Cell model for gas absorption with first-order irreversible chemical reaction and heat release in gas-liquid bubbly media

A model for mass and heat transfer during physical gas absorption in gas-liquid bubbly medium suggested in [1] is generalized for a case of chemical absorption accompanied by heat release. Diffusion and thermal interactions between bubbles are taken in to account in the approximation of a cellular model of a bubbly medium whereby a bubbly medium is viewed as a periodic structure consisting of identical spherical cells with periodic boundary conditions at a cell boundary. Distribution of concentration of the dissolved gas, temperature distribution in liquid and rates of mass and heat transfer during nonisothermal chemical absorption of a soluble pure gas from a bubble by liquid are determined. In the limiting case of chemical absorption without heat release the derived formulas recover the expressions for isothermal chemical absorption. In the limiting case of physical absorption with heat release the derived formulas recover the expressions for nonisothermal absorption obtained in [1].     

A model for the nonisothermal behavior of viscoelastic media

Summary A new constitutive model is derived for the viscoelastic behavior of polymers under non-isothermal loading. The model extends the concept of adaptive links (entanglements) between polymeric molecules to thermoviscoelastic media. By using experimental data for Nylon-6 and polyisobutylene in the vicinity of the glass-transition temperature, we find parameters of the model and study their dependence on temperature. The model is employed for the numerical analysis of the material response to time-periodic loads under isothermal conditions and to time-varying loads under heating. The results of numerical simulation demonstrate fair agreement with experimental data. Accepted for publication 23 May 1996     

Propagation of combustion waves in bubbly media

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 165–166, March–April, 1992.     

Velocity field in isothermal turbulent bubbly gas-liquid flow through a pipe

Velocity field was measured by laser Doppler velocimetry in isothermal, turbulent bubbly gas-liquid flow through a 26.6 mm inner diameter vertical pipe. The measurements were made about 33 diameters downstream from the pipe entrance, gas injection being just upstream of the entrance. The gas phase radial distribution at the measurement plane exhibited influence of the injection device in that higher gas fraction existed in the central region of the pipe. For comparison, velocity field was also measured in isothermal, turbulent single-phase liquid flow through the same pipe at the same axial plane. Measured were the radial distributions of liquid mean axial and radial velocities, axial and radial turbulent intensities, and axial Reynolds shear stress. The radial distributions of gas bubble mean axial velocity and axial velocity fluctuation intensity were also measured by LDV. A dualsensor fiberoptic probe was used at the same time to measure the radial distributions of gas fraction, bubble mean axial velocity and size slightly downstream of the LDV measurement plane.List of Symbols an average gas bubble diameter - f, f _TP friction factor, friction factor for gas-liquid flow - k _L liquid turbulent kinetic energy - , gas, liquid mass flow rate - R inner radius of pipe - r, {sitR}^* radial coordinate; nondimensional radial coordinate (=r/R) - Re _L liquid Reynolds number - U _G mean axial velocity of gas bubble - U _L mean axial velocity of liquid - U _LO mean axial velocity for flow at the total mass velocity with properties of the liquid phase - u _L ⁺ nondimensional mean axial velocity of liquid in wall coordinate - friction velocity - axial velocity fluctuation intensity of liquid - axial velocity fluctuation intensity of gas bubbles - V_L mean radial velocity of liquid - v _L radial velocity fluctuation intensity of liquid - (uv)_L single-point cross-correlation between axial and radial velocity fluctuations of liquid ( axial Reynolds shear stress) - T _in mean liquid temperature at test section inlet - x flow quality - y normal distance from wall - y ⁺ nondimensional normal distance from wall in wall coordinate (=yu/v_L) - _G gas phase residence time fraction - _L rate of dissipation in the liquid - _L Kolmogorov length scale in the liquid - _L liquid kinematic viscosity - _L characteristic turbulence length scale in the liquid - _G, _L density of gas, liquid - _m gas-liquid mixture density This work was partly supported by National Science Foundation, Thermal Transport and Thermal Processing Program, Chemical and Thermal Systems Division, under Grant No. CTS-9411898.     

Propagation of nonlinear waves in gas-liquid media with a gas content variable in space

A study is made of the propagation of nonlinear waves in gas-liquid systems of bubble structure with a gas content which is variable in the direction of propagation of the wave. It is shown that it is possible in a number of cases to amplify the pressure waves. Restrictions on the degree of homogeneity of the gas content are obtained such that, when they are fulfilled, this amplification is possible. The study also covers the effect of inhomogeneity in the gas content on the structure of nonlinear steady waves of the soliton and shock wave types.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 180–183, January–February, 1986.     

Shock waves in polydisperse bubbly media with dissipation

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–34, September–October, 1991.     

