A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion ...equations. Their accuracy is of order (Δx)2 and Δt, except for the fractional version of the Crank–Nicholson method, where the accuracy with respect to the timestep is of order (Δt)2 if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods’ numerical solutions are obtained and compared against exact solutions.
A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the ...forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grünwald-Letnikov discretization of the Riemann-Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.
We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples ...for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.
The wine industry is extremely vulnerable to climate change given the impact of temperature, water or soil conditions on wine production. Transformation strategies toward more sustainable models ...would help the industry to overcome the weak points. The present study evaluates the environmental impact of the so-called CO2-alcoholic fermentation processes (CO2-AFP) strategy, a recently developed eco-innovative strategy, which offers a new pathway toward a greener wine-making production with a 16.79% reduction of the carbon footprint considering scopes 1 (direct emissions), 2 (emissions generated directly in the production of electricity), and 3 (indirect emissions embodied in the organization's value chain). This strategy is based on a novel Carbon Dioxide Utilization approach from a biogenic CO2 released by fermentation processes to produce a fully marketable and environmentally friendly chemical product at a global level, i.e., sodium carbonate.
This paper presents the strategy development corresponding to a real case: a medium-size winery and distillery in Spain, where the “CO2-AFP Strategy” has been successfully tested and scaled-up. Detailed and tested carbon capture and utilization schemes are used to evaluate the overall carbon footprint balance (carbon calculation, capture potential, and carbon balance) via an improved hybrid multiregional input-output-lifecycle assessment model (MRIO-LCA). The benefits go beyond the reduction of carbon footprint in the fermentation industry. The application of the “CO2-AFP Strategy” implies a revolution, in terms of Circular Economy, in the sodium carbonate industry, as the symbiotic process between the different stages of the value chain will allow downstream carbon footprint reductions, facilitating a greener sodium carbonate production.
•The AFP-Strategy proposes a method for carbon capture from alcoholic fermentation.•A real-case cost-effective carbon capture and utilization process is assessed.•The corporate carbon footprint is estimated through a tiered hybrid method.•Multiregional Input-Output and Life-cycle Assessment techniques are combined.•The hybrid model provide a footprint detailed into three scopes.
We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). ...Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.
A comparison of simulation results with the prediction of the structural properties of square-shoulder fluids is carried out to assess the performance of three theories: Tang-Lu's first-order mean ...spherical approximation, the simplified exponential approximation of the latter and the rational-function approximation. These three theoretical developments share the characteristic of being analytical in Laplace space and of reducing in the proper limit to the Percus-Yevick result for the hard-sphere fluid. Overall, the best agreement with the simulation data is obtained with the simplified exponential approximation.
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