2 dimensional analysis

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. 

Consider the classical problem of pressure drop during flow in a smooth straight pipe, ignoring the inlet effects. The first step is to list all possible variables or quantities that are related to the problem under consideration. In this case, we have  

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. 

In this example there are m = 6 variables and since one only finds mass, length and time one can say that one has n = 3 dimensional quantities. Hence one can generate m — n = 3 dimensionless groups denoted by ni, n2 and n3. From the list above n = 3 repeating variables are selected. These variables can appear in all the dimensionless numbers. When selecting the repeating variables it is important that 

Here one can choose D, p, and p as the repeating variables. The first dimensionless group that was generated involves Ap. One can write the product of the repeating variables and Ap, where each of the repeating variables has an exponent that will render the whole product dimensionless 

Dimensional analysis is rarely taught in pure chemistry, rather in engineering chemistry. Most of the treatises on dimensional analysis content themselves to show how to convert units. Perhaps the most well-known application of dimensional analysis in pure chemistry is one dealing with the conversion of units, that is a part of dimensional analysis in the law of corresponding states [1]. The chapter is intended to show that there are other fields in chemistry that would benefit from dimensional analysis. In particular, it is shown how to establish Stokes law in a simple way using dimensional analysis. Further, dimensional analysis can be used to find rather unexpected relations. This should be used to motivate the reader to think about the context of various scientific quantities. 

There are several papers on dimensional analysis [2-5], but most of them are dealing with the conversion of units that is a part of dimensional analysis. Dimensional analysis in the more advanced sense is presented rather in the older literature [6-8] or in textbooks [9, 10]. One paper is dealing with the theorem of the corresponding states [1], which is a more advanced form of dimensional analysis. In summary, dimensional analysis is used rather in engineering sciences, including technical chemistry, than in pure chemistry. 

In an introductory part, we will discuss the issue and origin of physical units. Some historical facts in particular concerning the origin of units in Europe and the US are published online [11]. The history of the Bureau International des Poids et Mesures is documented elsewhere [12]. 

The process of dimensional analysis can be carried out in three deceptively simple steps. Some comments are necessary to ensure the proper application of this scheme  

Step 1. List all the variables that affect the system behavior. This is by far the most important and difficult step and confronts the user with the task of deciding which independent variables to include in the analysis. There is no easy recipe for carrying out this step, but the following suggestions may be foxmd useful  

Step 2. Write down the dimensions cf these variables. The basic dimensions that are commonly chosen in dimensional analysis are those of mass (M), length (L), time (0), and temperature (T). All other quantities are expressed in terms of these fundamental dimensions. Thus, force has the units of MLQ by virtue of Newton s law. Joule (J) is not a fundamental dimension but is instead expressed as ML 0 .  

Step 3. Combine the variables into a functional relationship involving dimensionless groups or some other dimensionally consistent form. The ultimate aim of this step, and of the analysis as a whole, is to express system behavior in terms of the functional relationship  

it is convenient to solve this relationship for the dependent variable. We then obtain 

Sometimes you can catch an error in the form of an equation or expression, or in the dimensions of a quantity used for a calculation, by checking for dimensional consistency. Here are some rules that must be satisfied  

In this book the differential of a funetion, such as d/, refers to an infinitesimal quantity. If one side of an equation is an infinitesimal quantity, the other side must also be. Thus, the equation d/ = a dx + bdy (where ax and by have the same dimensions as /) makes mathematical sense, but d/ = ax + b dy does not. 

Derivatives, partial derivatives, and integrals have dimensions that we must take into account when determining the overall dimensions of an expression that includes them. For instance  

Some examples of applying these principles are given here using symbols described in Sec. 1.2. 

Example 2. What are the dimensions of the quantity nRT n. p/p°) and of p° in this expression The quantity has the same dimensions as nRT (or energy) because the logarithm is dimensionless. Furthermore, p° in this expression has dimensions of pressure in order to make the argument of the logarithm, p/p°, dimensionless. 

At this point, it is appropriate to introduce the important subject of dimensional analysis of the double reciprocal plots, a subject that was mentioned in the preceding chapter (Section 2.9). In the scientific and technical literature, the presentation of double reciprocal plots falls into two main categories (Fig. 6). 

