ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT velocity a t 3 megacycles per second has been measured a t various mole fractions in 16 binaries of varying degrees of nonideality in studies conducted in this laboratory. Earlier experimental studies of this kind have been reviewed by Wada (6). The free volume theory is useful for interpreting and correlating sonic velocity data for liquid solutions. The application of the simple free volume theory to solutions has been well presented by Prigogine and Bellemans (4). For the consideration of sonic velocity data, the theory is extended as f o l l o ~ s . The internal pressure, Pi,is given by the free volume equation of state
James (S), and Furth (I) have been used for solids in recent years. They are described and compared with data of Bridgman and with computations made by the Fermi-Thomas method. These equations are used with the hydrodynamic theory to obtain expressions for the pressure, temperature, and entropy increase in various metals as a result of shock wave passage through the solids. For the Pack-Evans-James equation of state, which appears to be adequate for interpolating between the Bridgman data and the Fermi-Thomas computations, the difference between Hugoniot and adiabatic pressure to fourth order in the compreasion is
(3) where V and V Onow refer to the solution molar volume and incompressible molar volume, respectively. The ratio ( V0/V)1/3 is evaluated by means of Equation 1. By the use of the partial molar concept
v~,
where N A and N B are the mole fractions and VA, and V ~ are B the partial molar volumes and incompressible molar volumes of the components A and B, respectively. The two right-hand terms in Equation 4 may be individually determined by the wellknown slope-intercept method from a plot of Pi from Equation 3 versus mole fraction. The values of v&41’3 and V O B ~which /~ are thus obtained vary slightly with mole fraction in the direction predicted by Prigogine and Bellemans (4). The excess internal pressure for the nonideal solutions does not correlate directly with the heat of mixing. The relation between the behavior of the excess internal pressure and the heat of mixing may be useful in future .studies of the molecular internal degrees of freedom in solution, as implicitly suggested by Bondi (11.
Literaiure Cited (1) Bondi, A., J . Phys. Chem., 58, 929 (1954). (2) Collins, F. C., and Navidi, M. H., J . Chem. Phys., 22, 1254 (1954). (3) Collins, F. C., and Raffel, H., Ibid., 22, 1728 (1954). ngogine, I., and Bellemans, A., Trans. Faraday Soc., 49, 80 (4) (1953). (5) Wada, Y . , J . Phys. SOC.J a p a n , 4, 280 (1949). Supported in p a r t by Wright Air Development Center of U. 9. Air Force under Contract No. AF33 (616b373.
ENTROPIC EQUATIONS OF STATE AND THEIR APPLICATION TO SHOCK WAVE PHENOMENA GEORGE E. DUVALL AND BRUNO J. ZWOLlNSKl Poulfer laborafories, Stanford Research Institute, Stanford, Calif.
T
H E propagation of a shock wave through an inviscid fluid is described by a set of jump conditions, representing conservation of mass, momentum, and energy, which are the same for all mediums, and by an equation of state and the specific heat characteristic of the particular medium. If P P / d V z is positive, only compressive shocks can exist and the entropy increases discontinuously in the shock transition The hydrodynamic theory of shock waves in solids is based on the assumption that the shear modulus is completely negligible a t very high pressures and that the transition from the undisturbed to the shocked state occurs discontinuously a t the shock front. The region behind the shock front is then treated as an ideal fluid with a hydrostatic equation of state. Three equations of state that bear the names of Murnaghan ( 2 ) , Pack-Evans1182
The increase in entropy acrose the shock front to the same approximation is
where a, p are parameters of the Pack-Evans-James equation, VO = l / p o is specific volume a t room temperature, To, and zero pressure, C, is specific heat, g( Po)= ( ~ P / ~ T and ) T ~p ,is density behind the shock front.
Liieraiure Cited (1) Furth, R., Proc. Roy. Boc. (London), A183, 87 (1944). (2) Murnaghan, F. D., “Finite Deformation of an Elastic Solid,” Wiley, New York, 1951. (3) Pack, D. C., Evans, W. M., and James, H. J., Proc. Phys. Soc., 60, 1 (1948).
A N INSTRUMENT TO STUDY RELAXATION RATES BEHIND SHOCK WAVES E. L. RESLER, JR., AND M. SCHEIBE lnsfitute for Fluid Dynamics and Applied Mathemafics, University of Maryland College Park, M d .
A
N INSTRUMENT is described which combines the schlieren
technique, a photomultiplier tube, and an oscilloscope to measure the density distribution behind shock waves (in gases) produced in a shock tube. The use of shock waves and of optical techniques to study the chemical kinetics of reactions has been reported (1, 2 ) . This instrument however, is simpler in many respects than those previously used and is capable of better space resolution and higher sensitivity while still recording the same information. Basically the instrument functions in the following manner. A photomultiplier tube intercepts all the light clearing the knife edge of a schlieren system, the light beam of which passes through a section of the shock tube. A property of the schlieren system is that light will escape the knife edge in an amount proportional to the density derivative a t each point in the field of the light beam. Since the photomultiplier tube gathers all the light passing the knife edge it integrates the density derivative over the light beam or puts out a signal proportional to the density difference a t the edges of the light beam. Thus the system has the sensitivity of a schlieren and photomultiplier combination and the space resolution is determined not by the beam width but by how well the edges of the light beam are defined or to what degree the light in the beam is parallel. Therefore, if in the shock tube a shock wave with a relaxation zone behind i t characterized by a density distribution passes through the light beam of the schlieren system and if the length of the relaxation zone behind the shock wave is less than the width of the schlieren light beam, then the recorded signal from the photomultiplier tube will be the density
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47,No. 6
PULSATION AND VIBRATION distribution behind the shock wave (assuming the index of refraction of the gases do not change in the transition zone.) as the shock wave plus transition zone enter the light beam. By measuring these density distributions the way the gas or gases relax t o equilibrium after the enthalpy of the gas is increased suddenly a calculable amount by a shock wave can be determined. The theoretical aspects of the instrument and its predicted performance were verified experimentally by measuring vibrational heat capacity relaxation times behind shock waves in oarbor! dioxide containing water vapor. The instrument in these tests demonstrated a sensitivity sufficient to record a change in atmospheric density of 0.5% over I-mm. distance and a space resolution of the density in the shock tube of 0.1 mm. corresponding to times of the order of 0.1 microsecond. *
Literature Cited (1) Carrington, Tucker, and Davidson, Sorman, J . Phys. Chem., 57,
418 (1953).
