Academic literature on the topic 'Time Dependent Diffusivity (TDD)'

An ecological model for prey-predator planktonic species has been considered, in which the growth of prey has been assumed to follow a Holling type II function. The model consists of two reaction-diffusion equations and we extend it to time-varying diffusivity for plankton population. A comparative study of local stability in case of constant diffusivity and time varying diffusivity has been performed. It has been found that the system would be more stable with time varying diffusivity depending upon the values of system parameter.

It is shown that the velocity diffusivity of electrons in strong Langmuir turbulence is linearly proportional to the root-mean-square (rms) value of the electric field. The time-dependent diffusivity previously identified is a transient phenomenon. In 2-D spatial diffusion due to ExB velocity turbulence, time-dependent intermediate diffusivity emerges also followed by a well-behaved diffusivity proportional to the rms amplitude of the turbulent field.PACS Nos.: 52.25Fi, 52.35Ra, 52.65Cc

Fluid–solid adsorption processes are mostly governed by the adsorbate transport in the solid phase and surface diffusion is often the limiting step of the overall process in microporous materials such as zeolites. This work starts from a concise review of concepts and models for surface transport and variable surface diffusivity. It emerges that the phenomenon of hindered surface diffusion for monolayer adsorption, which is common in zeolites, and models able to fit a non-monotonic trend of surface diffusivity against adsorbate solid phase concentration, have received limited attention. This work contributes to the literature of hindered diffusion by formulating a time-dependent equation for surface diffusivity based on fractal dynamics concepts. The proposed equation takes into account the contributions of both fractal-like diffusion (a time-decreasing term) and hopping diffusion (a time-increasing term). The equation is discussed and numerically analyzed to testify its ability to reproduce the possible different patterns of surface diffusivity vs. time.

Reinforced concrete (RC) is a widely used construction material around the world. RC has many advantages in terms of structural stability. However, the reinforcement of RC requires extensive labour costs. Steel fibre reinforced concrete (SFRC) has been widely studied to replace steel bars in concrete structures over the decades. However, most underground structures, such as tunnel lining, are usually designed using conventional RC for long-term stability due to unexpected geotechnical characteristics, such as directional and depth-dependent varied lateral pressure, earthquakes, groundwater, and time-dependent swelling behaviour. In this paper, an alternative design of shaft structure using SFRC, based on the original RC designed data in the Toronto region, was studied to evaluate the feasibility of SFRC replacing conventional RC. A key geological feature of the site is that the bedrock is comprised of Georgian Bay shale, which exhibits long-term time-dependent deformation (TDD). The capacities of RC and SFRC for the shaft lining were calculated based on the Canadian concrete design codes CSA A23.3 and RILEM TC 162-TDF, to assess the benefit of adding steel fibre, and several analytical solutions were used to calculate the applied load on the lining. A specialised TDD constitutive model in Fast Lagrangian Analysis of Continua (FLAC) 2D was developed to estimate whether the optimum installation time of the shaft lining, based on the geological reports, is appropriate under swelling behaviour, and evaluate the resultant long-term stability. The calculated hoop thrust and bending moment for several loading cases were within the capacity of the SFRC shaft lining. The numerical analysis demonstrated that the proposed lining installation time could be reduced, despite consideration of the long-term TDD behaviour.

