Particle surface gradient

The magnitude of charge on a particle can be expressed in any one of several ways total charge (Jp), surface charge density (i pj), specific particle surface gradient or field (Sps charge-to-mass ratio (J c), or specific particle potential (Eps). These are related by the following identities ... 

Fio. la. Relationship between specific particle surface gradient and charge-to-mass ratio for various particle diameters. 

In order to provide a better means for visualizing the physical significance of both the specific particle surface gradient, 8ps, and the specific particle... 

Pig. 6. Maximum stable charge on particles expressed as specific particle surface gradients. 

Maximum field intensity, V/meter Sps Specific particle surface gradient for particle of size Dp, V/meter, Bp/TreS Dp2... 

Ga.s-to-Pa.rticle Heat Transfer. Heat transfer between gas and particles is rapid because of the enormous particle surface area available. A Group A particle in a fluidized bed can be considered to have a uniform internal temperature. For Group B particles, particle temperature gradients occur in processes where rapid heat transfer occurs, such as in coal combustion. 

When the mass transfer resistance within the particle is significant, a concentration gradient of reactant is established within the particle, with the concentration, and hence the reaction rate, decreasing progressively with distance from the particle surface. The overall reaction rate is therefore less than that given by equation 10.195. 

Next, consider the gradients of temperature. If the reaction is exothermic, the center of the particle tends to be hotter, and conversely for an endothermic reaction. Two sets of gradients are thus indicated in Figure 8.9. Heat transfer through the particle is primarily by conduction, and between exterior particle surface (Ts) and bulk gas (Tg) by combined convection-conduction across a thermal boundary layer, shown for convenience in Figure 8.9 to coincide with the gas film for mass transfer. (The quantities T0, ATp, A7y, and AT, are used in Section 8.5.5.)... 

D(external surface)(gradient at the inlet to the pore) k(volume of the particle)Cg... 

Unlike charges attract and like charges repel each other, so there is a high concentration of counterions attracted to the particle surface whilst co-ions (those with the same sign charge as that of the surface) are repelled. Thermal motion, i.e. diffusion, opposes this local concentration gradient so that the counterions are in a diffuse cloud around the particle. Of course particles which have a like charge will also repel each other but the interaction of the particle surfaces will be screened by the counterion clouds between the particles. The interaction potential is a function of the surface potential, i]/o, and the permittivity of the fluid phase, e = r80, where r is the relative permittivity.12,27... 

Diffusion. The transport process may consist of two parts, diffusion and convection. When the liquid is stagnant and resting relative to the particle the transport is done by diffusion only. A steady state is quickly established in the solution around the particle (4 ). (Strictly it is a quasi-steady state since the particle is growing ( 5)). At the particle surface the concentration gradient becomes equal to (c-cs)/r, which leads to the growth rate... 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

We denote the stress tensor inside the curly brackets in Eq. (13) asT. Equation (13) shows that the solution for the potential and its gradient at the particle surface are all that are required to calculate the force on a particle via the linearized Poisson-Boltzmann equation. 

Even with an adequate description of molecular velocities near the particle surface, it is not possible to completely establish all variables influencing thermal force. This is because there also exists a so-called thermal slip flow or creep flow at the particle surface. Reynolds (see Niven, 1965) and others have pointed out that as a consequence of kinetic theory, a gas must slide along the surface of a solid from the colder to the hotter portions. However, if there is a flow of gas at the surface of the particle up the temperature gradient, then the force causing this flow must be countered by an opposite force acting on the particle, so that the particle itself moves in an opposite direction down the temperature gradient. This is indeed the case, known as thermal creep. Since the velocity appears to go from zero to some finite value right at the particle surface, this phenomenon is often described as a velocity jump. A temperature jump also exists at the particle surface. 

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