Biot condition

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 ... 

To illustrate the use of Elq. (4.1) in zeolite catalysis, we will discuss the results of theoretical and experimental studies of the hydroisomerization reaction of hexanel L In order to exclude diffusion limitation, two conditions have to be met. First, the Biot condition must be satisfied when the rate of desorption is limiting ... 

Analytical solutions of equation 9.44 in the form of infinite series are available for some simple regular shapes of particles, such as rectangular slabs, long cylinders and spheres, for conditions where there is heat transfer by conduction or convection to or from the surrounding fluid. These solutions tend to be quite complex, even for simple shapes. The heat transfer process may be characterised by the value of the Biot number Bi where ... 

H.,2010. Kinetic model for polyhydroxybutyrate (PHB) production by Hydrogenophaga pseudoflcwa and verification of growth conditions. Afr. J. Biot. 9, 3151-3157. 

Isothermal and adiabatic heat transfer conditions can be obtained with different values of the Biot number. The wall temperature, 0w, is assumed to be a piecewise linear function of the axial position and is treated as a known quantity based on experimental evidence. 

The geometry of the tubes allows the heat transfer being considered one dimensional, and each tube to be a lumped system in front of the ambient air. This two conditions are fulfilled when Bi < 0.1 (Biot number Bi = a /(/(2a ), where R is the radius of the sample, X its thermal conductivity and a the heat transfer coefficient between the tube and the environment). Once the temperature-time curves of the PCM and the reference substance are obtained (Figure 160), the data can be used to determine the thermophysical properties of the PCM. 

Modeling of the packed bed catalytic reactor under adiabatic operation simply involves a slight modification of the boundary conditions for the catalyst and gas energy balances. A zero flux condition is needed at the outer reactor wall and can be obtained by setting the outer wall heat transfer coefficients /iws and /iwg (or corresponding Biot numbers) equal to zero. Simulations under adiabatic operation do not significantly alter any of the conclusions presented throughout this work and are often used for verification... 

The field fulfills the condition V = 0. This result is consistent with the fact that the curve Cj is a closed loop. The integral written above is identical to Biot-Savart law used to calculate the magnetic field generated by a current loop, provided this integral is multiplied by 4nl/c. 

The absence of magnetic monopole implies the conditions V = 0 and B = -Vx = 0, which are consistent with the relations given above, since B =VAA =0. However, as with the A vector potential, the equality Bx = = Vx / 0 enables us to define a scalar potential x that can be calculated on the basis of Biot-Savart law for a filiform (filament-shaped) circuit... 

Many substances can rotate the plane of polarization of a ray of plane polarized light. These substances are said to be optically active. The first detailed analysis of this phenomenon was made by Biot, who found not only the rotation of the plane of polarization by various materials (rotatory polarization) but also the variation of the rotation with wavelength (rotatory dispersion). This work was followed up by Pasteur, Biot s student, who separated an optically inactive crystalline material (sodium ammonium tartrate) into two species which were of different crystalline form and were separately optically active. These two species rotated the plane of polarized light equally but in opposite directions and Pasteur recognized that the only difference between them was that the crystal form of one was the mirror image of the other. We know to-day, in molecular terms, that the one necessary and sufficient condition for a substance to exhibit optical activity is that its molecular structure be such that it cannot be superimposed on its image obtained by reflection in a mirror. When this condition is satisfied the molecule exists in two forms, showing equal but opposite optical properties and the two forms are called enantiomers. 

A number of authors [46 to 48] employ the single sphere model in which the packed bed is considered as a set of equal spheres that are under the same state of extraction, and the fluid flowing around them is solute-free. That is, equation (3.4-90) would be valid, but without the generation term [46], The transport at the solid-fluid interface obeys the boundary condition (Eqn. 3.4-94) with C = 0 (fluid-flows at a large velocity). Under these assumptions, there is an analytical solution to the above problem (without axial dispersion) in terms of the Biot number (Bi = k, R/De), included in the following equation ... 

As a fifth attempt, an increase of the heat transfer at the wall in the Thomas model is not practicable and would not be efficient, since the major part of the resistance to heat transfer is the conductivity in the product itself, as shown by the high value of the Biot criterion, 300, which is closer to Frank-Kamenetskii conditions than to Semenov conditions. 

The parameter R is applicable for the case of instantaneous change in surface temperature (infinite h) for conditions of rapid heat transfer R is for a relatively low Biot modulus ( jl< 2) for conditions of slow heat transfer R" is for a constant heating or cooling rate.88 defines the minimum temperature difference to produce fracture under conditions of infinite heat-transfer coefficient, i.e. A = 1. The parameter Ris inversely proportional to a. Alow value of a is therefore essential for good thermal stress resistance. The coefficient of thermal expansion normally increases with increasing temperature however, thermal conductivity decreases. 

