Convective timescale

Note that err = y (crr)a3/k Tand recall that in a concentrated dispersion the Peclet number is Pe = 67ry (crr)a3/k T. The use of the suspension viscosity implies that the particle diffusion can be estimated from an effective medium approach. Both Krieger and Cross gave the power law indices (n and m) as 1 for monodisperse spherical particles. In this formulation, the subscript c indicates the characteristic value of the reduced stress or Peclet number at the mid-point of the viscosity curve. The expected value of Pec is 1, as this is the point at which diffusional and convective timescales are equal. This will give a value of ac 5 x 10 2. Figure 3.15 shows a plot of Equation (3.57a) with this value and n = 1... 

In DSMC, there is a long-time requirement for achieving steady state. Let us consider a gas flow in a channel of length 1 mm and height 1 pm with an average speed of 1 mm/s. The convection timescale for the macroscopic disturbance to travel from the inflow to outflow is equal to 1 s. The viscous timescale for the problem is. For air (y = lO m /s),... 

Lise 10 s, which is three orders of magnitude less than convective timescale. The mean collision time of air at standard condition is the order of the 10 ° s. Hence, the DSMC step must run about 10 ° time steps for settling to a steady-state condition. 

Mass transport can be by migration, convection or diffusion. As discussed in chapter 1, in the presence of strong electrolyte migration can be neglected, as can convection if the solution is unstirred, at a uniform temperature and the timescale of the experiment is short (i.e. a few seconds). Thus, we can make the first distinction between electrode reactions that are dominated by step 1, diffusion-controlled, and those for which steps 1 and 2 contribute to the overall observed rate. 

PMS stars with M < 0.35 M0 have a simple structure - they are fully convective balls of gas all the way to the ZAMS. As the star contracts along its Hayashi track the core heats up, but the temperature gradient stays very close to adiabatic except in the surface layers. Li begins to burn in p, a reactions when the core temperature, Tc reaches c 3x 106 K and, because the reaction is so temperature sensitive (oc Tc16-19 at typical PMS densities) and convective mixing so very rapid, all the Li is burned in a small fraction of the Kelvin-Helmholtz timescale (see Fig. 1). 

What happens for cooler (i.e. less massive) stars on the red side of the Li dip As we shall see now, the stellar mass or the effective temperature of the dip is a transition point for stellar structure and evolution. First of all it is a transition as far as the rotation history of the stars is concerned. Indeed the physical processes responsible for surface velocity are different, or at least operate with different timescales on each side of the dip. At the age of the Hyades, the stars hotter than the dip still have their initial velocity while cooler stars have been efficiently spun down (Fig. 1). This behavior is linked to the variation of the thickness of the superficial H-He convection zone which gets rapidly deeper as Teff decreases from 7500 to 6000K (e.g. TC98). Below 6600 K, the stars have a sufficiently deep... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

For a concentrated system this represents the ratio of the diffusive timescale of the quiescent microstructure to the convection under an applied deforming field. Note again that we are defining this in terms of the stress which is, of course, the product of the shear rate and the apparent viscosity (i.e. this includes the multibody interactions in the concentrated system). As the Peclet number exceeds unity the convection is dominating. This is achieved by increasing our stress or strain. This is the region in which our systems behave as non-linear materials, where simple combinations of Newtonian or Hookean models will never satisfactorily describe the behaviour. Part of the reason for this is that the flow field appreciably alters the microstructure and results in many-body interactions. The coupling between all these interactions becomes both philosophically and computationally very difficult. 

The fact that transport limits the rate of the overall electrode reaction affects the fastest timescale accessible. Once transport controls the rate, faster reaction steps cannot be characterized. It is thus important to enhance mass transfer, for example, by increased convection with high flow rates [37, 38]. 

In many processes involving reactive flows different phenomena are present at different order of magnitude. It is fairly common that transport dominates diffusion and that chemical reaction happen at different timescales than convection/diffusion. Such processes are of importance in chemical engineering, pollution studies, etc. 

