No abstract

If a bubble of radius R 0 was initially placed into a supersaturated (at the given pressure and temperature) mixture, then, due to the concentration difference Dr ¼ r Ly À r Lw between the gas dissolved in the liquid far away from the bubble and the appropriate equilibrium value at the bubble surface, there arises a directional diffusion flux of the dissolved substance toward the surface. At the interface, the transition of substance from the liquid to the gaseous state takes place. The result is the increase in the bubble volume. The growth of the bubble, in its turn, results in the increase of its lift velocity, as well as the increase of convective diffusion flux. The statement of the problem and the basic dynamic equations for a bubble in a solution were described in Section 6.8.Consider a multicomponent solution. The distribution of concentrations of the dissolved substances is described by the equations of convective diffusion qr iL qt þ u r qr iL qr þ u y 1 r qr iL qy ¼ D iL 1 r 2 q qr r 2 qr iL qr ; ð22:1Þwhere u r and u y are the radial and tangential velocity components of the flow that goes around the bubble, r and y are spherical coordinates, r iL are the mass concentrations of dissolved components, D iL is the coefficient of binary diffusion. The mass balance equation for the gas in the bubble is d dtwhere r iG is the mass concentration of i-th gas component in the bubble. The dynamic balance equation for the bubble, known as Rayleigh's equation, looks likeHere r L is the liquid phase density; R -the radius of the bubble; n L -kinematic viscosity of the liquid; S -the coefficient of surface tension, p G -gas pressure inside the bubble; p y -gas pressure far away from the bubble.

The behavior of continuous medium is described by equations that follow from the laws of conservation of mass, charge, momentum, angular momentum, and energy. These equations should be completed by correlations reflecting the accepted model of a continuous medium. Such correlations are called constitutive equations or phenomenological relationships. Examples of constitutive relations are the Navier-Stokes law which establishes the linear dependence of a stress tensor on the rate of a strain tensor; the Fourier law, according to which heat flux is proportional to a temperature gradient; Fick's law, according to which mass flux is proportional to the gradient of a substance concentration; and Ohm's law, which states that the force of a current in a conducting medium is proportional to the applied electric field strength or to the potential gradient. These constitutive equations have been derived experimentally. The coefficients of proportionality, that is, coefficients of viscosity, heat conductivity, diffusion, and electrical conductivity, referred to as transfer coefficients, can be derived experimentally, and in some cases theoretically through the use of kinetic theory [1].Conservation equations together with constitutive equations describe a phenomenological model of the continuous medium (continuum).If one considers a liquid containing particles, then the disperse phase may be treated as a continuum, its description requiring a special phenomenological model with constitutive equations that differ from those relating to the continuous phase.The flux of mass and energy in a continuum occurs in the presence of spatial gradients of state parameters, such as temperature, pressure, or electrical potential. Variables such as the volume of a system, its mass, or the number of moles are called extensive variables because their values depend on the total quantity of substance in the system. On the other hand, variables such as temperature, pressure, mole fraction of components, or electrical potential constitute intensive variables because they have certain values at each point in the system. Therefore, constitutive equations express the connection between fluxes and gradients of the intensive parameters. In the constitutive equations listed above, the flux de-45 Separation of Multiphase, Multicomponent

Consider the process of liquid-gas mixture separation in a gravitational horizontal separator. The mixture enters the separator from the supply pipeline equipped with a device for preliminary separation of free gas. A study of dispersivity of liquid-gas flow at the entrance is presented in [7]. As is shown in this work, the preliminary separation of large bubbles results in that the bubbles entering the separator have diameters ranging within a narrow interval from D 1 to D 2 , with D 2 A 3D 1 , and with the variance s 2 @ 0:003. The distribution of bubbles over diameters D at the entrance can be approximated in the interval ðD 1 ; D 2 Þ by a uniform (in the simplest case) distribution,or the normal one,where N 0 ¼ W 0 =V av is the number concentration of bubbles at the entrance; D av is the mean diameter of bubbles at the entrance; D 1 and D 2 are the smallest and the largest diameter of bubbles; W 0 is the volume concentration of bubbles at the entrance.Despite the preliminary separation of gas in the supply pipeline, the liquid-gas mixture enters the separator with sufficiently high concentration of the gaseous phase ðW 0 @ 10 À2 -10 À1 Þ. High values of W 0 result in the necessity to take into account the constrained (hindered) character of bubble floating. As was experimentally shown in [7], the velocity of bubble's ascent, with the constraint taken into account, is well approximated by the expressionð24:3Þwhere the exponent index b depends on the dimensionless parameter K ¼ S 3=2 =g 1=2 n 2 L r 3=2 L as follows:b ¼ 4:9 þ 3:4 Á 10 À5 K: ð24:4Þ 743Separation of Multiphase, Multicomponent

Using the condition expressed by Eq. (16.5), one obtainsWe can now derive an equation for y 1y ðtÞ. Let the drops be characterized by a distribution over volumes nðV; tÞ. If we consider only the change of distribution 16.