Dimensional consistency

For liquid-liquid systems, the separations are isothermal and the objective function is one-dimensional, consisting of Equation (7-17). However, the composition dependence of the... 

E] Gas absorption aud desorption from water aud organics plus vaporization of pure liquids for Raschig riugs, saddles, spheres, aud rods, dp = nominal pacldug size, Cp = dry pacldug surface area/volume, = wetted pacldug surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. <3 = surface tension, dynes/cm. 

Note that with U.S. Customary units, the conversion factor may he required to make equations in this section dimensionally consistent g = 32.17 (lhm-ft)/lhf-s ). 

These units are dimensionally consistent any set of consistent units can be used. 

Eqs. 14-198, 14-199, and 14-200 are dimensionally consistent any set of consistent units on the right-hand side yields the droplet size in units of length on the left-hand side.)... 

For conditions approaching constant flow through the orifice, a relationship derivea by equating the buoyant force to the inertia force of the liquid [Davidson et al., Tran.s. In.stn. Chem. Engr.s., 38, 335 (I960)] (dimensionally consistent),... 

Therefore Eq. (5-47) is applicable to first-order and to second-order rate constants, it being understood that the arithmetic operations are carried out on pure numbers generated as shown. We have not evaded the requirement of dimensional consistency, which is provided by Eq. (5-43). 

In general, engineering equations should be dimensionally consistent. Physical properties are important considerations in any study of accidents and emergencies. A substance may exhibit certain characteristics under one set of physical conditions, but may become hazardous if Uie conditions are changed. 

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants. 

The requirement of dimensional consistency is that the combined term on the right-hand side will have the same dimensions as that on the left that is, it must have the dimensions of pressure. 

The conditions of dimensional consistency must be met for each of the fundamentals of M, L, and T and the indices of each of these variables may be equated. Thus ... 

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... 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction. Most length terms are vectors in the Cartesian system and may have components in the X, Y and Z directions which may be expressed as Lx, Ly and Lz. There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis. This means that the number of dimensionless groups which are formed will be less. 

A useful equation for the calculation of liquid phase diffusivities of dilute solutions of non-electrolytes has been given by Wilke and CHANG(I6). This is not dimensionally consistent and therefore the value of the coefficient depends on the units employed. Using SI units ... 

The duty coefficient is given by the following equation (in which SI units must be used as it is not dimensionally consistent) ... 

With compression moulding, to ensure dimensional consistency, it is necessary to allow the excess material to move away from the edge of the cavity so that the lands between the cavities can contact with minimum thickness of rubber (flash) between them. Spew grooves and channels are provided of sufficient dimensions to accommodate this excess, and also to allow the escape of air from the mould cavity. In some cases, where the shape is complex, it may be necessary to provide extra venting to allow air to escape from a blind area, where it is likely to be trapped. 

A physical unit system is implicitly defined by the choice of three underlying base units, which suffice to determine dimensionally consistent units for other measurable physical quantities. (Why three such base units are required is as yet an unanswered physical question.) Although the choice of units may superficially appear arbitrary, it was recognized by Gibbs (in his first scientific communication)1 that one can rationally address the question of the conditions which it is most necessary for these units to fulfil for the convenience both of men of science and of the multitude. ... 

Note that equation 4.20 is dimensionally consistent only if C4 has the dimensions T2 L and consequently the numerical value of C4 is different for different sets of units. 

All equations are given on a dimensionally consistent basis and can be used with any dimensionally consistent units. The illustrative units given in the following are based on the MKSA system, using the rational basis for electrical units. For the cgs-esu (irrational) system, the corresponding cgs-esu units would be used and the permittivity, e, would have a value ofl/(47r). Universal constants and defined values are represented by a symbol in Gothic (sans serif) or in bold face type. 

Here the film coefficients are in kW/m2 K and the fluidising velocity uc is in m/s. As equation 6.54 is not in dimensionally consistent units (because some of the relevant properties were not varied), the coefficient, 24.4, is valid only for the units stated. The value of the index m is given by ... 

Ellis(36) presented the following dimensionally consistent equation for the HETP (Zt) of packed columns using 25 and 50 mm Raschig rings ... 

In order for Eq. (2.3) to be dimensionally consistent, G must have units of [length] . 

The definition for 6y, must be developed. The definition requires dimensional consistency the gap cannot be calculated as for 6c- The dimensional analysis in the y direction involves 6c, and the remaining solid bed at any position z down the channel. The following equation was developed to meet these criteria. 

It is noteworthy that from a modeling perspective, 0j is also a scaling factor, since the expectation operator and the variance are of different dimensions. If it is desirable to obtain a term that is dimensionally consistent with the expected value term, then the standard deviation of z0 may be considered, instead of the variance, as the risk measure (in which standard deviation is simply the square root of variance). Moreover, 0i represents the weight or weighting factor for the variance term in a multiobjective optimization setting that consists of the components mean and variance. 

Finally, it is sufficient for our purposes to think of the remaining factors in Equation (10), (47reoC2)-I, as providing dimensional consistency to the expression. Taking a look at the SI units of the right-hand side of Equation (10), we obtain... 

The slope of the 6 = 0 line includes the factor k, which must have the units concentration 1 for dimensional consistency. Hence the slope given must be multiplied by 2000 cm3 g 1 to give the true slope to which the analysis applies. Since the true slope equals 2B according to Equation (68), we obtain... 

EXAMPLE 10.1 Relative Magnitudes of van der Waa/s Forces and Relation to Heat of Vaporization. The parameter must have units energy length6 in order to satisfy Equation (33). Verify these units as well as the dimensional consistency of each of the three terms in Equation (34). Taking fi = 1.0 debye and a = 10-39 C2 m2 J, calculate the amount of energy needed to separate a pair of molecules from 0.3 nm to . Scaled up by Avogadro s number, how does this energy compare with typical enthalpies of vaporization ... 

The membrane pore size can be calculated from the measured bubble point Pj, by using the dimensionally consistent Equation 10.9. This is based on a simphstic model (Figure 10.6) that equates the air pressure in the cyhndrical pore to the cosine vector of the surface tension force along the pore surface [6] ... 

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