Diffusion processes, because of their applications to a wide range of phenomena, have been a subject of great scientific interest ever since Einstein formulated the celebrated theory of Brownian motion. Brownian motion is the most commonly known class of diffusion and is the dominant form of molecular transport in physical systems which are usually driven by thermal noise e.g. dissolution of sugar in water. It is also the simplest case of a random process where it is assumed that the time scale of motion of diffusing particle is much larger than that of the solvent molecules. This causes an extreme separation of time scales- one associated with the slower diffusing particle, and the other associated with the faster solvent molecules. This in turn leads to two fundamental laws of Brownian motion : (1) the mean square displacement (MSD) of particle is proportional to the time lapsed, i.e. hx2i / T . It is usually referred to as Fickian motion and (2) the probability distribution function (pdf) of displacements is Gaussian with the width of distribution p-scaling as T (this is equivalent to say that the motion is Fickian). However, there are many other diffusion processes which can not be classified as Brownian motion and hence are termed as anomalous diffusion. A diffusion process can be termed as anomalous if any one or both the laws of Brownian motion are violated. There are a lot of phenomena in which is diffusion is anomalous, i.e. where the pdf is not Gaussian but a stable distribution with a functional form f(jxj=T =2) such that the width of distribution increases like T =2 with 6= 1. The Brownian motion, on the other hand, would lead to a Gaussian distribution with = 1. In the past, it has been usually assumed that if 6= 1, i.e. if the diffusion is non-Fickian, then the distribution would also be non-Gaussian. Conversely, if = 1, then the distribution would be Gaussian. This was so well accepted that it was almost never tested until recently. In a series of experiments from Granick's group [1, 2] where the environment undergoes structural rearrangement on a time scale less than that of observation of diffusion, non-Gaussian distributions have been realized. Even more interesting, coexisting with this non-Gaussian distribution was observed a MSD which was found to be vary linearly in time at all times irrespective of the actual form of the distribution. In these experiments, the pdf was found to be exponential at short times which then crossed over to being Gaussian at large enough time scales. Chubynsky and Slater [3] have analyzed the \di using diffusivity" model, in which dif-fusion coefficient changes as a stochastic function of time, because of the rearrangement of environment. Assuming an exponential distribution of diffusivity at small time scales, these authors showed analytically that (1) the diffusion is Fickian and (2) the distribution of displacements, after averaging the Gaussian pdf over the exponential distribution of diffusivity, becomes non-Gaussian (exponential). The width of this non-Gaussian distribution increases as T . At larger time scales, they performed simulations and the result was a cross over to Gaussian distribution. Following their work, we have proposed a class of \diffusing diffusivity" models which we have been able to solve analytically at all time scales, using the methods of path integrals [4]. In the thesis, we are interested in developing models of diffusing diffusivity that could be used to describe different kinds of anomalous diffusion processes. We show that our model of diffusing diffusivity is equivalent to another important class of physical processes, i.e. that of the Brownian motion with absorption, or the reaction- diffusion process. In reaction-diffusion models, the concentration of a chemical substance changes in space and time because of its reaction with another substance while the diffusion causes the spread in the concentrations of various substances. The connection of diffusing diffusivity model to the reaction-diffusion model is particularly useful as one can now have different models of diffusivity describing its diffusion while, interestingly the reaction term remains unchanged. In our first model, diffusivity is modeled as a simple Brownian process. More precisely, we take D(t) = 2(t) where is the position vector of an n-dimensional harmonic oscillator executing Brownian motion. For the case n = 2, the equilibrium distribution of diffusivity is an exponential, thereby making this particular case an ideal choice to compare our results with the numerical results of Chubynsky and Slater [3]. We have shown that our results are in very good agreement with theirs [5]. Further, our model is quite generic and it is possible to nd exact analytical solution with arbitrary value of n. The non-Gaussianity parameter, which is a measure of deviation from normality, has been evaluated exactly as a function of time and n. At short times, the value of parameter is non-zero, signifying non-Gaussian dynamics which eventually becomes zero in the large time limit, marking an onset of Gaussian dynamics. For larger values of n, the non-Gaussianity starts disappearing faster implying an earlier onset of Gaussian behavior. The model has been applied to the problem of calculating survival probability of a free particle in crowded, rearranging and bounded regions. We have obtained exact results for this problem where we have shown that for larger compartments and faster relaxation of the surroundings, diffusion inside a crowded, rearranging medium is similar to the diffusion in a homogeneous medium with a constant diffusivity. We have also studied the model for rotational diffusion process. We have obtained simple analytical expressions for the probability distribution and the mean square angular displacement in arbitrary dimensions. As in the case of translation diffusion, a non-Gaussianity parameter quanties the extent of deviation from Gaussian dynamics, we have defined in a similar fashion a non-normal parameter for rotational diffusion. This could be useful in analyzing the experimental data to find the extent of deviation from normal diffusion. In another study, we have used the model of diffusing diffusivity for the diffusion of a harmonic oscillator in crowded, rearranging environment. We have obtained two interesting results here namely (1) the expression for the MSD in case of diffusing diffusivity is of same kind as that for the case of constant diffusivity and (2) the probability distribution function remains non-Gaussian even in the limit of very large time unlike the previous cases where it eventually crosses over to become Gaussian. In our model of diffusivity, and also in the model of Chubynsky and Slater [3], the distribution of diffusivity decays to zero exponentially fast, implying that the probability of having a large value of D is rather small. However, there are cases where the distribution of D is broad and therefore D can occasionally have a large value with a sizable probability. We have analyzed a model of diffusivity where it evolves as a Levy flight process. More with this modelxiv is found to be a stable distribution with a time dependent width. The width of the p distribution increases as T , as in the case of Fickian dynamics but at longer times it increases at a much faster rate as T 1=2 . Thus, the dynamics is Fickian at short times and super diffusive at long times. After studying the models of diffusivity where it evolves as a Brownian process and as a Levy flight process, respectively, we have also studied a model of diffusivity where it evolves as a sub diffusive process. For that we have modeled diffusivity as a continuous time random walk (CTRW) process such that it attains an exponential distribution in the equilibrium limit. This model is actually a generalization of our first model of diffusing diffusivity with a parameter 2 (0; 1]. The problem of diffusing diffusivity, in this case, is shown to be equivalent to a class of models known as reaction-sub diffusion systems. We have analyzed two such models of reaction-sub diffusion. With both these models, we get all the results of our first model of diffusivity if = 1. Within the first model, the MSD is found to increase linearly in time at all the time scales and for all values of 2 (0; 1], thereby confirming a Fickian dynamics. Although the probability distribution function also becomes Gaussian in the limit of very large time for all values of as is our first model of diffusing diffusivity yet the evolution of pdf from a non-Gaussian function to Gaussian is a very slow process. Smaller is the value of , slower is the transition from non-Gaussian to Gaussian dynamics. The second model leads to sub diffusive dynamics in position space. The MSD here is shown to increase as T with a non-Gaussian pdf at all the time scales.