Dan = timescale of pore diffusion/timescale of reaction = (L2 x kal)/De(( (5.46) Bi = external mass transfer/intemal mass transfer = (kfx RJ/Dgff (5.47) Bo = convective mass transfer/dispersive mass transfer = (u x k)/D L (5.48)... 

Table 9.1 shows that the convective transport has a timescale on the order of 10-1 s. This time is the time it takes for gas to pass through the cell. Hence, for calculation times T much shorter than this duration, convective transport can be assumed stationary (constant), and Equations (9.10) and (9.11) can be ignored. On the other hand, for time steps At longer than this duration, the quasi-steady formula (where the left hand sides to (9.10) and (9.11) are both zero) can be used. For most cell performance problems, these model equations for the gas phase are normally integrated with the model equation for the sohd components (e.g., cell and interconnect) which is presented next. 

The Sierra Nevada provide an example of the perils of relating erosion rates to relief. They are thought to have persisted since 60-80 Ma with high (1.5 km) relief over long wavelengths ( 50 km), but erosion rates over these timescales are estimated to have been only 0.03-0.08 mm/yr (House et al. 1997, 2001 Clark et al. 2005). Incision rates may have been higher ( 0.2 mm/yr) in the Pliocene (Stock et al. 2004), possibly aided by convective instability at depth (e.g., Ducea and Saleeby 1996), but this is still slow relative to erosion rates in many other tectonically active and inactive regions, and similar to some with much lower relief (e.g., Heffern et al. 2007). 

The amplitude factor will vary on a timescale comparable to the transit time of a scattering element as it convects through the scattering volume. If this volume is characterized by a length scale, L, that timescale is... 

Nevertheless, it is important to refer here to the fact that forced convection alters the electrode response only in the case of this being at a timescale that is long in comparison with the electrode process. For short timescales the (fast) perturbation will be confined to a very short distance from the electrode surface the electrode reaction parameters are not affected by the convection, this being simply a way of achieving good reproducibility. 

Herwig F. (2001) The evolutionary timescale of Sakurai s object a test of convection theory Astrophys. J. 554, L71-L74. 

The first modern isotopic studies of continental lithospheric mantle (CLM) revealed that it must have been isolated from the convecting mantle for bUlion-year timescales (Kramers, 1977 Menzies and Murthy, 1980a Richardson et al, 1984). 

The oxidative capabihty of the atmosphere is not simply a function of chemistry. Convective storms can carry short-lived trace chemicals from the planetary boundary layer (the first few hundred to few thousand meters) to the middle and upper troposphere in only a few to several hours. This can influence the chemistry of these upper layers in significant ways by delivering, e.g., reactive hydrocarbons to high altitudes. Conversely, the occurrence of very stable conditions in the boundary layer can effectively trap chemicals near the surface for many days, leading to polluted air. Larger-scale circulations serve to carry gases around latitude circles on timescales of a few weeks, between the hemispheres on timescales of a year, and between the troposphere and stratosphere on timescales of a few years. 

The region of the atmosphere that is in direct contact with the surface (on a timescale of 1 h or less) is commonly referred to as the boundary layer or mixed layer. Technically, the boundary layer refers to the region of the atmosphere that is dynamically influenced by the surface (through friction or convection driven by surface heating). Less formally, the boundary layer is used to represent the layer of high pollutant concentrations in source regions. The top of the boundary layer in urban areas is characterized by a sudden decrease in pollutant concentrations and usually by changes in other atmospheric features (water vapor content, thermal structure, and wind speeds). 

One of the fundamental difficulties with the whole mantle convection model is that geochemical heterogeneities cannot be sustained over geological timescales. And yet, observations from trace element and isotope geochemistry show that the mantle is and has... 

However, the student should be aware that, when dealing with timescales which are short compared with the convective turnover time, or with layers... 

During the main hydrogen-core burning phase, hydrogen is converted to helium via the CNO-cycle. Consequently the star has a convective core, but this decreases in mass from 20% to 8% of the total mass during main-sequence burning. At the same time, the luminosity increases as hydrogen is depleted (see above) all on a nuclear timescale tnuc 6 x 107 years. 

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