Frank modulus

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

The use of 0.05% strain as the lower limit for determining modulus is a very stringent requirement and needs an extensometer accurate to at least I micrometer. There arc also implications for the test piece in terms of flatness or warping and for the precision of the gripping mechanism which, frankly, are unrealistic in some cases. It may be that some of these points will be addressed at the next revision of the standard. 

On the other hand materials deform plastically only when subjected to shear stress. According to Frenkel analysis, strength (yield stress) of an ideal crystalline solid is proportional to its elastic shear modulus [28,29]. The strength of a real crystal is controlled by lattice defects, such as dislocations or point defects, and is significantly smaller then that of an ideal crystal. Nevertheless, the shear stress needed for dislocation motion (Peierls stress) or multiplication (Frank-Read source) and thus for plastic deformation is also proportional to the elastic shear modulus of a deformed material. Recently Teter argued that in many hardness tests one measures plastic deformation which is closely linked to deformation of a shear character [17]. He compared Vickers hardness data to the bulk and shear... 

Why does the free energy density acquire this particular form First, in the curvature term with modulus Ku, we must use the second derivatives because the first derivatives correspond to a pure rotation of all the layers that does not cost energy. The higher derivatives are ignored for small distortions. For the compressibility term, the first derivative (du/dz) is sufficient. Second, both the compressibility and the curvature terms must be squared due to head-to-tail symmetry and parabolic form of the density increment gdisrgo as a function of distortion (Hooke s law). However, the question arises why is only splay modulus taken into account in (8.44) and not the other two Frank moduli K22 and 33. Considering the splay and bend distortions of the SmA phase in Fig. 8.24 we can see that only the splay distortion is allowed because it leaves the interlayer distance and the... 

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