Biot number ratio

Then, the ranges of internal and external heat generation functions P and p f calculated on the basis of these coefficients show that the Biot number ratio is usually large (>10) in gas-solid catalytic reaction systems ... 

This discussion is not applicable to catalytic liquid-solid reactions since the range of values is much lower and the resulting Biot number ratios are also rather low [13, 19] ... 

It has been shown that under realistic reaction conditions, the major transport resistances are external heat transport and internal diffusion. The pellet can be treated as isothermal and the external mass transfer resistance can be neglected. While these conclusions are valid for gas-solid systems, they are not valid for liquid-solid systems as evident from the range of the Biot number ratio (10 -10 ) given in Table 4.2 for the liquid-solid systems. 

The Biot number is essentially the ratio of the resistance to heat transfer within the particle to that within the external fluid. At first sight, it appears to be similar in form to the Nusselt Number Nu where ... 

It will be noted that the relevant characteristic dimension in the Biot number is defined as the ratio of the volume to the external surface area of the particle (V/Ae), and the higher the value of V/Ae, then the slower will be the response time. With the characteristic dimension defined in this way, this analysis is valid for particles of any shape at values of the Biot number less than 0.1... 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

Thomas [209] describes the effect of Biot number on the critical 8 in calculations of runaway temperatures. Biot numbers for right cylinders with various 1/d ratios are available. 

To determine whether the thin body approximation may be used, one should compare the surface heat transfer coefficient, and the thermal conductance of the solid, ksom/8. Their ratio is the Biot number,... 

In order to reduce the complexity of the model two additional simplifying assumptions were made, (a) With typical residence times of 1 second, particle Reynolds numbers of 800 and tube-to-particle diameter ratios of 3, one would expect small values of the wall Biot number thus, a small number of radial finite difference (or collocation) points should be adequate for the numerical solution of the equations (8). (b) It was assumed... 

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) ... 

Figure 15. Ratio of interphase temperature difference versus total intra-interphase temperature difference r r as a function of the observable variable ijDaa (spherical catalyst, ratio of Biot numbers rp = Bim/Bi as a parameter, after Carberry [19]).

In reality, the Biot number for mass transport in most cases is considerably higher than the Biot number for heat transport, i.e. the ratio r is frequently larger than 40-50 [19]. This means that the catalyst pellet can usually be treated as isothermal, at a temperature level which is controlled by the interphase heat transfer resistance. However, this leads to a reduction of the general problem to the case which has already been treated in the previous section. 

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. 

This criterion resembles much those for the temperature gradients on a particle level. The first term again represents the dimensionless activation energy yw, based on the reactor wall temperature Tw. The second term represents the ratio of the heat production rate and the heat conduction rate in radial direction. The last term accounts for the relative contributions of the radial conductivity and the heat transfer at the reactor wall. The latter contains the particle to bed radius ratio and the Biot number for heat transport at the wall, defined as ... 

When Re and dp/dt vary, while both the activation energy and the ratio between the radial heat transfer and heat generation rate at the inlet are kept constant, the values of 7 and T do not vary significantly. On the other hand, since the effect of the Biot number on the error is small, no variations in the difference between models are expected. 

The Biot number can be viewed as the ratio of the convecKon at the surface to conduction within the body. 

When a solid body is being heated by the hotter fluid surrounding it (such as a potato being baked in an oven), heat is first convected to the body and subsequently conducted within the body. The Biot number is the ratio of the internal resistance of a body to heat conduction (o its external resistance to heat conveetton. Therefore, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body. 

Note that the Biot number is tlic ratio of the convection at the surface to conduction within the body, and this number should be as small as possible for lumped system analysis to be applicable. Therefore, small bodies with high tlieniial conductivity are good candidates for lumped system analysis, especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless. Thus, Ihe hot small copper ball placed in quiescent air, discussed eailier, is most likely to satisfy the criterion for lumped system analysis (Fig. 4-6). 

The Biot number can be thought of as die ratio of (n) The conduction thermal resistance to the convective thermal resistance. 

Biot number, Bi, is the ratio of the mass transfer resistance inside the emulsion globules to that of the external phase [5] and is given by Equation 25.1 ... 

Further, we notice that the ratio of dc to Se provides the Biot number. That is. 

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

Interpreting Lo/X as the specific thermal conduction resistance of the solid and l/a as the specific heat transfer resistance at its surface also allows the Biot number to be interpreted as the ratio of the two resistances... 

The Biot number is decisive for the ratio of the thermal resistance of the cylinder to that of the air... 

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). 

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