Energy equation, bubbles

From the simplified model of Figure 5.17, the energy equation for the bubble layer is written in terms of average conditions across the liquid layer ... 

The performance equation for the model is obtained from the continuity (material-balanoe) equations for A over the three main regions (bubble, cloud + wake, and emulsion), as illustrated schematically in Figure 23.7. Since the bed is isothermal, we need use only the continuity equation, which is then uncoupled from the energy equation. The latter is required only to establish the heat transfer aspects (internally and externally) to achieve the desired value of T. 

The actual motion of the bubble deviates markedly from that predicted by Eq. (11) or (12) because the expansion allows vaporization to occur at the bubble boundary and this cools the liquid near the bubble. The heat transfer at the bubble boundary is found by solving the energy equation. In the liquid phase, if R is measured from the bubble center,... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

This is our first boundary condition. The second condition is that of energy balance at the bubble surface. It is obvious that the change of kinetic energy is small. Therefore the energy equation is ... 

The next question is whether this bubble is stable or not. The following equation shows the variation of the Gibbs free energy with bubble radius for an embryo formed in a superheated liquid due to a density fluctuation ... 

Figure 4.3 shows a bubble in contact with a horizontal plate of poor wettabiUty. The bubble has three types of energy potential energy due to gravitational force, surface energy on the gas-liquid interface, and interfacial energy on the gas-solid interface. The following relationship can be obtained [13]. from the energy equation for a droplet on a horizontal flat plate [13] ...

A bubble is considered to detach from a plate of poor wettability as soon as the buoyancy force acting on the bubble becomes larger than the interfacial force acting on the bottom periphery of the bubble. This condition is described by the relationship (pl - p%)gVs > jtDbctgl sin0a, where Db is the bottom diameter of the bubble. The value of Fb at which the left and right hand sides become equal to each other is defined as the critical value for the detachment of the bubble. Db should, of course, be evaluated from the energy equation for each model. 

The shape of a sessile bubble placed beneath a horizontal plate can be predicted satisfactorily by using an energy equation for the bubble, irrespective of the wettability of the plate. 

Asai [15] developed an inviscid theory for the generation of a slag droplet at the interface between molten slag and metal in the absence of bubbles based on the energy equation. The reported diameter, dpc, is given by... 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

Application of these equations gives the results in Table 13-12. A set of T is calculated from the normahzed by bubble-point calculations. Corresponding values of are obtained from y = K x. Once newA. andT are available, new values of Vn are calculated from energy balances by using data from Maxwell (Data Book on Hydiocaihons, Van Nostrand, Princeton, N.J., 1950). First, an estimate of condenser duty is computed from an energy balance around the condenser. 

This equation gives only the energy required to break up a bubble. The rate of breakage will also involve the number density of eddies of size A and a probability that the bubble will break up [20]. 

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