(2) Smiley; E.
E”.,
Winkler, E. H., and Slawsky, Z. I., J . Chem. Phys., 20, No. 5, 923 (1952).
This research was partially supported by the United States Air Force through the Office of Scientific Research of the Air Research and Development
Command.
ULTRASONIC UNMlXlNG OF ISOTOPIC SOLUTIONS S. G. BANKOFF’ AND
R. N.
LYON
shown that the steady-state composition gradient is, a t most, very small. Hence, for practically all types of ultrasonic waves, no appreciable steady-state separation can be achieved in the gaseous state. This statement applies specifically to isotopic mixtures and becomes less valid for mixtures of widely differing molecular weight. It is also possible that under conditions where the Chapman and Cowling assumptions of the continuity of the hydrodynamic medium break down, some separation might be achieved. Debye ( 2 ) showed that a potential wave, due to partial unmixing, should exist if an electrolytic solution is irradiated with a traveling ultrasonic wave. This effect was confirmed experimentally ( 5 )with a standing wave. I n this paper the magnitude of the unmixing associated with the Debye effect is shown to tie only about mole fraction at 100 megacycles and 0.1 watt per sq. em. This is true also for nonionic solutions. The unmixing is about the same for a standing wave as for a traveling wave, although in the former case the potential wave is a standing wave 90” out of phase from the velocity wave. No treatment has been found in the literature of the separation to be expected on passing either asymmetrical or damped sine waves through a liquid mixture. However, as Debye points out, the frictional coefficients are far larger than the dynamical coefficients in liquid systems. Hence, it is not probable that appreciable separations could be reached a t present ultrasonic frequencies with either distorted or damped waves. Despite the intense accelerative effects of ultrasonic radiation, it is shown that negligible steady-state separation can be expected by passing either symmetrical or distorted sound waves, standing waves or damped waves through gaseous, and probably also liquid mixtures of isotopic constituents.
Oak Ridge National laboratory, Oak Ridge, Tenn.
T
HE high local accelerations and the multiple stage nature of
ultrasonic radiation apparently make it attractive for separations based on small differences in mass. Despite this apparent attractiveness, several authors have reported negative results from theoretical and experimental investigations of this possibility. However, these analyses deal with the simpler cases; and some instances of relatively large unmixing have recently been reported (3, 4). It seemed advisable, therefore, to institute a more comprehensive analysis, with special reference to isotopic separations. A generalized theory for gaseous isotope separation is developed, based on integrating the binary diffusion equation ( 1 ) over one period a t cyclical steady state. This yields
Literature Cited
Chapman, S., and Cowling, T. G., “Mathematical Theory of Non-Uniform Gases,” Macmillan, New York, 1939. Debye, P., J . Chem. Phys., 1, 13 (1933). Frei, H., and Schiffer, AT., Phys. Rev.,71, 555 (1947). (4) Passau, P., Ann. SOC.Sci. Bruzelles, 62, Ser. I, 40 (1948). (5) Yeager, E., Bugosh, J., Hovorka, F., and McCarthy, J., J. Chem. Phys., 17, 411 (1949).
COMBUSTION OSCILLATIONS IN DUCTED BURNERS JOHN C. TRUMAN Aeronautical Engineering Dept., USAF lnsfitute of Technology, Wright-Patterson Air Force Base, Ohio
ROGER T. NEWTON
x
Small Aircraft Engine Dept., General Electric Co., Lynn, Mass.
where y is the mole fraction, z the distance along the column, t the time, X the wave length, c the velocity of sound, m the mass, p the pressure, k~ the thermal diffusion ratio, and T the temperature. These terms represent, respectively, the driving forces for diffusion due to concentration, pressure, and temperature gradients. Assuming a perfect gas adiabat, the thermal diffusion term vanishes. For isotopic mixtures, the coefficient of the pressure gradient is very nearly constant, and hence the pressure gradient term vanishes, or nearly so. Hence, the steady-state concentration gradient is zero, irrespective of wave form, for an undamped wave. This is to be expected from thermodynamic considerations. The analysis is somewhat more difficult for damped traveling waves, but b y a method of approximate series solution, it is 1
Permanent address, Rose Polytechnic Institute, Terre Haute, Ind.
A
PROBLEM of increasing importance in t,he development of modern aircraft propulsion systems is that of combustion oscillations or combustion instability. These terms refer to periodic, large amplitude variations in pressure which are maintained in some manner by the combustion process. Such variations usually occur in the audiofrequency range. Their effects include changes in the chemical and thermodynamic processes of combustion, and hence in burner performance, and structural damage to burner components resulting from high amplitude pressure oscillations and from locally increased heat transfer rates. Most cases of combustion instability which have been reported in the literature fall into one of three classes: ( 1 ) oscillations associated with failure of the flame to stabilize on a flame holder; (2) oscillations depending on the existence of a time lag between the injection of propellants into the burner, and their transformation to high temperature gases; and (3) oscillations, initiated or
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1955
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