Model of gas absorption in gas-liquid plug flow with first-order and zero-order chemical reaction

A model of mass transfer during gas absorption in gas-liquid plug flow accompanied by irreversible chemical reaction of the first order and zero order is suggested. The expressions for coefficients of mass transfer during chemical absorption from a single Taylor bubble are derived in the approximation of the thin concentration boundary layer in a liquid phase. Under the assumptions of a perfect liquid mixing in liquid plugs recurrent relations for the dissolved gas concentrations in the n-th liquid plug and mass fluxes from the n-th Taylor bubble are derived. The total mass fluxes in gas-liquid plug flow during chemical absorption are determined. In the limiting case of absorption without chemical reaction the derived formulas recover the expressions for mass transfer during physical absorption in gas-liquid plug flow. Theoretical results are compared with available experimental data.     

Multivelocity effects in high-pressure-gradient flows of dilute bubbly media

The multivelocity effects associated with the behavior of gas or vapor bubbles in a region with high pressure gradients typical of the flows around a cavity in which the pressure is higher than that in the surrounding space are considered. For a low volume bubble concentration, the problem of fluid flow perturbation by the bubbles is examined. For gas bubbles, it is shown that taking multivelocity effects into account considerably reduces the additional jet momentum. It is found that, with time, the temperature distribution in the wake behind a vapor bubble becomes nonmonotonic and the maximum temperature may even exceed the initial bubble temperature. It is demonstrated that the bubbles may accumulate and a flow regime with a sharply pronounced two-phase jet extending to the outer edge of the main liquid jet may develop. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–100, January–February, 1998. The work received financial support from the Russian Foundation for Fundamental Research (project No.96-01-01442).     

A coupled finite element model for the consolidation of nonisothermal elastoplastic porous media

A coupled finite element model for the analysis of the deformation of elastoplastic porous media due to fluid and heat flow is presented. A displacement-pressure temperature formulation is used for this purpose. This formulation results in an unsymmetric coefficient matrix, even in the case of associated plasticity. A partitioned solution procedure is applied to restore the symmetry of the coefficient matrix. The partitioning procedure is an algebraic one which is carried out after integration in the time domain. For this integration, a two-point recurrence scheme is used. The finite element model is applied to the investigation of nonisothermal consolidation in various situations.     

Kinetic model of bubbly flow

A kinetic approach based on the approximate calculation of the fluid flow potential and formulation of Hamilton’s equations for generalized coordinates and momenta of bubbles is employed to describe processes of collective interaction of gas bubbles moving in an inviscid incompressible fluid. Kinetic equations governing the evolution of the distribution function of bubbles are derived. These equations are similar to Vlasov equations. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 130–138, September–October, 2000.     

Numerical simulation of circulation in gas-liquid column reactors: isothermal, bubbly, laminar flow

The gas-liquid flow inside a circular, isothermal column reactor with a vertical axis has been studied using numerical simulations. The flow is assumed to be in the laminar, bubbly flow regime which is characterized by a suspension of discrete air bubbles in a continuous liquid phase such as glycerol water. The mathematical formulation is based on the conservation of mass and momentum principle for the liquid phase. The gas velocity distribution is calculated via an empirically prescribed relative velocity as a function of void fraction. The interface viscous drag forces are prescribed empirically. For some cases a profile shape is assumed for the void ratio distribution. The influence of various profile shapes is investigated. The results are compared with those where the void ratio distribution is calculated from the conservation of mass equation. The mathematical model has been implemented by modifying a readily available computer code for single-phase newtonian fluid flows. The numerical discretization is based on a finite volume approach. The predictions show a good agreement with measurements. The circulation pattern seems not to be so sensitive to the actual shape of the void fraction profiles, but the inlet distribution of it is important. A significantly different flow pattern results when the void fraction distribution is calculated from the transport equation, as compared to those with a priori prescribed profiles. When the void fraction is uniformly distributed over the whole distributor plate, no circulation is observed. Calculations also show that even the two-phase systems with a few discrete bubbles can be simulated successfully by a continuum model.     

Turbulent flow structure and bubble distribution in an axisymmetric nonisothermal impinging gas-liquid jet

The flow structure of a bubbly impinging jet in the presence of heat transfer between the two-phase flow and the surface is numerically investigated on the basis of the Eulerian approach. The model uses the system of Reynolds-averaged Navier–Stokes equations in the axisymmetric approximation written with account for the inverse effect of the bubbles on the average and fluctuating flow parameters. The influence of the gas volumetric flow rate ratio and the dimensions of the bubbles on the flow structure in a gas-liquid impinging jet is studied, In the presence of gas bubbles the liquid velocity is higher than the corresponding value in the single-phase flow. A considerable, more than twofold, anisotropy between the axial and radial turbulent fluctuations in the gas-liquid impinging jet is shown to exist. An addition of air bubbles leads to a considerable growth in the liquid velocity fluctuations in the two-phase flow (up to 50% compared with the single-fluid liquid impinging jet). An increase in the disperse phase dimensions leads to intensification of turbulence of the liquid.     