In this study, a shear-dependent continuum model for platelet activation, adhesion and aggregation is validated using computational fluid dynamics (CFD). To take the presence of red cells into account, a combination of excess-platelet boundary layer and enhanced mass diffusivity of platelets and large species is used to mimic this behavior. The model has been validated under three different shear conditions and two different heparin levels. Also three-dimensional simulations were carried out to evaluate the model’s prediction of thrombus growth rate for stenosed tubes under various flow conditions and stenosis degrees. For these cases, also the effect of change in platelet diffusivity has been investigated by using an empirical correlation for enhanced diffusivity of platelets. For all 3D simulations, results for thrombus growth rate as a function of local wall shear rate were compared to those of experiments and numerical studies in the literature and an acceptable agreement was achieved.

This paper describes measurements of the dependence of lean blowout limits upon fuel composition for H2/CO/CH4 mixtures. Blowout limits were obtained at fixed approach flow velocity, reactant temperature, and combustor pressure at several conditions up to 4.4 atm and 470 K inlet reactants temperature. Consistent with prior studies, these results indicate that the percentage of H2 in the fuel dominates the mixture blowoff characteristics. That is, flames can be stabilized at lower equivalence ratios, adiabatic flame temperatures, and laminar flame speeds with increasing H2 percentage. Various methods of correlating these data were evaluated, using combinations of Lewis number (Lemix), adiabatic flame temperature (Tad), flame speed (SL), and chemical time (τchem). These correlations clearly indicate the significance of the mixture diffusivity, heat content, and flame propagation speed upon blowout characteristics across a wide fuel spectrum. Two basic models of flame stabilization discussed in the literature were evaluated — a well-stirred reactor based approach that considers the ratio of chemical and flow times, and a propagative mechanism that considers the ratio of flame and flow speed. Both mechanisms were able to correlate some, but not all segments of the data set.

Thermal diffusivity is a thermophysical property that quantifies the ratio of the rate at which heat is conducted through a material to the amount of energy stored in a material. The pulsed laser diffusion (PLD) method is a widely used technique for measuring thermal diffusivities of materials. This technique is based on the fact that the diffusivity of a sample may be inferred from measurement of the time-dependent temperature profile at a point on the surface of a sample that has been exposed to a pulse of radiant energy from a laser or flash lamp. The standard approach to PLD is based on a simple model that produces an explicit relationship between the diffusivity and the time required for the temperature of the sample surface to reach a specified fraction of the peak temperature. However, the standard approach is based on idealizations that are difficult to achieve in practice, so models that represent a PLD measurement system with greater fidelity are desired. Assessment of the impact of the approximations made in the development of the standard approach showed that neglect of the spatial and temporal variations of the input power leads to significant errors in measurement of the thermal diffusivity. The objective of this paper is to present the Distributed Source Finite Absorption model which represents the spatial and temporal variations in the pulse with greater fidelity. The cost of the increased fidelity is an increase in the complexity of the algorithm used to determine values of the thermal diffusivity. A simple relationship between an easily determined characteristic of the measured temperature profile and the thermal diffusivity does not exist. Therefore, a new method of extracting values from measured time dependent-temperature profiles based on a genetic algorithm and on reduced order modeling has been developed. This paper also presents a numerical verification of this proposed new method for measuring the thermal diffusivity.

Transient conduction on a vertical, constant heat flux surface in a saturated porous medium is studied experimentally and analytically with a focus on determining near-wall thermal diffusivity. For combinations of different particulate solid and interstitial fluid, which give a range of conductivity ratios, ks/kf, from 0.5 to 2400, the present study finds that early-time transient temperature profiles can be analytically predicted using the thermal conductivity of the interstitial fluid because the near-wall porosity approaches 1.0. The conjugate heat transfer analysis accurately predicts the time the conductive front takes to travel through the impermeable wall. The present study also finds that conductive heat transfer along the wall is dependent on the wall thickness and must be taken into account when assessing measurement of local and overall Nusselt numbers. The present results raise the possibility of reinterpretation of much of the porous medium heat transfer experiments that make up the current database.

