Dimension, reference

To eliminate the ambiguities in the subject of electricity and magnetism, it is convenient to add charge q to the traditional I, m and t dimensions of mechanics to form the reference dimensions. In many situations permittivity S or permeabiUty ]1 is used in Heu of charge. For thermal problems temperature Tis considered as a reference dimension. Tables 2 and 3 Hst the exponents of dimensions of some common variables in the fields of electromagnetism and heat. 

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) ... 

If length /, mass and time / are chosen as the reference dimensions, from Table 1 the dimensional formulas for the variables F, M and are as follows ... 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

Theorem 4. The set of B-numbers associated with a physical phenomenon is invariant with respect to the choice of the reference dimensions provided that the reference dimensions are considered independent, and that the number of these reference dimensions is not altered. 

The implication of this theorem is important in that in computing a complete set of dimensionless products or B-numbers associated with a physical phenomenon, it does not matter which set of dimensions are chosen as the reference dimensions as long as they are independent and their number is not altered. 

Suppose now that force length and time t ate chosen as the reference dimensions. From Table 1 the new dimensional matrix D, becomes (eq. 19)... 

The matrix D , that will tiansfomi D, to D is the dimensional matrix of the variables force, length, and time with respect to the reference dimensions and t. Again from Table 1 equation 20 is obtained. 

Let Dhe the dimensional matrix of order mhy n associated with a set of variables of a physical phenomenon, where m is the number of chosen reference dimensions and n the number of variables of the set. Without loss of generaUty, it may be assumed that n > m. Consider the augmented matrix (eq. 21) ... 

Pxampk 2. A smooth spherical body of projected area Al moves through a fluid of density p and viscosity p with speed O. The total drag 8 encountered by the sphere is to be determined. Clearly, the total drag 8 is a function of O, Al, p, and p. As before, mass length /, and time t are chosen as the reference dimensions. From Table 1 the dimensional matrix is (eq. 23) ... 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

Nominal size of a duct or fitting The reference dimension used for the designation, calculation, and application of ducts and fittings. 

To construct a model which will give behavior similar to another bed, for example, a commercial bed, all of the dimensionless parameters listed in Eqs. (37) or (39) must have the same value for the two beds. The requirements of similar bed geometry is met by use of geometrically similar beds the ratio of all linear bed dimensions to a reference dimension such as the bed diameter must be the same for the model and the commercial bed. This includes the dimensions of the bed internals. The dimensions of elements external to the bed such as the particle return loop do not have to be matched as long as the return loop is designed to provide the proper external solids flow rate and size distribution and solid or gas flow fluctuations in the return loop do not influence the riser behavior (Rhodes and Laussman, 1992). 

Putting all unknown effects together in the notation [Pg.38]

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

Dimensional calculations are greatly simplified if a consistent set of units is employed. The three major reference dimensions for mechanics are length, mass, and time, but length can be measured in units of inches, feet, centimeters, meters, etc. Which should be used The scientific community has made considerable progress toward a common system of reference units. This system is known as SI from the French name Systeme International d Unites. In SI, the reference units for length, mass, and time are the meter, kilogram, and second, with symbols m, kg, and s, respectively. 

Dimensional calculations are simplified if the unit for each kind of measure is expressed in terms of special reference units. The reference dimensions for mechanics are length, mass, and time. Other measurements performed are expressed in terms of these reference dimensions units associated with speed contain references to length and time—mi/hr or m/s. Some units are simple multiples of the reference unit—area is expressed in terms of length squared (m2) and volume is length cubed (in3). Other reference dimensions, such as those used to express electrical and thermal phenomena, will be introduced later. 

This step can be accomplished by means of the pi theorem which indicates that the number of pi terms is equal to m - n, where m (determined in step 1) is the number of selected variables and n (determined in step 2) is the number of basic dimensions required to describe these variables. The reference dimensions usually correspond to the basic dimensions and can be determined by a careful inspection of the variables dimensions obtained in step 2. As previously noted, the basic dimensions rarely appear combined, which results in a lower number of reference dimensions than the number of basic ones. 

We could also use F, L, and T as basic dimensions. Now, we can apply the pi theorem to determine the required number of pi terms (step 3). An inspection of the variable dimensions obtained in step 2 reveals that the three basic dimensions are all required to describe the variables. Since there are five (m = 5) variables (do not forget to count the dependent variable, Ap/1) and three required reference dimensions (n = 3), then, according to the pi theorem, two pi groups (5 - 3) will be required. 

In this dimensional analysis problem, five pi terms are needed because we have seven variables and two reference dimensions. The first pi term is represented by the conversion of the main reactant because this variable is dimensionless. The construction of the second pi group begins with variable d. This has a length... 

By inspection, the number of reference dimensions is 3 thus, the number of pi variables is 6 - 3 = 3 (the number of variables minus the number of reference dimensions). Reference dimension is the smallest number of groupings obtained from grouping the basic dimensions of the variables in a given physical problem. Call the pi variables IIi, II2, and II3, respectively. Letting Hi contain , write [T/N] = (FL/T)/ (1/T) = FL to eliminate T [ ] is read as the dimensions of. To eliminate L, write [iP/NDa] = FLIL = F To eliminate F, write PINDadlpN D )] = F KFT II ) = 1. Therefore,... 

We may want to perform the dimensional analysis ourselves, but the procedure is similar to the one done before. In other words, Vj, is first to be expressed as a function of the variables affecting its value Vj = f(g, p, pi, Pg, a, f). Pg is the mass density of the gas phase (air). Each of the variables in this function is then broken down into its fundamental dimensions to find the number of reference dimensions. Once the number of reference dimensions have been found, the number of pi dimensionless variables can then be determined. These dimensionless variables are then found by successive eliminations of the dimensions of the physical variables until the number of pi dimensionless ratios are obtained. 

Reference dimension—The smallest number of groups of the groups formed from all the possible groupings of the basic dimensions of the physical variables in a given problem. 

The vector model has proven versatQe to this purpose. It employs the unit vectors of the graphene s two-dimensional unit ceU as a reference dimension. The vector C running in paraUel to the coiling direction is a Unear combination of integer multiples of the units vectors. It is an interconnection of two identical points on the graphene lattice (Figure 3.2). C describes a straight line that represents the uncoiled perimeter of the respective nanotube. It also defines the orientation of the nanotube as the tubular axis f is perpendicular to the tube s cross-section, which on its own part Ues in a plane defined by the perimeter (or the coUed C). 

Reference Dimensions (cm) Material Surface Coating (s) or Treatment t (K)... 

The inside diameter of the shell is used as the reference dimension for heat exchanger design. For example, the unit shown in Figure 2-4 has a shell with an outside diameter of 20 inches and an inside iameter of 19-1/4 inches. These are the dimen-sons of 20-inch, Schedule 20 pipe, which is .eaper and easier to use than flat steel plate which wDuld have to be rolled into a cylinder and welded together to form the shell. 

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