Biot diffusion

The last three pi groups are well known in chemical engineering (113 is recognized as the Fourier reaction number (Fo ), 114 is the famous Biot diffusion number (Bi(j) and Hs is the Sherwood number (Sh)). 

These expressions can also be used for the case of external mass transfer and solid diffusion control by substituting D, for 8pDpi/( p + ppK)) and/c rp/(ppK)D,i) for the Biot number. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

The relative importance of the boundary-film and intra-pellet diffusion in mass transfer is measured by the Biot number. On the assumption that there is no accumulation of adsorbate at the external surface of a pellet, then ... 

Mass Biot hpyil D hm mass transfer coefficient D mass diffusivity... 

Ma et al. [104] attributed a decrease in diffusivity with an increase in initial concentration to pore diffusion effects. Because zeolites are bi-dispersed sorbents, both surface and pore diffusions may dominate different regions. In micropores, surface diffusion may be dominant, while pore diffusion may be dominant in macropores. This, therefore, supports the use of a lumped parameter (De). To explore further the relative importance of external mass transfer vis-a-vis internal diffusion, Biot number (NBl — kf r0/De) was considered. Table 9 summarizes the NBi values for the four initial concentrations. The NBi values are significantly larger than 100 indicating that film diffusion resistance was negligible. 

The Biot number Bim for mass transport. This can be interpreted as the ratio of internal to external transport resistance (intraparticle diffusion versus interphase diffusion) ... 

From this figure, it can be concluded that the reduction of the effectiveness factor at large values of becomes more pronounced as the Biot number is decreased. This arises from the fact that the reactant concentration at the external pellet surface drops significantly at low Biot numbers. However, a clear effect of interphase diffusion is seen only at Biot numbers below 100. In practice, Bim typically ranges from 100 to 200. Hence, the difference between the overall and pore effectiveness factor is usually small. In other words, the influence of intraparticle diffusion is normally by far more crucial than the influence of interphase diffusion. Thus, in many practical situations the overall catalyst efficiency may be replaced by the pore efficiency, as a good approximation. 

Similar results are obtained when, in addition to intraparticle diffusion, interphase mass transfer is considered. Instead of eq 51, in this case eq 60 may be substituted into eq 123, which then introduces the Biot numbers for mass transport of the reactants A and A2 as additional parameters. For large values of x and (i.e. >10) and, after some rearrangement,... 

If we assume that the Biot numbers of the two species are roughly the same, we note from eq 170 that when the ratio Bim/fa is sufficiently large (i.e. compared to AAcl/2 and to unity), indicating that interphase diffusion effects are not likely to influence the effective reaction rate, then, with c2,o = 0, eq 170 essentially transforms to eq 167. However, if this is not the case, the overall selectivity will be further reduced with decreasing value of Bim/fa. 

Figure 24. Variation of the apparent selectivity with conversion for a Type III reaction. Comparison of the results obtained under kinetic and diffusion control (isothermal conditions, intrinsic selectivity factor Ak = 4, equal Biot numbers Bim, — Bim2, initial concentration C2,o = 0).

S/m and fl/ x = Biot numbers for mass and heat transfer 4 and 4 x = Thiele modulus Le = Lewis number A0 i = dimensionless adiabatic temperature rise t) = effectiveness factor kg =mass transfer coefficient (ms-1) Rp = radius of catalyst pellet (m) Da = effective diffusion coefficient (ms-2) r =rate of reaction (molm-3s-1) C —concentration of reactant (molm-3) ... 

In the set of relations (3.182)-(3.188), P represents the coefficient for the velocity increase due to the species transport through the wall, Bi is the heat transfer Biot number (Bi = (arj)/ ), Bip is the mass transfer Biot number for the gaseous phase (Bi[) = (kri)/DA) and Bip is the Biot number for the porous wall (Bip = (k5xx,)/DAw)- Two new parameters and D w, respectively, represent the wall thickness and the wall effective diffusion coefficient of species. The model described by the set of relations (3.182)-(3.188) can easily be modified to respond to the situation of a membrane reactor when a chemical reaction occurs inside the cylindrical space and when one of the reaction products can permeate through the wall. The example particularized here concerns the heat and mass transfer of a... 

