Dimensionless quantities

Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

The dimensionless quantities in brackets are, respectively, the reciprocal of the Froude number, the Euler number, and the reciprocal of the Reynolds number for the system. 

Dimensionless Quantities. Certain quantities, eg, refractive index and relative density (formerly specific gravity), are expressed by pure numbers. In these cases, the corresponding SI unit is the ratio of the same two SI units, which cancel each other, leaving a dimensionless unit. The SI unit of dimensionless quantities may be expressed as 1. Units for dimensionless quantities such as percent and parts per million (ppm) may also be used with SI in the latter case, it is important to indicate whether the parts per million are by volume or by mass. 

Let m be the rank of the Ot matrix. Then p = n — m is the number of dimensionless groups that can be formed. One can choose m variables [Pj] to be the basis and express the other p variables in terms of them, givingp dimensionless quantities. 

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. 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

The dimensionless quantity Sh is called the Sherwood number. The heat transfer factor a is defined bv... 

In the structure with all the surfactant molecules located at monolayers, the volume fraction of surfactant should be proportional to the average surface area times the width of the monolayer divided by the volume, i.e., Ps (X Sa/V. The proportionality constant is called the surfactant parameter [34]. This is true for a single surface with no intersections. In our mesoscopic description the volume is measured in units of the volume occupied by the surfactant molecule, and the area is measured in units of the area occupied by the amphiphile. In other words, in our model the area of the monolayer is the dimensionless quantity equal to the number of amphiphiles residing on the monolayer. Hence, it should be identified with the area rescaled by the surfactant parameter of the corresponding structure. 

The drag coefficient also depends on shape and 0(, but in addition, because drag is partially due to friction, and frictional effects in a flow arc governed by a powerful dimensionless quantity called Reynolds number, then Cu is also a function of the Reynolds number. Re ... 

Activity is a dimensionless quantity, and / must be expressed in kPa with this choice of standard state. It is inconvenient to carry f° = 100 kPa through calculations involving activity of gases. Choosing the standard state for a gas as we have described above creates a situation where SI units are not convenient. Instead of expressing the standard state as /° = 100 kPa, we often express the pressure and fugacity in bars, since 1 bar = 100 kPa. In this case, /0 — 1 bar, and equation (6.92) becomes4... 

It should be noted that it is permissible to take a function only of a dimensionless quantity. It is easy to appreciate this argument when account is taken of the fact that any function may be expanded as a power series, each term of which must have the same dimensions, and the requirement of dimensional consistency can be met only if these terms and the function are dimensionless. Where this principle appears to have been invalidated, it is generally because the equation includes a further term, such as an integration constant, which will restore the requirements of dimensional consistency. For f x dx... 

For a pipe of radius r and length /, the dimensions of r/l are L/Lr and hence (AP/R) (r/l) is a dimensionless quantity. The role of the ratio r/l would not have been established had the lengths not been treated as vectors. It is seen in Chapter 3 that this conclusion is consistent with the results obtained there by taking a force balance on the fluid. 

The dimensionless quantity toQ 2/(ghf/A is a characteristic for a particular type of centrifugal pump, and, noting that the angular velocity is proportional to the speed N. this group may be rewritten as ... 

A logarthmic plot of the dimensionless quantities and rj is shown in Figure 10.11. It may be seen that rj approaches unity at low values of , and becomes proportional to 4> 1 at high values of . 

The dimensionless quantity on the left-hand side of Eq. (7.3) labeled as M by the above-cited authors, allows for the comparison of heat transfer by axial conduction in the wall to the convective heat transfer in the flow. 

We introduce the most basic aspects of elasticity. We begin with Hookes law the change in length of a strut is proportional to the applied force, or 6L = FL/EA. Note that this is a linear relationship. Restated in a normalized way, a = Ee, where a is the stress (Pa or N/m ), E is Young s modulus (Pa) a property of the material, and e is the strain (6LjL) a dimensionless quantity. 

Strain is a dimensionless quantity, defined as increase in length of the specimen per unit original length. It represents the response of the material to the stress applied to it. 

FIGURE 26.23 Side force coefficient (dimensionless quantity S/fiL) as function of the quantity c (Equation 26.17c) showing the two components due to adhesion and shding. 

In any circumstances, it can be expected that and (5x are algebraic functions of turbulence length scale and kinetic energy, as well as chemical and molecular quantities of the mixture. Of course, it is expedient to determine these in terms of relevant dimensionless quantities. The simplest possible formula, in the case of very fast chemistry, i.e., large Damkohler number Da = (Sl li)/ SiU ) and large Reynolds Re = ( Ij)/ (<5l Sl) and Peclet numbers, i.e., small Karlovitz number Ka = sjRej/Da will be Sj/Sl =f(u / Sl), but other ratios are also quite likely to play a role in the general case. 

The effect of transport limitations can conveniently be evaluated by considering the spherical catalyst particle shown in Fig. 5.32. We will introduce a dimensionless quantity called the Thiele diffusion modulus (Og) [W. Thiele Ind. Eng. Chem. 31... 

Double-closure is the joint operation of dividing each element of the contingency table X by the product of its corresponding row- and column-sums. The result is multiplied by the grand sum in order to obtain a dimensionless quantity. In this context the term dimensionless indicates a certain synunetry in the notation. If x were to have a physical dimension, then the expressions involving x would appear as dimensionless. In our case, x represents counts and, strictly speaking, is dimensionless itself. Subsequently, the result is transformed into a matrix Z of deviations of double-closed data from their expected values ... 

To obtain general solutions, the problem is reformulated in terms of the following dimensionless quantities ... 

For the formulation of a general solution, the dimensionless quantities defined by Eqs. (21)-(25) are used, but the dimensionless rate constant is defined by Eq. (33) rather than Eq. (26). 

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