Closure Laws for the Transport Equation of Interfacial Area in Dispersed Flow

Abstract

Derivation of a transport equation for the interfacial area concentration. In two-phase flows, the interfacial area is a key parameter since it mainly controls the momentum heat and mass transfers between the phases. An equation of transport of interfacial area may be very useful, especially for the two-fluid models. Such an equation should be able to predict the transition between the flow regimes. With this aim in view, we shall focus our attention on pipe flow. Besides in a first step, our study will be limited to dispersed flows. Different models are used to predict the evolution of bubble sizes. Some models use a population balance that provides a detailed description of the bubble size distributions, but they require as many equations as diameter ranges (Coulaloglou & Tavlarides1). Some others use only one equation for the transport of the mean interfacial area (Hibiki & Ishii2). In that case the bubble size distribution is treated as it would be monodispersed, its mean diameter being equal to the Sauter diameter. An intermediate approach was proposed by Kamp et al.3, in which polydispersed size distributions can be taken into account. It is the starting point of the present study in which: • The choice of an interfacial velocity is discussed. • The sink and source terms due to bubble coalescence, break-up or phase change are established. The model of Kamp et al. consists of transport equations of the various moments of the density probability function P(d) of the bubble diameter. In many experimental situations, P(d) is well predicted by a log-normal law (with two characteristic parameters d00 the central diameter of the distribution and a width parameter): The different moments of order ? of P(d) may be calculated: Sγ=n∫P(d)dγd(d)(1) where n is the bubble number density, S1/n, the mean diameter and S2/?, the interfacial area. A transport equation can be written for each moment: ∂Sγ∂t+∇·(uGSγ)=φγ(2) The lhs of (2) is an advection term by the gas velocity uG and the rhs is a source or sink term due to bubble coalescence, break-up or mass transfer. Since the bubble size distribution is characterised by the two parameters d00 and σˆ, only two transport equations (for S1 and S2) have to be solved to calculate the space-time evolution of the bubble size distribution. These two equations are still too cumbersome for a two-fluid model. Under some hypotheses (σˆ ∼ constant), they are lead to a single equation for the interfacial area. In its dimensionless form the interfacial area ai+ (ai+ = π S2 D, where D is the pipe diameter) reads: d/dt+(ai+)=f(RG,Re,We,ai+)(3) where RG is the gas fraction, Re is the Reynolds number of the mixture, We the Weber number of the mixture and t+ a dimensionless time.