Advective fluxes in multiphase porous media under nonisothermal conditions

We consider the case in which more than one fluid phase occupies the void space of a porous medium. The advective flux law is formulated for a fluid phase, under nonisothermal conditions and with the presence of solutes in the fluid phases. The derivation of the flux laws is based on an approximated version of the averaged balance equation for linear momentum. Taking into account momentum transfer through the interface between the fluid phases, leads to coupling between the flow in adjacent phases. Fluxes are also shown to depend on the surface tension at the interface between the adjacent fluid phases. Since the latter depends on temperature and solute concentration in the two phases, the advective flux is shown to depend on both temperature and solute concentration gradients in the two phases. A preliminary order of magnitude analysis gives conditions under which the coupling phenomena are not negligible. The approach is applied to the unsaturated zone, as a typical example of a multiphase porous medium.The main conclusion is that the well known Darcy law for single phase flow, may have to be modified for a multi fluid phase system, especially when temperature and solute concentration are not uniform.     

Mass transfer during gas absorption in a vertical gas-liquid slug flow with small bubbles in liquid plugs

A model is developed for the analysis of mass transfer during isothermal absorption in a vertical gas-liquid slug flow at large Reynolds numbers with liquid plugs containing small bubbles. Simple formulas for mass flux from the N-th unit cell of gas-liquid slug flow and for total mass flux from N unit cells are derived. In the limiting case the derived formulas for mass transfer during gas absorption in a slug flow with liquid plugs containing small bubbles recover the derived expressions for mass transfer in slug flow without small bubbles in the liquid plugs. Using the developed model recommendations concerning the design of the absorber operating in a slug flow regime are suggested. Received on 28 July 1997     

Equations of nonisothermal filtration in fast processes in elastic porous media

The problem of the nonisothermal joint motion of an elastic porous body and the fluid filling the pores is considered for the case where the duration of the physical process is fractions of a second. A rigorous derivation of averaged equations (equations not containing fast oscillating coefficients) based on the Nguetseng two-scale convergence method is proposed. For various combinations of physical parameters of the problem, these equations include anisotropic nonisothermal Stokes equations for the velocity of the fluid component and the equations of nonisothermal acoustics for the displacements of the solid component or anisotropic nonisothermal Stokes equations for a single-velocity continuum. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 113–129, July–August, 2008.     

Configurations of gas-liquid two-phase bubbles in immiscible liquid media

The configuration of a “two-phase bubble” constituted of a gas phase and a liquid phase in an immiscible liquid medium is classified into three types: complete engulfing of a gas bubble inside a liquid shell, partial coalescence of a gas bubble and a liquid drop forming a three-phase contact line, and non-coalescence whereby a gas bubble and a liquid drop remain separated. Simple criteria have been presented by which the favorable type of configuration in a given system is predicted from the values of the spreading coefficients characterizing the system. Experiments using some combinations of liquids as well as air suggest the general validity of the criteria.     

Derivation of the equations of slow nonisothermal gas flows

In recent years, some new phenomena have been predicted theoretically on the basis of the Burnett approximation. These include thermal-stress and concentration-stress convection [1–3], and also effects due to the influence of a magnetic field in a multiatomic gas (viscomagnetic heat flux, etc., [4]). It has been shown theoretically (see [5]) that under certain conditions various terms of the Burnett approximation must be taken into account in the expression for barodiffusion. The conclusions relating to a viscomagnetic heat flux have recently been confirmed experimentally [4]. The predicted phenomena follow rigorously from the Burnett equations. However, many hydrodynamicists adopt a sceptical attitude to these equations, which is due partly perhaps to attachment to the classical Navier-Stokes equations, which have served theoreticians without fail for a century and a half. In this connection, we discuss the evolution of ideas relating to the validity of the Burnett approximation. We discuss the minimal assumptions which must be made in order to derive the equations of slow [Reynolds number R = 0(1)], essentially nonisothermal [ ln T = 0(1)] flows of a gas as a continuous medium (Knudsen number K O) in the case when the derivatives of the thermal Burnett stresses in the momentum equation have the same order of magnitude as the Euler and Navier-Stokes terms of this equation [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 77–84, November–December, 1979.We thank G. I. Petrov and L. I. Sedov for discussions that stimulated the above analysis.