2.9: Linearized source-sink terms

EXAMPLE 2.9 Degradation of 1,3-butadiene resulting from a spill (linearized source-sink terms)... 

A linear PDF, such as the diffusion equation, can only have first-order and zero-order source or sink rates. But what if the source or sink term is of higher order An example would be the generalized reaction... 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

The coefficients a consist of all the inflow contributions (convective as well as diffusive) while the coefficients b consist of all the outflow contributions. In the absence of any source or sink, the mass conservation equation dictates that the sum of inflow contributions is equal to the sum of outflow contributions. In the presence of linearized source terms, one can write. 

At this point it is important to notice that the interfacial source and dissipation terms occurring in the k- equation above are consistent with the source and sink terms in the two-phase k - equation developed by [74]. Lopez de Bertodano et al [93] recognized that the terms pitsi + are a linear superposition of shear and bubble induced dissipation. Similarly, the terms Pk,si + were found to denote the linear superposition of the shear and bubble induced sources of turbulence. Note that the latter term, i.e., by introducing the suggested parameterizations, is identical to the bubble induced source of turbulence (Pf,) which was taken to be proportional to the work done on the liquid by the bubbles as shown in equation (5.3). The k - equation used by [93] is thus identical to equation (5.3). The only modification... 

With the presence of the liquid-phase and interphase mass, momentum, and energy transport, additional source terms are added into the continuity, momentum, and scalar transport equations. As the droplets evaporate the heat of vaporization is taken from the gas phase and there is evaporative cooling of the surrounding gas. This gives rise to a sink term in the energy equation. By assuming adiabatic walls and unity Lewis number, the energy and scalar equations have the same boundary conditions and are linearly dependent [5]. 

At this point it is important to notice that the interfacial source and dissipation terms occurring in the -equation above are consistent with the source and sink terms in the two-phase -equation developed by [77]. Lopez de Bertodano et al. [104] recognized that the terms p esi -I- are a linear superposition of shear and... 

The solution for (Eq. 9.9) requires two boundary conditions on c, one on v an initial condition on c and similarly one initial condition on q. Finally we must prescribe the sink/source term for the adsorption. This can be done in the most general case by writing another pde to describe adsorption, which is the transport of the adsorbing species into the crystal structure of the formed adsorbent. This model must be sufficiently broad to allow us to calculate the uptake at any location in the packed bed and at any time during the process. In many cases it wiU be found expedient and quite satisfactory to prescribe the uptake term as some kind of linear driving force model (LDF). 

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