Bubble-induced turbulence

A k — Model for Bubbly Flows Bubble Induced Turbulence... 

Based on these observations [93] proposed a modified model containing two time constants, one for the liquid shear induced turbulence and a second one for the bubble induced turbulence. The basic assumption made in this model development is that the shear-induced turbulent kinetic energy and the bubble-induced turbulent kinetic energy may be linearly superposed in accordance with the hypothesis of [128, 129]. Note, however, that [82] observed experimentally that this assumption is only valid for void fractions less than 1 %, whereas for higher values there is an amplification in the turbulence attributed to the interactions between the bubbles. The application of this model to the high void fraction flows occurring in operating multiphase chemical reactors like stirred tanks and bubble columns is thus questionable. 

Nevertheless, in the first step in the model derivation a transport equation for the bubble induced turbulent kinetic energy was postulated ... 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

This relation was obtained using the well-known expression for steady interfacial drag and the two formulations for bubble induced turbulence production (i.e., the one given by [74], and the other one defined by [93]). 

Lopez de Bertodano [92] stated that this simple modification has a big effect on the dynamic and the asymptotic behavior of the model. At a later stage, [93] also stated that the bubble induced time constant, which is proportional to the residence time of a bubble, is usually very short compared to the time constant of the shear induced turbulence. They concluded that for most practical cases the transport equation for bubble induced turbulence (5.13) can be reduced to fcei = fceia-... 

Another important modification implemented is that [92] and [93] assumed that the idea of linear superposition may also be used for the viscosity. Sato et al [129] proposed that for bubbly flow the turbulent viscosity should be the sum of the single phase shear induced turbulent viscosity (j/si) and the bubble induced turbulent viscosity (j/bi) ... 

The heat transfer rates in bubble columns are much higher than that anticipated from single phase flow considerations. This enhancement is ascribed solely to the bubble-induced turbulence and liquid circulation. Little work has been reported on heat transfer, both at wall and to/from immersed surfaces, in bubble columns employing non-Newtonian media. Nishikawa et al. reported the first set of data on the effect of shearthinning viscosity of CMC solutions on jacket and coil heat transfer coefficients [7]. They reconciled their results for Newtonian and power law liquids by introducing the notion of an effective viscosity estimated via Equation 3, provided the gas velocity was greater than 40 mm/s. For superficial gas velocity lower than this value, the effective shear rate varies as for coil heat transfer... 

In a recent study Jakobsen et al. [71] examined the capabilities and limitations of a dynamic 2D axi-symmetric two-fluid model for simulating cylindrical bubble column reactor flows. In their in-house code all the relevant force terms consisting of the steady drag, bulk lift, added mass, turbulence dispersion and wall lift were considered. Sensitivity studies disregarding one of the secondary forces like lift, added mass and turbulent dispersion at the time in otherwise equivalent simulations were performed. Additional simulations were run with three different turbulence closures for the liquid phase, and no shear stress terms for the gas phase. A standard k — e model [95] was used to examine the effect of shear induced turbulence, case (a). In an alternative case (b), both shear- and bubble induced turbulence were accounted for by linearly superposing the turbulent viscosities obtained from the A — e model and the model of Sato and Sekoguchi [138]. A third approach, case (c), is similar to case (b) in that both shear and bubble induce turbulence contributions are considered. However, in this model formulation, case (c), the bubble induced turbulence contribution was included through an extra source term in the turbulence model equations [64, 67, 71]. The relevant theory is summarized in Sect. 8.4.4. 

A simphfied way is to decouple the two approaches first, the stabihty condition serves as the close law for the simphfied algebraic conservation equations, as described by the EMMS model for gas—hquid and gas—solid systems. The nonlinear optimization problem can therefore be solved to obtain the global or local structure parameters which are then be used to derive the closure law or correlations for the drag, bubble-induced turbulence and even the correction factors for the kernel functions of bubble coalescence and breakup for PBEs. 

There are three types of closure models in CFD simulation of gas—hquid flow in bubble columns, i.e., drag force, bubble-induced turbulence, and kernel functions of bubble breakup and coalescence. We will show how we utilize the EMMS approach to derive new models and integrate them into CFD simulation. 

---