Dimensionless numbers, physical

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

The advantage of the use of dimensionless numbers in calculations such as these is that they may be applied independent of the physical size of the system being described. 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

TT-theorem) or from the governing equations of the flow. The latter is to be preferred because this method will give a sufficient amount of dimensionless numbers. Furthermore, it will connect the numbers to the physical process via the equations and give important information in cases where it is necessary to make approximations. 

It is useful to take similarity principles and dimensionless numbers into consideration when planning experiments. Experiments may involve different levels of velocities and temperature differences. It is important to select values that give a large variation of Archimedes number (12,56) to obtain a high possibility of large physical effects in the measurements. 

Scaling by use of dimensionless numbers only is limited in two-phase flow to simple and isolated problems, where the physical phenomenon is a unique function of a few parameters. If there is a reaction between two or more physical occurrences, dimensionless scaling numbers can mainly serve for selecting the hydrodynamic and thermodynamic conditions of the modelling tests. In... 

Mole fraction, often symbolized by x or X followed by a subscript denoting the entity, represents the amount of a component divided by the total amount of all components. Thus, the mole fraction of component B of a solution, xb, is equal to hb/Xhi where Hb is the amount of substance B and Sni is the total amount of all substances in solution. In biochemical systems, usually the solvent is disregarded in determining mole fractions. The mole fraction, a dimensionless number expressed in decimal fractions or percentages, is temperature-independent and is a useful description for solutions in theoretical studies and in physical biochemistry. 

The second technique is physical insight into the problem, where ratios of forces or mass/heat transport determinants are factored to develop dimensionless numbers. This technique can also be found in most fluid mechanics texts. 

From the above example, we also learn that transformation of physical dependency from a dimensional into a dimensionless form is automatically accompanied by an essential compression of the statement the set of the dimensionless numbers is smaller than the set of the quantities contained in them, but it describes the problem equally comprehensively. In the above example, the dependency between five dimensional parameters is reduced to a dependency between only two dimensionless numbers. This is the proof of the so-called pi theorem (pi after Ft, the sign used for products), which states ... 

Is one model scale sufficient or should tests be carried out in models of different sizes One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called process point in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. 

This graph is extremely easy to use. The physical properties of the material system, the diameter of the vessel (D), and the desired mixing time (6) are all known and this is enough to generate the dimensionless number II2. 

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

Where no complete mathematical description of the process and no dimensionless-numbers equations are available, modeling based on individual ratios can be employed. This is the most characteristic case for a number of industrial processes, especially in the field of organic-chemicals technology. This method is referred to as scale-up modeling (Mukhyonov et al., 1979). In such cases, individual ratios for the model and the object, which should have a constant value, are employed. For instance, there should be a constant ratio between the space velocity of the reacting mixture in the model and the industrial object. Some of the dimensionless numbers mentioned in physical modeling are also employed in this case. 

Quantities of the square core matrix may eventually appear in all of the dimensionless numbers as fillers, whereas each element of the residual matrix will appear in only one dimensionless number. For this reason the residual matrix should be loaded with essential variables like the target quantity and the most important physical properties and pro-cess-related parameters. 

The dimensionless numbers are formed as fractions, where each physical quantity indicated in the residual matrix represents the numerator, while a product of all quantities of the core matrix (with the exponents indicated in the residual matrix) constitutes the denominator. This standard procedure yielded the following n set ... 

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. 

The classic technique to determine dimensionless numbers, described above, is cumbersome to use in cases where the list of related physical quantities becomes large. Pawlowski [8] developed a matrix transformation technique that offers a systematic approach to the generation of II-sets. 

With the above matrix set the dimensionless numbers can be generated, in this case, 4 dimensionless groups, by placing the physical quantities in the residual matrix in the numerator and the quantities in the core matrix in the denominator with the coefficients in the residual matrix as their exponent. Hence,... 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

The dimensionless numbers introduced above can provide, in the light of their physical meaning, some preliminary information about the system behavior ... 

Dimensionless numbers are combinations of variables in which all units cancel. These numbers are named after scientists and are used in the physical sciences to summarize causal relationships between variables and processes. One convenience of expressing these physical relationships in dimensionless terms is that large amounts of data may be collapsed into a single relationship. Cussler (1997, p. 232) describes what typically happens after first exposure to dimensionless numbers at first one thinks that a tool has been found that will solve all problems, followed by disillusionment after a first difficulty in application, and finally, renewed enthusiasm if one perseveres. 

Dimensionless numbers are very simple to calculate, although proper interpretation is dependent on the choice of variables such as which length to use for the characteristic length. The magnitudes of the dimensionless numbers are predictors of physical behavior. In addition, the functional relationships written in terms of the appropriate dimensionless numbers are extremely powerful because they will be valid for that geometry and boundary conditions for a wide range of variable combinations. In chemoreception there are six dimensionless numbers that are particularly important (Re, Pe, Sh, Fo, Sc, and Wo). 

In the physical literature, functional relationships for mass transfer can be found for a few cases of low Re, but unfortunately the actual range of dimensionless numbers for which the formulas are valid are often either not provided or can be somewhat misleading. For example, Clift et al. (1978) provide the following formula for a spheroid in creeping flow Re 1) ... 

In the following discussion, it will be understood that in algebraic expressions and equations all physical quantities are designated in a consistent set of units (e.g., pressure in dyne/cm2 and energy in ergs) in this case quantities of elements will be dimensionless (number of atoms). Numerical citations in text and tables, however, will be in more familiar laboratory units (e.g., pressure in atm or torr or Pa, energy in cal, and gas quantities in cm3 STP). 

Last but not least, in the final chapter it is demonstrated with a few examples that different types of motions in the living world can also be described by dimensional analysis. In this manner the validity range of the pertinent dimensionless numbers can be given. The processes of motion in Nature are subjected to the same physical framework conditions (restrictions) as the technological world. 

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. 

The nature of the steps which have to be carried out now makes this dimensional matrix less than ideal because it is necessary to know that each of the individual elements of the residual matrix will appear in only one of the dimensionless numbers, while the elements of the core matrix may appear as fillers in the denominators of all of them. The residual matrix should therefore be loaded with essential variables such as the target quantity and the most important physical properties and process-related parameters. Variables with an, as yet, uncertain influence on the process must also be included in this group. If, later, these variables are found to be irrelevant, only the dimensionless number concerned will have to be deleted while leaving the others unaltered. 

The dimensionless number IT does not usually occur as a target number for Ap. It has the disadvantage that it contains the essential physical property, kinematic viscosity v, which is already contained in the process number (which is where it belongs). This disadvantage can easily be overcome by appropriately combining the dimensionless numbers IT and II2. This results in the well-known Euler number... 

Note The pi-theorem only stipulates the number of the dimensionless numbers and not their form. Their form is laid down by the user, because it must suit the physics of the process and be suitable for the evaluation and presentation of the experimental data. 

This method of compiling a complete set of dimensionless numbers makes it clear that the numbers formed in this way cannot contain numerical values or any other constant. These appear in dimensionless groups only when they are established and interpreted as ratios on the basis of known physical interrelations. Examples ... 

It has already been pointed out that using dimensional analysis to handle a physical problem and, consequently, to present it in the framework of a complete set of dimensionless numbers, is the only sure way of producing a simple and reliable scale-up from the small-scale model to the full-scale technological plant. The theory of models states that ... 

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