Dimensions physical quantities

Dimensions are physical quantities such as mass (M), length (L), and time (T) and examples of units corresponding to these dimensions are the gram (g), metre (m) and second (s). If, for example, something has a mass of 3.5 g then we write... 

The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let (( i) represent a set of fundamental units, hke length, time, force, and so on. Let [Pj represent the dimensions of a physical quantity Pj there are n physical quantities. Then form the matrix Ot) ... 

The second physical quantity of interest is, r t = 90 pm, the critical crack tip stress field dimension. Irwin s analysis of the crack tip process zone dimension for an elastic-perfectly plastic material began with the perfectly elastic crack tip stress field solution of Eq. 1 and allowed for stress redistribution to account for the fact that the near crack tip field would be limited to Oj . The net result of this analysis is that the crack tip inelastic zone was nearly twice that predicted by Eq. 3, such that... 

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. 

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. 

The circulatory motion described is due to the buoyant force g Ap, where Ap = p — [p(l — e) + p e] = (p — p )e, p being the density of the liquid, p that of the vapor, and e the volume fraction of the vapor in the neighborhood of the surface. The velocity u0 of the liquid should depend on gAp, the kinematic viscosity v, and density p. Since the number of physical quantities is four, while that of the independent dimensions involved is three, it follows that a single dimensionless group can be formed, which must be a constant. Consequently,... 

Table 2 Often Used Physical Quantities and Their Dimensions According to the Currently Used SI in Mechanical and Thermal Problems...

Fundamental (primary) quantities that cannot be expressed in simpler terms include mass M), length (L), and time (7). Physical quantities may be expressed in terms of the fundamental quantities e.g., density is ML velocity is LT. In some instances, mass units are covertly expressed in terms of force (F) in order to simplify dimensional expressions or render them more identifiable. The MLT and FLT systems of dimensions are related by the equations... 

The next step was to quantitatively determine some of the parameters involved in the adsorption of ions. We started by comparing equations of states in three dimensions (gas in a cylinder) with those in two dimensions (adsorbed molecules) (Section 6.8.5). This led us to define adsorption isotherms in electrochemical systems They are relationships relating the physical quantities [number of adsorbed molecules (r or 0), activity of ions in solution (a), charge or potential of the electrode ( M or E) and... 

What are the dimensions and SI units for the following physical quantities ... 

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. 

Physical Quantity Name of Unit Symbol Dimensions SI Units... 

Most of the physical quantities considered up to now have dimensions. That is, one must specify a unit of measure for the physical quantity. For example, the length of an object is measured in units of meters or millimeters. These fixed measuring units are man-made and do not depend on the specific problem at hand. It may be advantageous to express the dimensional physical quantities in terms of units that are natural to the problem that is, the unit of measure depends on the characteristics of the problem such as boundary and initial conditions and physical constants. 

A plane-parallel stellar atmosphere is a semi-infinite medium. In the numerical calculation, we divide it into n finite elements and 1 semi-infinite element. Let us define a node as a point between two elements. Node 0 is defined as the boundary between the surface element and the vacuum. In total, we have n+1 nodes. The distribution of any physical quantity is represented by a vector of n+1 dimensions with its values at the n+1 nodes as elements. The mean intensity of radiation J is written in the ordinary expression as... 

Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models. 

Units are a necessary part of the specification of a physical quantity. When physical quantities are subjected to mathematical operations, the units must be carried along with the numbers and must undergo the same operations as the numbers. Quantities cannot be added or subtracted directly unless they have not only the same dimensions but also the same units, for example ... 

Dimensional analysis, often referred to as the II-theorem is based on the fact that every system that is governed by m physical quantities can be reduced to a set of m - n mutually independent dimensionless groups, where n is the number of basic dimensions that are present in the physical quantities. The II-theorem was introduced by Buckingham [1] in 1914 and is therefore known as the Buckingham II-theorem. The II-theorem is a procedure to determine dimensionless numbers from a list of variables or physical quantities that are related to a specific problem. This is best illustrated by an example problem. 

Once we have defined all the physical quantities, also referred to as the relevance list, we write them with their respective dimensions in terms of mass M, length L, time T and temperature 0, and in some cases force F, i.e. 

Table 4.1 presents various physical quantities with their respective dimensions in an MLT0 system and in an FLTQ system, respectively. 

Note When primes are used on thermodynamic potentials, it is important to indicate in the context the intensive variables that have to be specified. This also applies when primes are used on equilibrium constants, amounts, or numbers like the number of components, degrees of freedom, and stoichiometric numbers. SI units are indicated in parentheses. When a physical quantity does not have units, no units are given. Dimensions of matrices are also indicated in parentheses. 

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