Bulk diffusion path

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

By comparing the relative magnitude of the mean free path (z) and the pore diameter (27), it is possible to determine whether bulk diffusion or Knudsen diffusion may be regarded as negligible. Using the principles of the kinetic theory... 

From an order-of-magnitude analysis, when the mean-free path of a molecule is less than 0.01 times the pore radius, bulk diffusion dominates, and when it is greater than 10 times the pore radius, Knudsen diffusion dominates. This means that Knudsen diffusion is significant when the pore radius is less than about 0.5 fim. For reference, a typical carbon gas-diffusion layer has pores between 0.5 and 20 /rm22 229 in radius, and a microporous layer contains pores between 0.05 and 2 Thus, while Knudsen... 

Bulk path at moderate to high overpotential. Studies of impedance time scales, tracer diffusion profiles, and electrode microstructure suggest that at moderate to high cathodic over potential, LSM becomes sufficiently reduced to open up a parallel bulk transport path near the three-phase boundary (like the perovskite mixed conductors). This effect may explain the complex dependence of electrode performance on electrode geometry and length scale. To date, no quantitative measurements or models have provided a means to determine the degree to which surface and bulk paths contribute under an arbitrary set of conditions. 

These are depicted schematically in Figure 18.4 in the case of metal A deposited on metal B. Bulk diffusion, as noted above, is the transfer of B into A or A into B through the crystal lattice. This is characterized by the coefficient D in the figure. Defect path diffusion is the migration along lattice defects such as grain boundaries, characterized by the coefficient D in the figure. Ordered A B, possible phases are indicated between the metals. Finally, Kirkendall void porosity is indicated and will be expected to be present if the interdiffusion rates from one metal to the other are not equal in both directions. 

Fig. 1. Model of subarachnoidal space, CSF flow, and molecular flux (N1). After CSF production in choroid plexus of the ventricles (1,2,3), CSF passes the aperture (4,5), reaches the cistemae (6-9), and divides into a cortical and a lumbar branch of the subarachnoidal space. Finally, CSF drains through the arachnoid villi into venous blood. The illustration represents an idealized cross section through the subarachnoid space. Molecules diffuse from serum with a concentration C(ser) flu ough tissue along the diffusion path x into the subarachnoid space with a concentration C(csF)- Th molecular flux J depends on the local gradient Ac/Ax or dddx and the diffusion constant D. The CSF concentration increases with decreasing volume exchange, i.e., decreasing CSF volume bulk flow (F= 500 ml/day). The flow rate of a molecule in CSF is r= FIA, where A is the varying cross section of the subarachnoid space.

Knudsen Diffusion Only Is Occurring. For a very fine pore material in which the effective pore diameter is less than the mean free path of the molecules, bulk diffusion and Poiseuille flow do not occur. For this case, the change in volume given when C + CO2 —> 2CO has no influence on the rate of diffusion of carbon dioxide into the rod, and is not dependent on the total pressure in the pores. Considering a wedge of carbon (Fig. Al),... 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

Bulk diffusion coefficients in binary gas mixture are almost independent of the ratio of components of the mixture. Therefore, it was supposed that if diffusion in the measurements described above is of the bulk type, i.e., the free path of molecules is much lesser than the diameter of pores, then the first gas diffuses into the second gas at the same rate as the second gas diffuses into the first. 

In industry large pellets of a catalyst were employed (e.g., 6-8 mm in size), and the rate of the process was essentially affected by the slowness of the diffusion of ammonia in the pores of the catalyst the efficiency factor at this size of pellets is about 0.5. The effect of diffusion retardation of the ammonia synthesis was studied both at high pressures (99), when the free path of molecules is much smaller than the radius of catalyst pores so that the bulk diffusion is operative, and at pressures near to 1 atm (116), where there is a transition from the bulk to the Knudsen diffusion. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

Under this condition, there is complete depletion of O2 at the electrode next to the porous layer of Fig.6a or inside the cavity of Fig.6b. The constant depends on the diffusion constant of O2 (in its particular carrier gas), Dq, and the geometrical characteristics of the diffusion barrier. In the device of Fig.6b (sensor with integral cavity), the diameter of the aperture C (usually greater than 50 microns) is much larger than the mean free path of the gas molecules at 1 atm (about 1 micron) and bulk diffusion dominates. In this case(ll-12). D0 - K Ta/P and cr - (DgA)/(kTd), where K] is a constant, P is the absolute pressure, a is a constant having a value between 1.5 and 2 and A and d are the cross-sectional area and length of the aperture C. Representative values for D0 are about 1.5 cm2/s at 700 °C and 0.15 cm2/s at 20 °C. Since Pg — cP with c the percentage of O2 molecules in the gas, we have... 

In the device of Fig.6a (sensor with distributed cavity), depending on the dimensions of the pores of the diffusion barrier, the oxygen diffusion can be bulk diffusion or Knudsen diffusion(ll). In the former case, the sensor output (limiting current Ig) is given by Eq. (9). Knudsen diffusion occurs when the average pore diameter is much smaller than the mean free path of the gas molecules, in which case collisions between molecules and pore walls are the dominant events. In this case(121. Dq -k T /2,and... 

It should be emphasised that the main kinetic equations presented in this chapter hold for any mechanism of transfer of the atoms across the bulk of a growing compound layer since the assumption the longer the diffusion path, the greater the time to overcome it is clearly always fulfilled. A knowledge of the details of this mechanism is only important when establishing a relationship between the transport properties of the layer of a given phase and the diffusion characteristics of its components. 

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