Biot’s number

The dimensionless quantity, R km/Dei, is called the Biot s number for mass transport ... 

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... 

In Eqs. (4.77) and (4.78) A is the thermal conductivity of the material of the hollow cylinder, a the coefficient for convective heat transfer, and Bi Biot s number on the inner (i) and outer (a) surfaces of the hollow cylinder, respectively, with... 

During each simulation time step the temperature is uniform in each particle (Biot s number well below unity). 

Coefficient is a dimensionless quantity its value can vary between one and zero. The more ( differs from unity, the greater is the inequality in the field of temperature. This coefficient is a function of Biot s number Bi = La/X. Coefficient y/ decreases to zero as constant Bi increases to infinity. Let us assume that the same body is observed under various conditions of heat transfer from the body to the environment and that as a result there are various values of the surface heat transfer coefficient a. For the sequence of increasing values of a... 

The uptake curves according to Eq. (5-47) are given in Fig. 5.6.a with Biot s number as a parameter (Suzuki and Chihara,1982). 

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. 

Using the appropriate equations and correlations of Sections 3.3.5 and 3.3.6, we can calculate the power consumption per unit volume of liquid and thus we can have an approximation of the actual mass transfer coefficient in the liquid film. N = 6 for this type of impellers and tlius k is about 0.1 cm/s. It is obvious that k is about 20 times the minimum value used above for the Biot number. 

H [reciprocal of modified Lewis number], s[Biot number] and the equilibrium parameter p. 

Figure 9 illustrates the effect of the Biot number, s, on the uptake curve for a given r) and p. Figure 10 shows the corresponding dimensionless temperature profiles at x = 0. It may be seen from Figure 9 that s has small effect on the uptake curve at low t but it strongly affects the... 

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 these parameters s designates some characteristic dimensions of the body for the plate it is the half-thickness, whereas for the cylinders and sphere it is the radius. The Biot number compares the relative magnitudes of surface-convection and internal-conduction resistances to heat transfer. The Fourier modulus compares a characteristic body dimension with an approximate temperature-wave penetration depth for a given time r. 

K N Nb Speed of agitator Number of tubes in a vertical row or number of tubes in a bundle Ni for number of baffles Np for total number of tubes in exchanger Nc for number of tubes in one cross-flow section N o for number of cross-flow rows in each window Biot number, hp Ax/k rad/s r/h... 

Dimensionless distance from the center Dimensionle.s.s heat transfer coefficient (Biot number) Dimensionless time (Fourier number)... 

This agrees with (2.199), because for the plate of thickness 2R, V/A = R. The same method can be used to show that for other shapes of solids (2.199) agrees with the exact solution for Bi = 0, corresponding to A — oo, with q 0, If however, with a finite value for A the Biot number would be zero when a = 0, it follows from (2.171) that = 1. The insulated plate keeps its initial temperature o, or in other words no equalization of the temperatures o and s takes place. 

The time dependent difference A + from (2.201) assumes a maximum value A +ax for each body for a given Biot number, and this maximum appears at a certain time. This maximum deviation of the approximate solution has been calculated for different solid shapes. For Bi = 0.1 the maximum difference lies without exception at less than 2% of the characteristic temperature difference o — s- The Biot number in this case was evaluated with the cylinder or sphere radius R as its characteristic length (Bi = aR/A). In parallelepipeds and prisms the characteristic length was taken to be half of the smallest side length 2X. In bodies whose dimensions X, Y, Z or R and Z do not differ greatly, the maximum error from the approximation solution lies around 1% of o — s The error increases rapidly with larger Biot numbers. The error of less than 2% of o — s calculated for Bi = 0.10 could presumedly be tolerated. 

Here, the Biot number Bi = aa/U provides a measure of the relative importance of kinetic desorption relative to interfacial convection, cs and cs are the bulk-phase concentrations evaluated in the limit as we approach the surface of the drop, k = c,Xj/a is a measure of the ratio of adsorption to desorption to the exterior fluid and k = c.Xj /a is the same quantity for the drop. We recall that S and a are the adsorption and desorption rate constants defined following (2-150). 

---