Particle scale

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

Figure 6.2. (a). Colloidal silica network on the surface of spores from Isoetes pantii (quill wort). Scale = 20 pm. (b). Polystyrene networks and foams produced as a biproduct of colloidal latex formation. Both types of colloidal system are typical of the diversity of patterns that can be derived from the interactions of minute particles. Scale (in (a)) = 50pm. 

Fig 2 Immunogold negative staining, with a monoclonal antibody (JIM 5 (13)) that recognises a relatively unesterified pectic epitope, of rhamnogalacturonans extracted from onion cell walls. Arrows indicate 5 nm colloidal gold particles. Scale bar represents 200nm. 

What is actually going on at the particle scale (in terms of, e.g., heat and mass transfer, or mechanical load on particles as a result of particle-particle and particle-impeller collisions) and how are these microscale events affected by the larger-scale phenomena ... 

Ten Cate et al. (2004) were able to learn from their DNS about the mutual effect of microscale (particle scale) events and phenomena at the macroscale the particle collisions are brought about by the turbulence, and the particles affect the turbulence. Energy spectra confirmed that the particles generate fluid motion at length scales of the order of the particle size. This results in a strong increase in the rate of energy dissipation at these length scales and in a decrease... 

The terms Jga and Jsa are the diffusive fluxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to particle-scale turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the... 

In general, measuring beads requires less laser power than measuring cells because of their higher index of refraction (n 1.5 for polystyrene beads vs. n 1.37 for cells).15 The optical force imparted to a particle scales with the difference in index of refraction between the particle and the fluidic medium.16 For bead measurements, we typically operate at a laser power of 2.5 W, whereas for cell measurements the laser is operated at 10 W to obtain similar displacements. These relative power levels are in line with the comparative refractive index differences between the two different particle types and water. 

It is emphasized by several authors that an all-round mathematical model describing the thermochemical conversion process in the conversion zone needs to take both the micro- and the macro-perspective into account [25,26]. The micro-scale perspective in this context will refer to the single particle scale, whereas the macro-scale corresponds to an overall fuel-bed perspective. 

Dispersion in a flow through a porous media occurs due to heterogeneity in the media (i.e., the conductivity of the soil varies with space). This is shown on three levels in Figure 6.15. On the particle scale, a thread of tracer will be split a number of times as it moves through the media. Each split of the tracer thread will move through the media at a speed corresponding to the resistance that it encounters. If you take a number of tracer threads coming out of the media at different times and collected them in an outlet pipe, what you would see at the end of the pipe would be a dispersed... 

Two flow paths on the particle scale move through the media at different rates. 

We define N as the total number of atoms, which is constant, so that the number of particles is N/n. The number of surface atoms on a given particle scales as the square of the particle radius, or as Thus, is proportional to the total surface area and scales as n [second part of Eq. (1)]. The surface energy per area (or the energy per surface atom) depends on the surface radius of curvature. The surface energy is taken to be a constant plus a term that is inversely proportional to the radius of curvature. Thus, the energy per surface atom (p — jl ) scales as C -I- where C and D are constants. Thus, we finally get Psuri = A l... 

Figure 14. An ADF STEM image of a typical catlyst particle. Scale bar 20nm.

As a result of Brownian motion, continual fluctuations of concentration take place on a molecular or small-particle scale. For this reason, the second law of thermodynamics is only valid on the macroscopic scale. 

For front particles at the average position of the diffusion front, the mass M of the front particles scales with radius r as... 

In the preceeding section, we have shown how fluid flows through a packed bed may be observed at various levels, leading to completely different interpretations. Actually, when modelling a transport process, it is always necessary to consider at least two observation levels the bed scale and the particle scale. 

The bed scale corresponds to the whole bed or to a volume containing a large number of particles. That is the level at which we want to derive models for the investigated transport processes. However these processes are generally ruled by gas-liquid-solid interactions occurring at the particle scale. That is the reason why it is necessary to model these processes at the particle scale. The change of scale or volume averaging between both levels is ruled by the percolation process, i.e., by the velocity distribu-... 

To describe the transport processes at the particle scale, we have to adopt a representation of the transport cell which is associated to each bond in the lattice defined by the percolation process (see Figure 3). This cell is assumed to be exactly the same at any position within the bed. The randomness of the process is indeed accounted for by the percolation process, i.e. by the connections between the pores. 

Figure 3. Particle scale modeling of gas-liquid cocurrent downflow through a packed bed.

In this study, we have shown how gas-liquid flow through a random packing may be represented by a percolation process. The main concepts of percolation theory allow us to account for the random nature of the packing and to derive a theoretical expression of the liquid flow distribution at the bed scale. This flow distribution allows us to establish an averaging formula between the particle and bed scales. Using this formula, we propose the bed scale modelling of some transport processes previously modelled at the particle scale. 

We firstconducted a preliminary kinetic study in order to determine the intrinsic and particle scale apparent reaction rates. A powder catalyst was used to determine the intrinsic reaction rate while cylindrical particles were used both for the parti-... 

The particle scale apparent reaction rate has been determined in a discontinuous Carberry type reactor and in a continuous micro-trickle bed reactor in which catalyst particles... 

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