PULSATION AND VIBRATION
FLAME FRONT STABILITY IN LIQUID FUEL DROPLET COMBUSTION C. C. MIESSE Aeroief-General Corp., Azusa, Calif.
H E previous analysis on the oscillation of the flame front between two unlike droplets in a liquid bipropellant system (3) has been extended to the case of a single fuel droplet burning in a n oxidizer atmosphere. By determining the radial concentration distribution of the oxidizing vapor, applying Fick’s first law of diffusion, and following the method of Burke and Schumann (Z), the mass flow rate of the oxidizing vapor to the spherical flame front varies inversely as the radius of the fuel droplet. Under these conditions, the steady-state flame front position varies directly with the radius of the fuel droplet, and stability is ensured if the flame front radius is at least three halves of the droplet radius. This condition can be expressed explicitly in terms of the stoichiometric mixture ratio and the physical properties of the two constituents, exclusive of the size of the fuel droplet. If, however, it is assumed that the mass flow rate of the oxidizing vapor is independent of the radius of the fuel droplet, then the ratio of the steady-state flame front radius to the droplet radius is a function of the droplet radius, and the stability criterion involves the radius of the fuel droplet. Although this assumption was made without theoretical confirmation, the experimental observations of Burgoyne and Cohen ( 1 )indicate that the ratio of flame front to droplet radii varies in a manner similar to that derived under the assumption of constant oxidizer flow rate. Under this assumption, stability is increased as the size of the droplet decreases. The following conclusions are common to both assumptions: 1. Stability is increased by an increase in the vaporization ratio of the fue1 droplet. 2. Stability increases with an increase in the stoichiometric mixture ratio. 3. Stability is increased as the diffusion rate of the oxidizer flow is decreased. literature Cited (1) Burgoyne, J. H., and Cohen, L., Proc. Rov. SOC.(London), A225,
375 (1954).
(2) Burke, S.P., and Schumann, T. E. W., IND. ENG.CHEM.,20,9981004 (1928). (3) Miesse, C. C., Fifth Symposium (International) on Combus-
length and 0.5 inch in diameter, were held in a vertical position. The lower end of the sample was placed in the distilled water of the transducer and the upper end was placed in the heat treating furnace. The ultrasonic frequencies used were 400 and 1000 kilocycles. The ultrasonic treated samples of 0.07% carbon hypoeutectoid steel had finer grain size and greater hardness than the reference samples which had undergone the same heat treating cycle but with no ultrasonic treatment. When either the intensity or the frequency of the ultrasonic energy was increased, the grain size was further decreased. The grain size of the ultrasonic treated samples of 1.05% carbon hypereutectoid steel w,s coarser than that of the reference sample. The samples treated a t the louest intensity had the coarsest grain size while higher intensities produced grain size which approached the grain size of the nontreated samples. Samples treated a t 1000 kilocycles showed a grain size slightly smaller than the samples treated at 400 kilocycles. Hardness values showed a relationship similar to that of the grain size. The lamellar layers of ferrite and iron carbide making up the pearlite crystals in the 1.05% carbon steel were definitely thicker in the ultrasonic treated samples than in the untreated ones.
SONIC VELOCITY MEASUREMENTS IN STUDY OF LIQUID AND LIQUID SOLUTION PROPERTIES F. C. COLLINS, M. H. NAVIDI, AND L. P. FRIEDMAN Departmenf o f Chemisfry, Polytechnic Institute of Brooklyn, Brooklyn I , N. Y.
S
ONIC velocity is a convenient measurement for the study of liquid properties in addition to the customary measurements of specific volume, vapor pressure, specific heat, heat of vaporization, and dielectric constant. The adiabatic compressibility so obtained is related to other thermodynamic variables and is useful for correlations among liquids and for interpreting the microscopic structure of the given liquids. By use of the free volume theory, the following equation can be obtained which gives the ratio of the incompressible molar volume V oto the actual volume V without additional assumptions ( 3 )
tion, Reinhold, New York, 1955.
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ULTRASONIC EFFECT O N POLYMORPHIC TRANSFORMATION OF STEEL H. V. FAIRBANKS AND F. J. DEWEZ, JR.’ Chemical hgineering Dept., Wesf Virginio University, Morgantown, W. Va.
In Equation 1 us is the measured sonic velocity and the other symbols have their customary meaning. Data of this kind have been used in connection with a free volume theory of viscosity and diffusion for simple liquids (3). The viscosity term vCwhich arises from the transport of momentum across the bodies of molecules upon collision is 2 u(MRT)1’2 5 T ” V [ l - (VO/V)1’3]
Ilc = - _ _
T
HE purpose of this investigation was to determine the effects of ultrasonic energy on 0.07 and 1.05% carbon steels during polymorphic transformation, or annealing. The steel samples used were slowly cooled during the transformation in which the iron changed from a face-centered cubic structure to a hodycentered cubic structure. The ultrasonic energy was produced b y means of a bowlshaped crystal of barium titanate and was coupled to the steel samples through distilled water. The steel samples, 8 inches in 1
Present address, U. S. Steel Corp., Pittsburgh, Pa.
June 1955
where u is the molecular collision diameter. Equation 2 does not take account of attractive intermolecular interactions but, nevertheless, yields numerical values for the viscosity coefficient of the order of one third of the experimental value?. I n the case of binary solutions, deviations from regular behavior are due to differences in intermolecular forces between unlike species, variation in number of nearest neighbors, and nonrandom mixing. Sonic velocity provides an additional experimental variable for the evaluation of these several effects. Sonic
INDUSTRIAL AND ENGINEERING CHEMISTRY
1181
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 part 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
