Measurement dimensional analysis

So far, we have learned the evaluation of heat transfer by analytical means and by the analogy between heat and momentum transfer. When an analytical solution is beyond our reach, or when there exists no analogy between momentum and heat, we rely on experimental measurements. Dimensional analysis provides an effective way of organizing experimental data. The next section is devoted to a review of the methods of dimensional analysis, arranged in a manner particularly suitable to heat transfer studies. 

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for... 

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Derived from dimensional analysis with various constants and indices being deduced from spray data for water and kerosene measured using light-scattering technique... 

When working with measurements, you often have to convert units before performing other calculations. There are two methods of converting measurements. One is using proportions and the other is using a scientific method called dimensional analysis. 

Divide 7,620 by 609.6 to get 12.5 inches per second. As you can see from this example, dimensional analysis is an efficient way to convert measurement units when there are several conversions to be made. 

There are two methods for converting measurement units the proportion method and dimensional analysis. 

The velocity u0 and the laminar path length x0 can be related to measurable physical quantities by using dimensional analysis. Indeed, the circulatory motion is induced by the buoyancy force gAp, where Ap is the difference between the density near the wall (assumed to be equal to that of the liquid) and the density of the bubble bed... 

It is obvious that this conclusion is wrong Dimensional analysis is a method based on logical and mathematical fundamentals (2,6). If relevant parameters cannot be listed because they are unknown, one cannot blame the method. The only solution is to perform the model measurements with the same material system and to change the model scales. 

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. 

Scale-up of the tableting process in the pharmaceutical industry is still an empirical process. Dimensional analysis, a powerful method that has been successfully used in other applications, can provide a solid scientific basis for tableting scale-up. It is a method for producing dimensionless numbers that completely describe the process. The analysis should be carried out before the measurements are made, because dimensionless numbers essentially condense the frame in which the measurements are performed and evaluated. It can be applied even when the equations governing the process are not known. 

The disadvantages of using empirical correction factors, which lump many parameters together, becomes clear when one considers that a and 0 have been found to change depending on not only the concentration and type of contaminants, but also on the hydrodynamics of the system. Clearly, a better understanding of the relationship between physical properties and kLa and the quantification of these physical properties in (waste-)water is necessary, so that correlations based on dimensional analysis can be made. However, from the practical point of view, the empirical correction factors have proven their worth, when measured and used appropriately. 

Dimensional analysis shows that the measurements of the smallest eddy LK and the size of the strained molecule L lie within the same order of magnitude. 

The goal of many centrifugation experiments is the measurement of s. This value is important because it can be used to calculate the size (molecular weight, kilo base pairs, etc.) of a molecule or cell organelle. The units of r are not obvious from Equation 7.8. Dimensional analysis shows the following v in cm/sec, w in radians/sec, r in cm, m0 in grams, v in cm3/g, p in g/cm3, and f in g/sec. Therefore, the unit for s is second. 

Because many experiments involve numerical calculations, it s often necessary to manipulate and convert different units of measure. The simplest way to carry out such conversions is to use the dimensional-analysis method, in which an equation is set up so that unwanted units cancel and only the desired units remain. It s also important when measuring physical quantities or carrying out calculations to indicate the precision of the measurement by rounding off the result to the correct number of significant figures. 

You probably could have determined how many seconds there are in 1 week without using dimensional analysis, but for more difficult problems this strategy can be a most valuable resource. Table 4.1 lists some standard measurements conversions. 

This relationship is shown in Fig. 88 for four boat types and four Championships and proves an excellent agreement between the prediction given by dimensional analysis and the measured data. 

There are a variety of problem-solving strategies that you will use as you prepare for and take the AP test. Dimensional analysis, sometimes known as the factor label method, is one of the most important of the techniques for you to master. Dimensional analysis is a problem-solving technique that relies on the use of conversion factors to change measurements from one unit to another. It is a very powerful technique but requires careful attention during setup. The conversion factors that are used are equalities between one unit and an equivalent amount of some other unit. In financial terms, we can say that 100 pennies is equal to 1 dollar. While the units of measure are different (pennies and dollars) and the numbers are different (100 and 1), each represents the same amount of money. Therefore, the two are equal. Let s use an example that is more aligned with science. We also know that 100 centimeters are equal to 1 meter. If we express this as an equation, we would write ... 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

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