Base dimensions

Base quantity Base dimension Base unit... 

The base dimension of temperature 0 appears only in two parameters. They can, therefore, produce only one dimensionless quantity ... 

The rows of the matrix are formed of base dimensions, contained in the dimensions of the quantities, and they will determine the rank r of the matrix. The columns of the matrix represent the physical quantities or parameters. 

A dimensional system consists of all the primary and secondary dimensions and corresponding measuring units. The currently used Systeme International d unites (SI) is based on seven base dimensions. They are presented in Table 1 together with their corresponding base units. 

If, in the formulation of a problem, only the base dimensions [M, L, T] occur in the dimensions of the involved quantities, then it is a mechanical problem. If [0] occurs, then it is a thermal problem and if [N] occurs it is a chemical problem. 

Can this dependency be dimensionally homogeneous No The first thing that now becomes clear is that the base dimension of mass [M] only occurs in the mass m itself. Changing its measuring unit, e.g. from kilograms to pounds, would change the numerical value of the function. This is unacceptable. Either our list should include a further variable containing M, or mass is not a relevant variable. If we assume - by simplification - the latter, the above relationship is reduced to ... 

Both 1 and g contain the base dimension of length. When combined as a ratio 1/g they become dimensionless with respect to L and are therefore independent of changes in the base dimension of length ... 

The dependency between four dimensional quantities, containing two base dimensions (L and T) in their dimensions, is reduced to a 4 - 2 = 2 parametric relationship between dimensionless expressions ( numbers ) ... 

By the quotient o/gp [L2] two base dimensions [M, T are eliminated at once, so that by the division of this expression by d2 the dimensionless number... 

In this example, five dimensional quantities produce two dimensionless numbers. This was to be expected because their dimensions contain three base dimensions 5-3 = 2. 

As a rule, more than two dimensionless numbers are necessary to describe a phys-ico-technological problem they cannot be produced as shown in the first three examples. The classical method to approach this problem involved a solution of a system of linear algebraic equations. They were formed separately for each of the base dimensions by exponents with which they appeared in the physical quantities. J. Pawlowski [5] replaced this relatively awkward and involved method by a simple and transparent matrix transformation ( equivalence transformation ) which will be presented in detail in the next example. 

With the dimensions of these quantities a dimensional matrix is formed. Their columns are assigned to the individual physical quantities and the rows to the exponents with which the base dimensions appear in the respective dimensions of these quantities (example Ap [M1 L-1 T"2]). This dimensional matrix is subdivided into a quadratic core matrix and a residual matrix, whereby the rank r of the matrix (here r = 3) in most cases corresponds to the number of the base dimensions appearing in the dimensions of the physical quantities. 

The following example has been chosen because it impressively demonstrates the scale-invariance of the pi-space. Besides this, in the matrix transformation we will encounter a reduction of the rank r of the matrix. This will enable us to understand why, in the definition of the pi-theorem (section 2.7), it was pointed out that the rank of the matrix does not always equals the number of base dimensions contained in the dimensions of the respective physical quantities. 

Reduction of the number of parameters necessary to define the problem. The pi-theorem states that a physical relationship between n physical quantities can be reduced to a relationship between m = n - r mutually independent pi-numbers. Herein, r represents the rank of the dimensional matrix which is formed by the physical quantities in question and corresponds, in most cases, to the number of base dimensions contained in their dimensions. 

In association with four base dimensions contained in their dimensions, these 13 x-quantities yield the following 13-4 = 9 pi-numbers ... 

In fact, this actually means that the pi-set could be reduced if one succeeds to enlarge the base dimensions of the dimensional system. However, it must be considered that in the enlargement (reduction) of a dimensional system the relevance list must also be enlarged (reduced) by the corresponding dimensional constant by which the number of the resulting pi-numbers is not changed. However, it can turn out that in the enlargement of the dimensional system the additional dimensional constant is, a priori, irrelevant to the problem. In this case, it need not be incorporated into the relevance list and the number of pi-numbers is, in fact, reduced by one. 

These six quantities contain four base dimensions [L, T, , H], wherein H means the amount of heat with calorie as measuring unit. According to the pi-theorem, a dependence between two pi-numbers will result. Rayleigh obtained the following two pi-numbers which are today named The Nusselt number Nu and the Peclet number Pe, the latter being the product of Reynolds and Prandtl numbers, Pe = RePr ... 

In dealing with Boussinesq s problem, Lord Rayleigh used the amount of heat H (measuring unit calorie) as one of the then used base dimensions. Only since the introduction of SI (Systeme International d Unites) it was required to make no distinction between heat and mechanical energy, because both were considered to be equal. In order to comply with this requirement, the Joule equivalent of heat J [M L2 T2 H-1] had to be introduced as a natural constant in the relevance list. If we proceed from the assumption of an inviscid , ideal liquid, no mechanical heat can be converted into heat. In this case, J is irrelevant. 

Dimensional analysis of this example is associated by a reduction of the rank of the matrix, because the base dimension of mass is only contained in the density, p. From this it does not follow that the density wouldn t be relevant here, but that it is already fully considered in the kinematic viscosity v, which is defined by v = p/p. Therefore... 

Fluid mechanics and mixing operations in various types of equipment, agglomeration as well as disintegration and mechanical separation processes, just to mention a few, are described by parameters, the dimensions of which only consist of three base dimensions Mass, Length and Time. An isothermal process is assumed The physical properties of the material system under consideration are related to a constant process temperature. The process relationships obtained in this way are therefore valid for any constant, random process temperature to which the numerical values of the physical properties are related. This holds true as long as there is no departure from the scope of the validity of the respective process characteristic verified by the tests. 

The base dimension M is contained only in the density p. Therefore, this quantity has to be deleted from the relevance list. 6-2 = 4 numbers will be produced. The pi-set reads ... 

Heat transfer processes are described by physical properties and process-related parameters, the dimensions of which not only include the base dimensions of Mass, Length and Time but also Temperature, , as the fourth one. In the discussion of the heat transfer characteristic of a mixing vessel (Example 20) it was shown that, in the dimensional analysis of thermal problems, it is advantageous to expand the dimensional system to include the amount of heat, H [kcal], as the fifth base dimension. Joule s mechanical equivalent of heat, J, must then be introduced as the corresponding dimensional constant in the relevance list. Although this procedure does not change the pi-space, a dimensionless number is formed which contains J and, as such, frequently proves to be irrelevant. As a result, the pi-set is finally reduced by one dimensionless number. 

From the point of view of dimensional analysis, a chemical engineering problem presents itself with the appearance of chemical parameters containing an additional base dimension, namely the amount of substance N in their respective dimensions (base unit mole). Consequently, we then have to deal with a 5-parametric dimensional system [M, L, T, , N]. 

Only 9 numbers are formed from these 15 dimensional parameters if the amount of heat, H, is added to the five primary quantities [M, L, T, Q, N] as the sixth base dimension 15-6 = 9. If L/d, cout/cin, E/RT0, and AT/T0 are anticipated as trivial numbers, the other five pi-numbers can be obtained using the following simple dimensional matrix ... 

Consider a sealed 20-cm-high electronic box whose base dimensions are 40 cm X 40 cm placed in a vacuum chamber. The emissivity of the outer surface of the box is 0.95. If the electronic components in the box dissipate a total of 100 W of power and the outer surface temperature of the box is not to exceed 55 C, deicrmi ne the temperature at which the surrounding surfaces must be kept if this box is to be cooled by radiation alone. Assume the heat transfer from the bottom surface of the box to the stand to be negligible. 

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