Dimensional homogeneity

for illustrative purposes, the equation relating the distance s, covered by an object in time t, travelling at an initial velocity u, when subject to a constant acceleration a  

Thus Equation (2) is said to be dimensionally homogeneous or dimensionally consistent. 

Any equation relating physical quantities must be dimensionally homogeneous or dimensionally consistent. It is not permissible to add a length to a velocity because they are quantities of different types. Even Einstein s E = me2 is dimensionally homogeneous. 

Note that in some equations, the values of constants are a function of the system of units used. These constants have dimensions and are known as dimensional constants. For example, many flowmeters measure volumetric flow rates, Qv, by measuring heads of fluids, Ah. Typically, Qv oc a/Ah or Qv = csfKh. Here, as Qv[f y[Ah, c is a dimensional constant. Dimensionally, L3T-1[=] cs/L or c [=] L2 5T 1, and the value of c will depend upon the units of both Qr and Ah. 

E = modulus of elasddty of the material What are the units for modulus of elasticity  

For Equation (6.1) to be dimendonally homogeneous, the units on the left-hand side of the equation must equal the units on the right-hand side. This equality requires the modulus of elasticity to have the units of N/m, as follows  

Solving for the units of E leads to N/m, newton per squared meter, or force per imit area, (b) The heat transfer rate throv a solid material is governed by Fourier s law  

WTiat is the appropriate unit for the heat transfer rate q  

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As an example, we look at tire etching of silicon in a CF plasma in more detail. Flat Si wafers are typically etched using quasi-one-dimensional homogeneous capacitively or inductively coupled RF-plasmas. The important process in tire bulk plasma is tire fonnation of fluorine atoms in collisions of CF molecules witli tire plasma electrons... 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

This result was first discussed by Buckingham (8) and stated in its present form by Langhaar (23). It states in effect that an equation is dimensionally homogeneous if and only if it can be reduced to a relationship among a complete set of B-numbers. Buckingham s result (8) was originally stated as Theorem 2. 

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

Umashankar, K., Taflove, A., and Rao, S.M., 1986, Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects, IEEE Trans. Antennas. Propagat. 34(6) 758-766. 

Dimensional analysis is based on the recognition that a mathematical formulation of a physicotechnological problem can be of general validity only when the process equation is dimensionally homogenous, which means that it must be valid in any system of dimensions. 

The aim of dimensional analysis is to check whether the physical content under examination can be formulated in a dimensionally homogeneous manner or not. The procedure necessary to accomplish this consists of two parts ... 

In the second step, the dimensional homogeneity of the physical content is checked by transferring it in a dimensionless form. Note A physical content that can be transformed in dimensionless expressions is dimensionally homogeneous. 

Applying the rule of dimensional homogeneity and making c and e the unrestricted constants leads to... 

Equation (7) gives a relationship between velocity, rotation rate, and cylinder radius that can be used to complete the dimensional analysis discussed earlier. Applying dimensional homogeneity and solving leads to... 

The solution to every dimensionally homogeneous physical equation has the form %2, %2,. ..), = 0, in which Ui, nj, nj o represent a complete set of dimensionless groups of the variables and the dimensional constants of the equation. 

The solution to every dimensionally homogeneous physical equation has the form... 

Homogeneous system fall Ja are zero). An example of a two-dimensional homogeneous system is the first-order reversible reaction between two chemical species discussed in Chapter 12 using the example of the hydration of an aldehyde (see Eq. 12-16). Again, matrix theory provides us with a very useful mle which states that for such systems the resulting matrix is singular (that is, its determinant is zero, see Box 21.8) and thus at least one eigenvalue must be zero. Furthermore, in Eq. 21-48 all ) , are zero. 

Some of the considerations of this section have been discussed by the late Gianni Astarita, whose death in 1997 cut short a tremendously productive career. In a paper dedicated to M. M. Sharma on his sixtieth birthday,1 Astarita points out that Euclid recognized the need for dimensional homogeneity and Ptolemy wrote a book entitled On Dimension, and that their overly strict rules of manipulation were not relaxed until Newton s time. The modem... 

Two-dimensional homogeneity in the Chi a-DPL mixed monolayer system was confirmed by the II - A isotherms as well as by the observed enhancement of the fluorescence yield of the mixed mono-layer upon decreasing the Chi a surface concentration (66). 

The anisotropy in g(ri, r2) may be determined by the use of akinematic argument. Consider a bulk of particle subjected to a mean shear flow. The radial distribution function, which is spherical in equilibrium, becomes distorted into an ellipsoidal distribution as a result of the presence of the mean shear. Hence, in order for g(n, r2) to exhibit an anisotropy, g (ri, r2) should depend not only on ap, r, and r2 but also on Tc, vi, and v2. For dimensional homogeneity, g can only be a function of ap, k U2i/Tcl/2, and U x/Tc. For a small deformation rate (or when the magnitude of U2i is small relative to Tc1/2), it is assumed that g(ri, r2) takes the form [Jenkins and Savage, 1983]... 

We saw earlier that Equation (2), s = ut + jat2, was dimensionally homogeneous each term has the same dimensions. Ratios of the terms in this equation will therefore be dimensionless. So we can make the original expression dimensionless by, for example, dividing each term... 

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