Abstract The present study experimentally investigates the behavior of thermal diffusivity of a sheared granular suspension with neutrally buoyant particles. The study uses a Taylor-Couette cell with a rotating outer cylinder and fixed inner cylinder to create a uniform shear flow while suppressing fluid turbulence for lower Reynolds numbers. Spherical PMMA beads of 2 mm were used to make a suspension with a glycerol-water mixture as the base fluid. For a small fixed volume fraction of 5%, changing the rotation speed, and therefore the shear rate, particle Reynolds numbers are varied from 0–30, going from Stokes flow to time-dependent flow with increasing inertia. We study the thermal diffusivity of the suspension by examining the decay of the inner wall temperature after a sharp thermal pulse. The thermal diffusivity is extracted from the observed temperature decay using a model for one-dimensional diffusion into the particle suspension. A non-monotonic behavior of thermal diffusivity with volume fraction is observed.

An extended definition of the effective thermal diffusivity is posed via an analogy to acoustic and EM wave propagation in discretely inhomogeneous media. Specifically, the propagation of a periodic, plane thermal wave of frequency ω, through an inhomogeneous medium consisting of spherical particles embedded in a continuous matrix, is theoretically examined. An exact solution for the time–harmonic conduction equation, for the multiple sphere system, is developed by use of the scalar wave harmonic functions and the addition theorem for the harmonics. An effective medium model, which is based on the Quasi–Crystalline approximation (QCA) for acoustic and EM wave propagation, is developed, and a formulation for the frequency–dependent effective thermal diffusivity is derived. In the limit of x = Rω/α0→0, where R is the sphere radius and α0 the matrix thermal diffusivity, it is shown that formulation reduces to that derived from a static model.

For high rate-capability and low cost lithium-ion batteries, the prevention of capacity loss is one of major challenges facing by lithium-ion batteries today. During electrochemical processes, lithium ions diffuse from and insert into battery electrodes accompanied with the phase transformation, where ionic diffusivity and concentration are keys to the resultant battery capacity. In the current study, we first compare voltage vs. capacity curves at different C-rates (1C, 2C, 6C, 10C). Second, lithium-ion distributions and intensity are quantified via the Time-of-Flight Secondary Ion Mass Spectroscopy (ToF-SIMS). The result shows that voltage vs. capacity relations are C-rate dependent and larger hystereses are observed in the higher C-rate samples. Detailed quantification of lithium-ion intensity for the 1C sample is conducted. It is observed that lithium-ions are distributed uniformly inside the electrode. Therefore, the current study provides a qualitative and quantitative data to better understand C-rate dependent phenomenon of LiFePO4 battery cells.

Models based on one-dimensional solution of Fickian diffusion equation are commonly used to characterize diffusion coefficients of polymers. These methods are known to have errors due to edge effects, which are believed to be compensated by correction factors. In order to better understand these errors and performance of various correction factors, non-dimensional solution of diffusion equation in a three-dimensional rectangular domain is used to simulate artificial mass diffusion data. The simulated data is then utilized to analyze the errors associated with commonly used methods of diffusivity determination for a variety of geometries with aspect ratios ranging from 1 to 100. The diffusivity values are shown to be highly dependent on the approach used in determining the initial slope of the percent mass gain versus root square time curve. The previously proposed correction factors failed to yield accurate diffusion coefficients and are found to have up to 70% error. A simple, yet accurate method for determination of mass diffusivity of isotropic materials based on the solution of Fickian equation in three-dimensional domain is proposed. The applicability of the method is demonstrated with experimental data and good agreement is observed.

Many soft materials of industrial importance such as pastes, gels, concentrated dispersions and emulsions fail to attain thermodynamic equilibrium over practical time scales due to jamming of constituent entities. Such soft materials, known as soft glassy materials, have very high viscosity and show extremely slow relaxation behavior. In this work heat transport behavior of a model soft glassy material, namely aqueous suspension of laponite is studied. Laponite is synthetic nanoclay with particles of disc-like shape, diameter 25 nm and thickness 1 nm. For concentrations above 1 volume % the aqueous system forms a space filling gel that supports its weight. In the present work, thermal diffusivity of laponite suspension in deionized water is investigated with laser interferometry. A sample is placed in an octagonal cavity comprising fixed copper plates on the top and bottom sides. Initially, the sample is at a uniform temperature. Collimated laser light beam generated from He-Ne laser with 632.8 nm wavelength is passed through the test cell. The interferometer is aligned in the infinite fringe setting, obtained by balancing the refractive index of the laponite suspension against a reference cell filled with glucose solution. At certain time instant (t = 0), the temperature of top surface is increased by 1–3°C. The interference patterns are analyzed to obtain the time dependent temperature field. This data is regressed against the analytical solution of the unsteady heat conduction problem to determine the thermal diffusivity of the solution. Results show a strong dependence of thermal diffusivity on laponite concentration.