Mechanical dimensionless

Hamiltonian function in Hamiltonian mechanics Dimensionless function in particle drag expression (—) Interfacial heat transfer due to phase change (J/m s) Specific enthalpy of ideal gas mixture (kJ/kg)... 

Remember the units involved here For f they are length time for N, length and for t, time. Therefore the exponent is dimensionless, as required. The form of Eq. (4.24) is such that at small times the exponential equals unity and 6 = 0 at long times the exponential approaches zero and 0 = 1. In between, an S-shaped curve is predicted for the development of crystallinity with time. Experimentally, curves of this shape are indeed observed. We shall see presently, however, that this shape is also consistent with other mechanisms besides the one considered until now. 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

The physical properties of argon, krypton, and xenon are frequendy selected as standard substances to which the properties of other substances are compared. Examples are the dipole moments, nonspherical shapes, quantum mechanical effects, etc. The principle of corresponding states asserts that the reduced properties of all substances are similar. The reduced properties are dimensionless ratios such as the ratio of a material s temperature to its critical... 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Heat Exchangers Since most cryogens, with the exception of helium 11 behave as classical fluids, weU-estabhshed principles of mechanics and thermodynamics at ambient temperature also apply for ctyogens. Thus, similar conventional heat transfer correlations have been formulated for simple low-temperature heat exchangers. These correlations are described in terms of well-known dimensionless quantities such as the Nusselt, Reynolds, Prandtl, and Grashof numbers. 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Mechanism N Dimensionless rate equation Constant pattern Refs. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

As I mentioned above, it is conventional in many engineering applications to seek to rewrite basic equations in dimensionless form. This also applies in quantum-mechanical applications. For example, consider the time-independent electronic Schrodinger equation for a hydrogen atom... 

X(( = Correlation parameter, dimensionless (tt refers to the turbulent-turbulent flow mechanism)... 

X = correlating parameter, dimensionless for turbulent-turbulent flow mechanism, Eigure Rf ... 

Rock Mechanical Properties. In the previous section (Figure 4-313), the wear of the bit teeth can be determined in shales by plotting the dimensionless bit torque (T, ) versus the dimensionless ROP (R,). By introducing a new parameter, namely the apparent formation strength, the bit effects can be separated from the lithology effects. 

The behavior of the gas as it flows down the tube is controlled by fluid mechanics and a complete investigation wouldbe lengthy and outside the scope of this book. It is enough to say that the Reynolds number, which is a dimensionless parameter that characterizes the... 

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

This dimensionless number measures the breadth of the molecular weight distribution. It is 1 for a monodisperse population (e.g., for monomers before reaction) and is 2 for several common polymerization mechanisms. 

The fundamental scattering mechanism responsible for ROA was discovered by Atkins and Barron (1969), who showed that interference between the waves scattered via the polarizability and optical activity tensors of the molecule yields a dependence of the scattered intensity on the degree of circular polarization of the incident light and to a circular component in the scattered light. Barron and Buckingham (1971) subsequently developed a more definitive version of the theory and introduced a definition of the dimensionless circular intensity difference (CID),... 

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