Solid cube expression

We demonstrated that solid cube expression, a rule-expression format used in the semantic refinement of the design object, can be effectively applied to a wide variety of other refinement operations. 

This is often referred to as the contracting volume (cube or sphere) equation and is the simplest example of a more general family of expressions [28—31,432,453,458,459], which includes consideration of different rates of interface advance in different crystallographic directions and of variations in crystallite dimensions and shapes. The approach is readily extended, by use of solid geometry, to allow for angles between planar surfaces. Some examples of characteristic behaviour are conveniently discussed with reference to the expression... 

Since m is the mass of solid remaining at time t, the quantity m/m0 is the fraction undissolved at time t. The time to total dissolution (m/m0 = 0) of all the particles is easily derived. Equation (49) is the classic cube root law still presented in most pharmaceutics textbooks. The reader should note that the cube root law derivation begins with misapplication of the expression for flux from a slab (Cartesian coordinates) to describe flux from a sphere. The error that results is insignificant as long as r0 8. 

FIG. 3 Water loss (WL) and solid gain (SG) expressed on initial dry matter (idm) of strawberry (ST) slices (Brambilla et al., 2000) and apple (AP), carrot (CA), and pumpkin (PU) cubes (Kowalska and Lenart, 2001) after 60 min osmotic dehydration in a 60% (w/w) sucrose solution at 30 °C at atmospheric pressure. 

FIG. 5 Effects of varying raw material treatments during osmotic dehydration on moisture (MC) and solid (SC) content expressed on initial dry matter (idm). Apple cubes, ultrasound (U) (Simal et al., 1998) apple slices, vacuum (V) (Salvatori et al., 1998b) and bell pepper disks, high temperature (Ade-Omowaye et al., 2002b). 

Person 2 What is the equivalent expression for the second term in Eq. (3.34) for the cube Recall that y is the interfacial energy for the liquid-solid interface. 

The tendency of textile polymers to accumulate static electrical charges is related to their electrical resistance. With solid materials, the electrical resistance is defined as the resistance between opposite faces of a 1 m cube. With textile polymers, for reasons similar to those given earlier, it is more convenient to express the resistance of fiber filaments by... 

The cube root law results from a combination of exponential growth with mass transport limitations in the solid phase, which is expressed in the concept of dp as shown by Pirt (1975). 

For cubes, b = 6. Other regular convex solids have intermediate values of b. Note that when this relationship between area and volume is scaled downward, we eventually find a situation where solids do not occupy space in a way that is compatible with this model. That is because the actual dimensions of an atom are best expressed in terms of probability density functions rather than as absolutely fixed spatial dimensions. 

The given expression presents the basis for understanding the thermodynamic reasons for solid-phase amorphization, and of solving the first phase problem at reaction-diffusion in general. The first phase, to be formed in the diffusion zone, must have a maximal product of mobility and free formation energy cubed. In case of amorphous layer formation, it means that the increased mobility of atoms in the disordered phase compensates for the low transformation driving force. 

The more extended the specific surfaces of precipitates are, the more impurities they tend to adsorb. The specific surface is defined as the surface of the precipitate of unit mass. Usually, it is expressed in cm /g. For a given mass of solid, the specific surface increases considerably when the particule sizes decrease. For example, a 1-cm-long cube exhibits a specific surface of6x 1 x 1 = 6 cm. If its mass is 2 g, its specific surface is 6/2 = 3cm /g. If the same mass is distributed over 1000 cubes that are 0.1cm long, the total surface is 0.1 x 0.1 x 6 x 1000 = 60 cm and its specific surface is 30 cm /g. In order to clarify this point, let s consider 2 g of a colloid containing about 10 particules, whose length is about 10 cm. They exhibit a specific surface of 30 m /g. 

The application of this formula, and subsequent studies on the extension of Maxwell s work to more condensed media (i.e. media with large /) in which the shape of the solid is not necessarily spherical has been discussed in detail by Meredith and Tobias [2]. If one tries to represent the Maxwell formula by an array of prisms as shown in Figure 1, in which the electrical current passes vertically through a cube of width and height 1 cm, application of Ohm s law leads indeed to an expression for the conductivity, k, of the suspension in terms of the conductivities k and k of the... 

---