A diffusion controlled film model that accounts for botii the continuous phase and membrane phase resistance in the form of a Biot number. 

According to Sontheimer s theory [41], two typical shapes of breakthrough curves may exist. In the case of the porous diffusion predomination (carbon F - 100) curve of breakthrough is vertical and exhibits convex shape to X - axis or when layer diffusion predominated (carbon N) concave shape of elongated S letter (it is marked by BIOT (Bi) - number). It was stated, that elongated shape of breakthrough curve is connected with dilution of adsorption face, what makes the achievement of equili-birium state difficult. Thus, adsorption described by curve of vertical shape is most convenient. 

Biot number The Biot number characterizes the rate of the film mass transfer kinetic in the general rate model, Bi = (kfdp)/(ICpDp), with kf the fUm mass transfer coefficient, dp, the particle size, and Dp the diffusion coefficient inside the particles. 

A general experimental result is the difference between measured rates of diffusion in macroscopic experiments and measurements of self-diffusion by spectroscopic techniques such as gradient field NMR [80-84]. The difference between the microscopic measurement and the macroscopic experiment is desorption and reentry of molecules in zeolite microcrystallites. In this respect, it is important to remember the Biot condition, which states the condition when the measured rate of diffusion is independent of the rate of desorption ... 

Biot, M. A., 1984. New variational-Lagrangian irreversible thermodynamics with application to viscous flow, reaction-diffusion, and solid mechanics. Advances in Applied Mechanics 24, 1-91. 

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... 

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

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

The Biot s number gives the ratio between the diffusion resistance in the fluid film and in the catalyst particle. Usually BiM 1 for porous catalyst particles. Equation (9.152) can now be rewritten as... 

Consider mechanical excitation frequencies in a liquid-saturated porous medium (Fig. 10). Two formalisms were conventionally used for high frequencies, Biot-Gassmann wave mechanics (Biot 1956) is used for low frequencies, Darcy theory is used (inertial effects assumed negligible). However, there clearly must be approximately three orders of magnitude where both diffusion and... 

Biot and Darcy theory shortcomings have been largely overcome by development of a coupled diffusion-dynamic formalism (de la Cruz et al. 1993, Spanos 2001, Spanos et al. 2(X)3). Porosity is treated as an explicit thermodynamic variable, so that dnumerical model development. Nevertheless, if they are solved subject to the assumption of the incompressibility of a liquid saturant, the existence of a slow wave is predicted. It is called the porosity dilation (PD) wave it is not a strain wav, it is a coupled liquid-solid displacement wave, and it has some interesting properties. 

When fluids can seep through pores, interacting mechanically with the solid skeleton, the material is composed of more than one constituent thus we need to use a mixture theory in which we could clearly make out each part filled by different constituents on a scale which is rather large in comparison with molecular dimensions so we put forward a new continuum theory of an immiscible mixture consisting both of a continuum with ellipsoidal microstructure (the porous elastic solid) and of two classical media (see, also, the conservative case examined by Giovine (2000)). In accordance with Biot (1956), we consider virtual mass effects due to diffusion we also introduce the microinertia associated with the rates of change of the constituents local densities, as well as the one due to the deformation of the pores close to their boundaries. 

ABSTRACT With the increase of mine exploitation depth and appliance widely of large-scale full-mechanized equipment, coal block gas emission has been one of the most gas effusion source. Base on unsteady diffusion theory and mass transmission fundamental, the mathematical and physical model of gas diffusion through coal particles with third type boundary condition was founded and its analytical solution was obtained by separate variableness method. The characteristics of gas through coal particles was analyzed according as mass transmission theory of porous material. The results show that the Biot s criterion of mass transmission can reflect the resistance characteristic of gas diffusion and the Fourier s criterion of mass transmission can represent the dynamic feature of diffusion field varying with time. 

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