Dimensional analysis defined

Dimensional Analysis. In the design of rather simple devices or systems, dimensional analysis can be used in conjunction with physical model experimental investigations to gain insight into the performance of a particular design concept. It is usually possible to define the performance of a simple device or system with a certain number of well chosen geometric and performance related variables that describe the device or system. Once these variables have been selected, dimensional analysis can be used to ... 

Although the absorption of a gas in a gas-liquid disperser is governed by basic mass-transfer phenomena, our knowledge of bubble dynamics and of the fluid dynamic conditions in the vessel are insufficient to permit the calculation of mass-transfer rates from first principles. One approach that is sometimes fruitful under conditions where our knowledge is insufficient to completely define the system is that of dimensional analysis. 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

A related concept to dimensional analysis is quantity calculus, a method we find particularly useful when it comes to setting out table header rows and graph axes. Quantity calculus is the handling of physical quantities and their units using the normal rules of algebra. A physical quantity is defined by a numerical value and a unit ... 

To approach the analysis of, and to be able to comprehend, the complex phenomena of thermochemical conversion of solid fuels some idealization has to be made. For a simplified one-dimensional analysis, there is an analogy between gas-phase combustion and thermochemichal conversion of solid fuels, which is illustrated in Figure 41. Both the gas-phase combustion and the thermochemical conversion is governed by a exothermic reaction which causes a propagating reaction front to move towards the gas fuel and solid fuel, respectively. However, there are also some major differences between the conversion zone and the combustion zone. The conversion front is defined by the thermochemical process closest to the preheat zone, which is not necessarily the char combustion zone, whereas for the flame front is defined by the ignition front. In practice, many times the conversion zone is so thin that the ignition front and the conversion front can not be separated. 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

Normalized potential sweep voltammetry (NPSV) involves a three-dimensional analysis of the LSV wave where the normalized current (I/Ip) is taken as the Z axis, theoretical electrode potential data as the X axis, and experimental electrode potential data as the Y axis, with the potential axes defined relative to Ep/2. The method is illustrated by the voltammogram in Fig. 15. The projection of the wave on to the X—Y plane results in a straight line of unit slope and zero intercept if the theoretical and experimental data describe the same process. In practice, NPSV analysis simply involves the linear correlation of experimental vs. theoretical electrode potentials at particular values of the normalized current. 

The organization of this chapter is as follows. In Sect. 7.1 wo carefully define the continuous chain limit and we introduce the appropriate modification of the Feynman rules. We. then establish the two parameter scheme by dimensional analysis. Section 7.2 is devoted to the question whether the continuous chain limit exists. The analysis is presented on the diagrammatic level. It exploits the field theoretic representation, which also is derived on the level of diagrams. All the analysis is based on the cluster expansion. Extension to the loop expansion is not difficult, but will not be considered, since it is not needed in the sequel. 

In analogy with the one-dimensional analysis, the Jj are defined over complete periods of the orbit in the (qj,Pj) plane, Jj = ptdq j. If one of the separation coordinates is cyclic, its conjugate momentum is constant. The corresponding orbit in the (qj,Pj) plane of phase space is a horizontal straight line, which may be considered as the limiting case of rotational periodicity, for which the cyclic qj always has a natural period of 2-k, and Jj = 2irpj for all cyclic variables. 

The constant k must be defined in such a way that the quantities on the two sides of the equation are expressed in the same units. Such dimensional analysis demands that k have the dimensions of a force times a distance squared. In terms of mass (M), length (L) and time (T), k has dimensions of ML3T-2. 

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... 

Before considering the dry spin process in a dimensional analysis, some terms have to be explained and defined. 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

It is also possible to derive the Reynolds number by dimensional analysis. This represents a more analytical, but less intuitive, approach to defining the condition of similar fluid flow and is essentially independent of particular shape. In this approach, variables in the Navier-Stokes equation (relative particle-fluid velocity, a characteristic dimension of the particle, fluid density, and fluid viscosity) are combined to yield a dimensionless expression. Thus... 

One of the goals of the experimental research is to analyze the systems in order to make them as widely applicable as possible. To achieve this, the concept of similitude is often used. For example, the measurements taken on one system (for example in a laboratory unit) could be used to describe the behaviour of other similar systems (e.g. industrial units). The laboratory systems are usually thought of as models and are used to study the phenomenon of interest under carefully controlled conditions. Empirical formulations can be developed, or specific predictions of one or more characteristics of some other similar systems can be made from the study of these models. The establishment of systematic and well-defined relationships between the laboratory model and the other systems is necessary to succeed with this approach. The correlation of experimental data based on dimensional analysis and similitude produces models, which have the same qualities as the transfer based, stochastic or statistical models described in the previous chapters. However, dimensional analysis and similitude do not have a theoretical basis, as is the case for the models studied previously. 

The dependence of Vik on mass may now be obtained from a dimensional analysis. The friction coefficient rik depends on two parameters defining the state of the system, say kT and the overall particle concentration c, and on the parameters of the cited equation of relative motion. These parameters are the reduced mass niik = rriimk/ rrii -f rrik) and a set of ( -values corresponding to a set of d,fc-values. Planck s constant h does not enter into our classical calculation. The dimensions of the quantities involved are given in Table IV. 

Classical techniques have relied heavily on dimensional analysis, the combining of the many variables into physically meaningful non-dimensional groups, supported with experiments to quantify heat transfer for various geometries. For most drying applications of pharmaceutical relevance, the most important of these non-dimensional groups are the Nusselt number (Nu), the Prandtl number (Pr) and the Reynolds number (Re), defined as follows ... 

Entrainment may be defined as the carryover of ejected particles, while selective entrainment of finer or less dense particles is often referred to as elutriation. In most industrial processes, neither entrainment nor elutriation are desirable, which is in sharp contrast to this particular application. Consequently, there is very little research aimed specifically at enhancing the selective removal of less dense material from fluidised beds. Most research on entrainment is based on dimensional analysis applied to experimental data either with no or very limited consideration of the underlying physics Predictions made from these correlations are limited to very simple geometries. They may vary widely even for reactor airangements close to the experimental conditions they are based on, and are often completely unreliable when conditions are markedly different. In several intemal studies they have been found inadequate for entrainment and elutriation predictions in the fluidised bed system under investigation. The problem is too complex to be adequately represented by a small number of ordinary equations that would simply require substitution of a few parameters to obtain the rales of entrainment of the different particle size ftactions. 

For the correct representation of the viscoelastic behavior of a fluid from the viewpoint of dimensional analysis, however, the ratio of the normal stress coefficients to the shear stress is used. The so-called Weissenburg number is defined as ... 

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