Solid phase Biot

The recent experimental study by Melanson and Dixon [13] provides estimates of k g and the solid phase Biot number Big for spheres, cylinders and rings in beds of low tube to particle diameter ratio. It also throws up important questions about the mechanisms of solid conduction and wall heat transfer in such beds which seem to defy current theories. 

However, the Nusselt number refers to a single fluid phase, whereas the Biot number is related to the properties of both the fluid and the solid phases. 

Constant diffusivity (D ) and the solid-liquid mass-transfer coefficient are assumed for QX and QY. The model equations are nondimensionalized in terms of the Thiele parameter (f>, Biot number for mass transfer Bi, and nondi-mensional time and distance. An important conclusion from the subsequent analysis of the model simulations is the importance of the solid phase on the conversion of the organic substrate in the organic phase. Results of their simulation are shown in Figure 11. It can be noticed that at low (f>, corresponding to low diffusional limitations, the overall organic reagent conversion is lower than at higher values o <. This result is the exact opposite of what is observed in analysis of... 

Soil is a mixture of a solid phase, a liquid phase (i.e., water) and a gaseous phase. Saturated soil is a two-phase material which consists of a solid phase and a liquid phase. The formulation of a theory for a saturated soil commenced with the concept of the effective stress of Terzaghi at the beginning of the twentieth century, and Terzaghi s concept was extended by Biot to a three dimensional consolidation theory. 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model 1561 can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model 1541 should be used. 

The fluid-phase temperatures predicted from a two-phase pseudo-homogeneous model were shown to give reasonable agreement with experimental measurements, without explicitly adjusting the model parameters. It was demonstrated that more refined experimental measurements will be needed to determine the parameters of the model in particular, the solid and fluid phase wall Biot numbers were mutually adjustable. 

Biot number for mass transfer concentration of species i,kmol/m liquid-phase diffusivity,m7ls effective diffusivity within solid or catalyst phase, Damkohler number... 

Generally, the solids are not structurally homogeneous, but the solid and liquid nevertheless will be called phases and leaching will be treated as a two-phase, mass transfer process- The solid consists of a matrix of insoluble solids, the mare, and the occluded solution. It may also contain undissolved solute and a nonextractable secondary phase, for example, coffee oil in water-soaked coffee grounds. This secondary phase is treated as part of the mate. Dimensionless parameters that can affect solnta transfer include the solute equilibrium distribution coefficients, m and M tha Pick number, v the strippirg factor, a the Biot number, Bi and the Peclet number, Be. These parameters are defined more precisely in the Notation section. 

Variation of phase velocity over a range of solid-volume percentages are calculated for the above models for two types of particles, i.e., glass beads and kaolins (with acoustic impedances of 21.12 x 10s and 10.66 x 105 g/cm2-s, respectively). Calculated results are shown in Fig. 5-27 for glass beads and in Fig. 5-28 for kaolins. All models, except Biot-2, show decreasing phase velocity at lower volume fractions, then increasing phase velocity at higher volume fractions. 

In Biot formalism, there exist several assumptions that restrict its generality and make true liquid-solid coupling impossible. Biot assumed that for a REV in a multiphasic porous medium, a single energy functional could be stipulated to define the energy state. It has been shown that for N continuous contiguous phases, N functionals are needed to fully describe behavior. (For example, simultaneous countercurrent flow of two immiscible liquids is evidence that at least two separate energy functionals are needed.)... 

Because of the constant porosity assumption and a single energy functional, it is impossible to stipulate wave attenuation in Biot wave mechanics from first principles of thermodynamics, which should be part of any complete theory. Biot approached wave attenuation (not spatial spreading) empirically, rather than accounting for energy losses through compression and rarefaction cycles of the solid and liquid phases. 

Lan] studied transfer processes and phase boundary reactions during the carbon exchange between the carbon containing gas phase and solid Cu-Fe alloy with 4 at.% Cu at 900 to 1000°C in H2-CH4 mixtures. The diffusion coefficient, transition coefficient and the Biot index were determined. 

Now we need to express s in terms of the stress in the liquid using a constitutive equation. Various authors have does this by using empirical (nonlinear elastic) equations [24,78], or by assuming elastic behavior with the solid and liquid phases compressible [63,58] or incompressible [79,64], or by allowing the network to be purely viscous [67] or viscoelastic [73,59,60,61,80] Biot [61] allows for anisotropy of the mechanical properties as well as for viscous dissipation within both the solid and liquid phases. 

The Sherwood number can be viewed as describing the rabo of convective to diffusive transport and finds its counterpart in heat transfer in the form of the Nusselt number. It is high ( 1) when flow is turbulent or the boundary layer "film" is very thin. The Biot number has the same form as the Sherwood number but refers to two adjacent phases or media. In one of these, which we term internal, transport is usually by diffusion. This can be in a gas bubble, a liquid drop, a porous solid parbcle, or some other enbty. The adjacent ("external") phase is a liquid or a gas in relative motion to the particle and has an attendant "film resistance." Like the Sherwood number, Bi is high ( 1) when the external phase is in turbulent motion (caused, for example, by stirring) or the boxmdary layer is very thin. We will have more to say about it in Illustration 5.1. 

The presence of a liquid phase and a liquid-solid interface in multiphase reactors results in added transport resistances. For instance, the effective diffusivity in liquid-filled pores (of the order of 10 to 10 cmVsec) is much smaller than that in gas-filled pores (of the order of 10 cmVsec). The solubility of the gaseous reactant is an important factor since the gaseous reactant has to be dissolved into the liquid reactant for the reaction to take place on the catalyst surface. As emphasized in Chapter 4, the Biot number for heat is much larger than the Biot number for mass for liquid-solid systems the opposite is true for gas-solid systems. Therefore, the major external resistance lies in the mass transport, and the pellet is not necessarily isothermal. In many cases, however, the equilibrium gas concentration in the liquid is quite low and, thus, the heat evolved is small in spite of high heats of reaction. The pellet can be considered isothermal in such a case,... 

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