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Sound propagation in polydispersed air fogs
J. Acmsol Sci. Vol. 29, Suppl. 1. pp. S8174818, 1998 8 1998 Published by Elwier Science Ltd. All rights resewed Printed in Great Britain 0021-85W98 $19.00 + 0.00
SOUND PROPAGATION
IN POLYDISPERSED
AIR FOGS
D.A. Gubaidullin Institute of Mechanics & Mechanical Engineering, Russian Academy of Sciences, 2/3 1, Lobachevsky str., Kazan, 4201 1 1, Russia, tel/fax:7-8432
365289. e-mail:gubajdullin@@ci.kcn.ru
Kazan Branch of Moscow Power Engineering Institute
KEYWORDS
Vapor-Gas-Droplet Mixtures, Drop Size Distribution, Acoustic Waves A linear theory on the propagation of plane waves in the polydispersed gas-droplets suspensions is presented. The case when the gas carrier phase is a homogeneous mixture of two components. The first one is a vapor of the liquid of the droplets, and the second is a neutral gas. Both non-steady and non-equilibrium cl‘fccls of’ the phase interaction (mass. momentum, energy interface exchange) are taken into acct>unt. An effect of phastransformations (evaporation and condensation) influenced by diffusion of vapor through the neutral gas is studied. Some calculations is done for the polydispcrsed air-water droplets fogs. Dynamics of weak impulse of pressure in polydispersed air fogs with continuous drop size distributions and in two-fractional aerosols are studied. Let us consider the plane one-dimensional motion of a polydispersed vapor-gasdroplets mixture in the acoustic field when disturbances of the suspension parameters are small. In usual for mechanics of gas-particle mixture assulnptions prl>posed by Nigmatulin (1990) the linearized system of equations of planar one-dimensional motion of diluted polydispersed vapor-gas-droplet mixture has the form w-ritien by Nigmatulin and Gubaidullin (1996). It is supposed that the aerosol carrier gaseous phase consists of two components (inert gas and vapour), and each nf them can be accounted as a calorifically perfect gas. The particled droplets in liquid are considered to bc noncompressible. Interface interaction depends on the oscillation frequency. The intensity of the nonequillibrium mass transfer is determined with the help of well known Hertz-Knudsen-Langmuir formula. General dispersion relation that describe the propagation of weak monochromatic disturbances in polydispersed tv\o-component vapor-gas-droplet mixtures have been derived by Gubaidullin & Ivandaev (1994). Consider the propagation and attenuation ol weak pulse perturbation in vapor-gasdrops systems. For calculations by computer we use the I‘ast Fourier transforn (FFT) algorithm proposed by Tikhonov et al (1984). allilwing one to recluse sabstantially the number of arithmetic operations required. The propagation of low-amplitude perturbation pulses. represented in the form of superpositions of monochromatic harmonics, XCL~IS sccording to the propagation laws of weak monochromatic waves (Gubaidullin. 1O%,. In this context and according to the results S817
S818
Abstracts
of the 5th International
Aerosol
Conference
1998
obtained above for monochromatic waves the attenuation of long-wave pulses in gas suspensions with phase transformations depends nonmonotonically on the mass content of the suspended phase m. Thus, for m = 0.01 the attenuation is not only higher than for m = 0. 1, but also higher than for m = 1.0. ‘The interphase mass exchange can strongly affect the perturbation attenuation. For m = 0.01 the contribution of mass exchange to pulse attenuation is substantial. If this drop mass content is sufficiently high (m =l), the presence or absence of mass exchange affects weakly both the propagation rate and attenuation coefficients of weak pulses. The attenuation of weak pulse perturbations in gas suspensions without phase transformations is proportional to the particle mass content m, and always increases with increasing m. A paradox of the damping crisis is the fact that within a certain range of m and of disturbance frequency the damping coefficient decreases with the elevation of droplets content which a sources and the main cause of wave dissipation. The propagation and attenuation pulsing perturbations in polydispersed fogs strongly depend from a kind of drop size distribution function. So the attenuation of a pulse in polydispersed aerosols with continuous drop size distribution higher, than in case of monodispersed fog., however can be considerably less. than in two-fractional gas particle suspensions. In this connection for considered functions of distribution the maximum effect of influence of drops on attenuation of pulses in polydispersed vapor-gas-drop systems is reached by use of two-fractional distribution of drops. Impulse, having originally the rectangular form, at propagation under effect dispersion and attenuation accepts the form of pulse of Gaussian shape. ACKNOWLEDGEMENT‘S The research described in this publication was made possible by Grant No 96-15 96905 for young doctors of sciences from the President of Russian Federation and by Grant No 97-02- 16043 from Russian Basic Research Foundation. REFERENCES Nigmatulin R.I. (1990) Dynamics of Multiphase Media. vol. 1. Hemisphere, N.Y. Nigmatulin RI., Gubaidullin D.A. (1996) Influence of phase transformation in acoustics of polydispersed fogs, Doklady Akudemii Natrk Rossii., 347. 3,330-333. Gubaidullin D.A., Ivandaev A.I. (1994) Influence of polydispersivity to propagation of sound in vapor-gas-droplets mixture, .J. qf Applied Mechanics and Technical Physics., 4, 75-83. Gubaidullin D.A. (1996) An influence of heat and mass transfer in aerosols acoustics, Proceeding of the Znd European Thermal-Sciences and 141h UIT National Heat Transfer Conference. Rome. Italy. 363-367. 1996. Tikhonov A.I., Goncharskii A.V., Stepanov V.V., Yagola A.G. (1983) Algorithms and a Priori Information (in Russian). Nauka. Moscow.
