100 times as large, the required Fl would be Fl = 1.39m2, and the temperature after the mixing would be T3 = (873 + 100(288 + v2/2000))/101 = 3180 K in order to make the shroud thrust P become P =2qF(iT3/T2 1) = 81.3 kg that is, 8.5 percent of the propeller thrust. The natural drag of the highpressure shroud was disregarded as well as that of the entire piston power plant; a possible direct thrust of the exhaust gas jet remains practically unaffected by the additional equipment. For lower flight speeds, this thrust ratio deteriorates, for higher ones, it improves. IV. Summary The admixing of surrounding air to exhaust gas jets of power plants may have a thrustincreasing effect, if it takes place at a pressure other than the surrounding pressure. NACA TM 1557 Admixing in lowpressure mixing chambers, it is true, is limited to such moderate speeds of motion that the flight impact pressure does not yet have too much of a disturbance effect on the low pressure. In these cases, there result very considerable thrust increases which make the lowpressure mixing interesting for fixed installations in subsonic and supersonic wind tunnels, altitude test chambers, and water or land vehicles. Admixing in highpressure chambers, in contrast, becomes the more effective the higher the speed of motion so that this type is suitable especially for aircraft in the high subsonic or in the supersonic range. The relative thrust increase, under otherwise equal circumstances, is the greater, the smaller the Mach number of the original exhaust gas jet. This increase rises therefore sharply in the following order: rocket, turbojet power plant, pulse jet power plant. For the last type of power plant, the operation of which is sensi tive against high approach flow velocities and low air densities, one may in highpressure mixing chambers attain an improvement in climate by which the flight speed range and flight altitude range for pulse jet power plants are extended. One may expect from this type of power plant, under these circum stances and at high flight speeds, total efficiencies which approach those of the turbojet power plants. The relative thrust increase for lowpressure mixing chambers is greatest in the static case; with increasing speed of motion it drops very rapidly and disappears at fractions of the Mach number 1. The relative thrust increase for highpressure mixing chambers is zero in the static case, rises very rapidly with increasing speed of motion, and attains maximum values in the supersonic range. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1557 27 References 1. Melot, H. F.: French patents 522163, 1919; 523427, 1920; 571863, 1922. 2. Busemann, A.: Schriften der Deutschen Akademie fur Luftfahrtforschung, Heft 1071/43. Berlin 1943. 28 NACA TM 1557 F2 2 V2 " P2  T2 I 0 I "H Figure 1. Free mixed jet surrounding air in motion. NACA TM 1557 U 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2. Propulsive mixing efficiency of constantpressure mixing. NACA TM 1557 Mixing chamber Figure 3. Constantpressure mixing chamber. NACA TM 1557 I I A S Diffuser F Fe P P2 v V2 P PT T T2 Core jet ch (m2) Fo (kg/m2) Po (m/s) vo (kgsec 2/m4) P0 (K) To Mixing amber Nozzle F3 F4 P3 P4 V3 V4 P3 P4 T3 T4 Figure 4. Ramjet power plant with momentum heating. NACA TM 1557 P/Po Figure 5. Dependence of the thrust quotient P/PO of a lowpressure mixing chamber in the static case on the mixing chamber pressure with and with out reverse dissociation of the rocket jet at 17500 K. 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