Class A geometries

For three special geometries, the Laplacian operator can be written in a simpler form. These are the class A geometries the infinite slab, the infinite circular cylinder, and the sphere. For these we can write... 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

The whole procedure to perform the adiabatic self-heating test by means of the closed cell is explained in detail in Chapter 6, in connection with the procedure to calculate the T for a powdery chemical of the TD type, having some one of several specific shapes including the class A geometries as well as an arbitrary size, confined in a fiber drum, and placed in the atmosphere under isothermal conditions. 

Individual values of for the several specific shapes including the so-called class A geometries... 

The infinite slab, the infinite circular cylinder and the sphere were called the class A geometries by T. Boddington et al. As stated repeatedly above, the values of dc for the several specific shapes including the class A geometries have already been calculated by T. Boddington et al., respectively. Table 13 is quoted from page 486 of their paper [58]. 

It is seen in Eq. (79) that, once the value of in addition to the heat generation data, a and b, is fixed for a powdery chemical of the TD type, having some one of the several specific shapes including the class A geometries as well as an arbitrary value of r, confined in a closed container of the corresponding shape and size, and placed in the atmosphere under isothermal conditions, only the value of r remains unfixed in the individual values of the variables, a, b, d g, dc and r, apart from the constant, A T, in Eq. (79) which defines the for the powdery chemical, because the shape of the chemical is specified, so that the value of ( c is also fixed. ... 

However, in the same manner as a powdery chemical of the TD type, if the sawdust heap, which is placed in the atmosphere under isothermal conditions, has some one of several specific shapes including the class A geometries as well as an arbitrary size, i.e., an arbitrary value of r, the calculation of the 71 for the sawdust heap becomes very simple, because the values of for these specific shapes have already been calculated by T. Boddington et al., respectively [58], with the result that, as explained in Section 6.2, the heat transfer data required to calculate the 71. for such a sawdust heap as specified above is only the effective thermal diffusivity, (3 e, of the sawdust. For the values of dc calculated by T. Boddington et al., refer to Table 13 in Section 6.5. 

Procedure to calculate the Tc for a sawdust heap, having some one of the several specific shapes including the class A geometries as well as an arbitrary value of r, placed in the atmosphere... 

Flgon 2 StatUmary-state temperature profiles for class A geometries, sbA, cylinder, and sphere. Dimensionless temperature excess =(T— as a fimctUm of... 

Collocation Methods.— In two papers in 1972, Hardee et a/, presented a polynomial method for predicting critical temperature profiles in multidimensional bodies. Initially the method was applied to the three class A geometries but later it was extended to derive critical conditions for rectangular bricks, finite cylinders, and (right) cones. 

General Series Solution.— This approach is due to Boddington et al. ° and is based on the derivation of a unified series solution of equation (6) for class A geometries. 

The solution ensures that the temperature attains its maximum value at the body s centre and in general it is only at that point that the total heat balance equation is exactly satisfied. However, for class A geometries the solution is globally exact, even though the three bodies differ in shape so widely. 

The method using a quadratic heat generation rate was applied by Sherrington to class A geometries, to Cartesian bodies (cube, rod, etc.), and to the right cylinder and cone. The results are included as variational (1) in the Table. Many extensions are possible. Boundary conditions other than 6 = 0 can be covered, e.g. cases where heat is transferred by convection or radiation at the boundaries. 

Recently Wake has applied a variational treatmrait to the stationary problem, deriving critical conditions both for the class A geometries and for the cube, square rod, and equicylinder in systems where the heat transf(H is resisted by conduction in the interior and by convection at the surface. Here the condition at the boundary becomes dO/dp + N6 = 0, where N is the Biot number hLIk The limit as bf- oo corresponds to the Frank-Kamenetskii solutions. Wake uses trigonometric, rather than polynomial, expressions for this tempoature field and proceeds to derive the conditions under which solutions of the time-dependent variational equations are just possible, associating these with a critical value of 6. Results for N = oo are listed as variational (2) in the Table. For the more rorai conditions of finite Biot numbers Wake compares his results for class A geometries with the analytical forms due to Thomas. Errors are less than 0.1 % though the computational effort required is substantial. 

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