Regularizing
SOUND PROPAGATION
IN POLYDISPERSED
AIR FOGS
D.A. Gubaidullin Institute of Mechanics & Mechanical Engineering, Russian Academy of Sciences, 2/3 1, Lobachevsky str., Kazan, 4201 1 1, Russia, tel/fax:7-8432
365289. e-mail:gubajdullin@@ci.kcn.ru
Kazan Branch of Moscow Power Engineering Institute
KEYWORDS
Vapor-Gas-Droplet Mixtures, Drop Size Distribution, Acoustic Waves A linear theory on the propagation of plane waves in the polydispersed gas-droplets suspensions is presented. The case when the gas carrier phase is a homogeneous mixture of two components. The first one is a vapor of the liquid of the droplets, and the second is a neutral gas. Both non-steady and non-equilibrium cl‘fccls of’ the phase interaction (mass. momentum, energy interface exchange) are taken into acct>unt. An effect of phastransformations (evaporation and condensation) influenced by diffusion of vapor through the neutral gas is studied. Some calculations is done for the polydispcrsed air-water droplets fogs. Dynamics of weak impulse of pressure in polydispersed air fogs with continuous drop size distributions and in two-fractional aerosols are studied. Let us consider the plane one-dimensional motion of a polydispersed vapor-gasdroplets mixture in the acoustic field when disturbances of the suspension parameters are small. In usual for mechanics of gas-particle mixture assulnptions prl>posed by Nigmatulin (1990) the linearized system of equations of planar one-dimensional motion of diluted polydispersed vapor-gas-droplet mixture has the form w-ritien by Nigmatulin and Gubaidullin (1996). It is supposed that the aerosol carrier gaseous phase consists of two components (inert gas and vapour), and each nf them can be accounted as a calorifically perfect gas. The particled droplets in liquid are considered to bc noncompressible. Interface interaction depends on the oscillation frequency. The intensity of the nonequillibrium mass transfer is determined with the help of well known Hertz-Knudsen-Langmuir formula. General dispersion relation that describe the propagation of weak monochromatic disturbances in polydispersed tv\o-component vapor-gas-droplet mixtures have been derived by Gubaidullin & Ivandaev (1994). Consider the propagation and attenuation ol weak pulse perturbation in vapor-gasdrops systems. For calculations by computer we use the I‘ast Fourier transforn (FFT) algorithm proposed by Tikhonov et al (1984). allilwing one to recluse sabstantially the number of arithmetic operations required. The propagation of low-amplitude perturbation pulses. represented in the form of superpositions of monochromatic harmonics, XCL~IS sccording to the propagation laws of weak monochromatic waves (Gubaidullin. 1O%,. In this context and according to the results S817
S818
Abstracts
of the 5th International
Aerosol
Conference
1998
obtained above for monochromatic waves the attenuation of long-wave pulses in gas suspensions with phase transformations depends nonmonotonically on the mass content of the suspended phase m. Thus, for m = 0.01 the attenuation is not only higher than for m = 0. 1, but also higher than for m = 1.0. ‘The interphase mass exchange can strongly affect the perturbation attenuation. For m = 0.01 the contribution of mass exchange to pulse attenuation is substantial. If this drop mass content is sufficiently high (m =l), the presence or absence of mass exchange affects weakly both the propagation rate and attenuation coefficients of weak pulses. The attenuation of weak pulse perturbations in gas suspensions without phase transformations is proportional to the particle mass content m, and always increases with increasing m. A paradox of the damping crisis is the fact that within a certain range of m and of disturbance frequency the damping coefficient decreases with the elevation of droplets content which a sources and the main cause of wave dissipation. The propagation and attenuation pulsing perturbations in polydispersed fogs strongly depend from a kind of drop size distribution function. So the attenuation of a pulse in polydispersed aerosols with continuous drop size distribution higher, than in case of monodispersed fog., however can be considerably less. than in two-fractional gas particle suspensions. In this connection for considered functions of distribution the maximum effect of influence of drops on attenuation of pulses in polydispersed vapor-gas-drop systems is reached by use of two-fractional distribution of drops. Impulse, having originally the rectangular form, at propagation under effect dispersion and attenuation accepts the form of pulse of Gaussian shape. ACKNOWLEDGEMENT‘S The research described in this publication was made possible by Grant No 96-15 96905 for young doctors of sciences from the President of Russian Federation and by Grant No 97-02- 16043 from Russian Basic Research Foundation. REFERENCES Nigmatulin R.I. (1990) Dynamics of Multiphase Media. vol. 1. Hemisphere, N.Y. Nigmatulin RI., Gubaidullin D.A. (1996) Influence of phase transformation in acoustics of polydispersed fogs, Doklady Akudemii Natrk Rossii., 347. 3,330-333. Gubaidullin D.A., Ivandaev A.I. (1994) Influence of polydispersivity to propagation of sound in vapor-gas-droplets mixture, .J. qf Applied Mechanics and Technical Physics., 4, 75-83. Gubaidullin D.A. (1996) An influence of heat and mass transfer in aerosols acoustics, Proceeding of the Znd European Thermal-Sciences and 141h UIT National Heat Transfer Conference. Rome. Italy. 363-367. 1996. Tikhonov A.I., Goncharskii A.V., Stepanov V.V., Yagola A.G. (1983) Algorithms and a Priori Information (in Russian). Nauka. Moscow.
Regularizing