For suspension-to-gas (or bed-to-gas) heat transfer in a well-mixed bed of particles, the heat balance over the bed under low Biot number (i.e., negligible internal thermal resistance) and, if the gas flow is assumed to be a plug flow, steady temperature conditions can be expressed as... 

However, the boundary condition at the external pellet surface is now defined by eq 36 instead of eq 38. As a consequence, a different expression for the integration constant C results, which is not only a function of the Thiele modulus , but also depends on the Biot number for mass transport Bim. Hence, a complete characterization of this problem already requires two parameters. 

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).

For Rep < 100 and 0.05 < rp/r, < 0.2, wall Biot numbers range between 0.8 and 10 [28], so this means that wall effects cannot be neglected a priori [38]. Also this criterion contains procurable parameters. For the wall heat transfer coefficient hw and the effective heat conductivity in the bed Abc(r, the correlations in Table 2, eqs. 44-47 can be used [8, 39]. These variables are assumed to be composed of a static and a dynamic (i.e. dependent on the flow conditions) contribution. Thermal heat conductivities of gases at 1 bar range from 0.01 to 0.5 Js m l K l, depending on the nature of the gas and temperature. 

The limits of the Biot heat number Bi are 0.01 and 50 [8], so it will depend on the particular conditions which criterion is the most severe. In the laboratory reactors it is often <1. It is obvious that decreasing the particle size will shift the largest gradient to the film layer around the particle. 

Table 1 gives a summary of calculated times for 50% extraction of solute (t 50%)- Simulations conditions are P = 20 MPa, T = 313 K and Re = 40 for the system DCB/small cylinders and downflow operation. From Table 1, it is seen that for a constant value of Kq, a decrease of De and K lead to a significant increase of 150% irrespective of the Biot number. On the other hand, the influence of Kq is found to be significant only in the higher range of Biot number (Bi = 250) where the external mass transfer gradient may become limiting (Kq is very small). 

From the analysis of Equation 18> it follows that the main variables that affect the error in the reaction rate are E and P due to their effect on T and TC Thus, very good responses are obtained from a one-dimensional model when reaction conditions are mild (moderate values of E and P). It can also be seen that for these conditions, the influence of the distribution of the radial heat transfer resistances between the bed and the wall, given by the Biot number, is small. E.g., for T = 673°K, Tw = 643°K and E = 12.5 kcal/mol, the maximum er, found for Big -> < is 2.8%. 

At the inflow and at the top of the computational domain one calculates the flow variables, as induced by the freestream vortex via Biot-Savart interaction rule. At the outflow, fully developed condition is applied for the wall-normal component of the velocity ( = 0) and using the same in SFE, one can obtain the vorticity boundary condition at the outflow from Equation (3.4.2). At the top frame of Fig. 3.8, one sees incipient unsteady separation on the wall. In subsequent frames, one notices secondary and tertiary events induced by the primary instability. In these computed cases, one does not notice TS waves and the vortices formed on the wall are essentially due to unsteady separation that is initiated by the freestream convecting vortex. These ensemble of events have been noted as the vortex-induced instability or bypass transition in Sengupta et al. (2001, 2003), Sengupta Dey (2004) and in Sengupta Dipankar (2005). 

There is a simple method for speeding the calculation of conversion that should be pointed out. After the conditions in the upstream end of a reactor have been calculated, including a region well beyond the point where the temperature is a maximum, the conditions in the rest of the reactor can be calculated accurately enough with only 2 or 3 radial increments, or, if the Biot number is not too large, with the one-dimensional approximation. In this region, the temperature has no intrinsic interest, and it is only necessary to estimate the average conversion. 

The Biot numbers of heat and mass transfer could be obtained from the boundary conditions of a third kind ... 

This indicates that one part of the heat does not flow directly from wall to fluid. A longitudinal heat flow exists and Agostini [30] and Commenge [31] give a rule to estimate whether or not the conditions required for a purely transversal heat flow are fulfilled. They define a Biot number which allows us to compare the convective heat flow and the conductive longitudinal heat flow. The former gives the definition... 

In the thermal conduction theory, such a distribution in general is thought to be caused on condition that the rate of thermal conduction in the self-heating solid chemical placed in the atmosphere under isothermal conditions is far less than the rate of heat transfer from the solid chemical through the whole surface to the atmosphere. In other words, this condition is expressed as [/> > A, which is equivalent to that the Biot number takes a large value. 

Besides, it is self-evident, as stated in the preceding section, that the spatial distribution of temperature, in particular, in the early stages of the self-heating process, or of the oxidatively-heating process, in a small-scale chemical of the TD type, including every small-scale gas-permeable oxidatively-heating substance, subjected to either of the two kinds of adiabatic tests, is the very ultimate of the Semenov model, because the condition, the Biot number = Ur A = 0, holds strictly in such a chemical. 

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