In the first case (A), the reciprocal value of the initial rate of reaction (i , dimension mol liter s ) is shown as the function of the reciprocal substrate concentration (A, dimension mol liter ). The slope of this plot is equal to KjVmax and has a dimension of time (s). The advantage of this presentation is that the knowledge of the actual concentration of enzyme (,) is not required. In the second case (B), the reciprocal value of the apparent catalytic constant = Uo/ oidimension s ) is plotted against the reciprocal substrate concentration. In this case, the slope of the plot is equal (remember that 

The first plot (A) is easily transformed into the second plot (B), simply by dividing all initial rates of reaction, Vo, with the concentration of enzyme, Eo. Therefore, the difference between the two presentations appears trivial. However, this is not the case, because both presentations appear in the literature a failure to appreciate the difference can produce serious inconveniences with bisubstrate and trisubstrate reactions and especially with the interpretation of replots of slopes and intercepts on ordinate, because of the difference in dimensions (lUBMB, 1992). 

Comish-Bowden, A. (1976) Principles of enzyme kinetics, Butterworths, Lxjndon. 

Let us use a control volume approach for the fluid in the boundary layer, and recognize Newton s law of viscosity. Where gradients or derivative relationships might apply, only the dimensional form is employed to form a relationship. Moreover, the precise formulation of the control volume momentum equation is not sought, but only its approximate functional form. From Equation (3.34), we write (with the symbol implying a dimensional equality) for a unit depth in the z direction 

Dimensional homogeneity is fundamental to equations relating variables in the description of natural processes. The recognition of this basic attribute is the substance of dimensional analysis, which results in the reduction of relevant parameters to the essential minimum. Physical quantities comprise combinations of one or more basic dimensions. Table 2.2 shows some commonly used physical quantities in engineering expressed in terms of basic dimensions (mass, M length, L time, T, temperature, 0). 

The requirement of dimensional consistency places a number of constraint on the form of the functional relation between variables in a problem and forms the basis of the V chmqu 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. 

The application of the principles of dimensional analysis may best be understood by considering an example. 

It is found, as a result of experiment, that the pressure difference (AP) between two ends of a pipe in which a fluid is flowing is a function of the pipe diameter d, the pipe lengtii I, the fluid velocity u, the fluid density p, and the fluid viscosity /x. 

The form of the function is unknown, though since any function can be expanded as a power series, the function may be regarded as the sum of a number of terms each consisting of products of powers of the variables. The simplest form of relation will be where the function consists simply of a single term, or  

The requirement of dimensional consistency is that the combined term on the right-hand side will have the same dimensions as that on the left that is, it must have the dimensions of pressure. 

Both heat and mass transfer in turbulent flow are generally not amenable to analytical treatment. It is customary in these cases to resort to what is termed dimensional analysis. This device consists of grouping the pertinent physical parameters of the system into a number of dimensionless groups, thus reducing the number of variables that have to be dealt with and evaluating the undetermined coefficients experimentally. It is a powerful tool for arriving at a first qualitative description of complex systems and eases enormously the experimental work required to quantify the relationship. 

It is often necessary to convert a given result from one system of units to another. The best way to do this is by a method called the unit factor method or, more commonly, dimensional analysis. To illustrate the use of this method, we will consider several unit conversions. Some equivalents in the EngUsh and metric systems are listed in Table 1.4. A more complete list of conversion factors given to more significant figures appears in Appendix 6. 

Consider a pin measuring 2.85 cm in length. What is its length in inches To accomplish this conversion, we must use the equivalence statement 

This expression is called a unit factor. Since 1 inch and 2.54 cm are exactly equivalent, multiplying any expression by this unit factor will not change its value. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

The pin has a length of 2.85 cm. Multiplying this length by the appropriate unit factor gives 

Have you ever calculated your car s gas mileage If so, you know that you divide the miles driven by the gallons of gasoline used so that you have a number that is the miles per gallon for you car. This is a simple example of dimensional analysis. Knowing the units of the number that you want to calculate, you can figure out what calculation you need to do. Following is another example. 

After making a measurement in a particular unit, it may be necessary to express it in some alternate unit. For example, consider homeowners who are 

Any zero located between 2 significant figures is significant. 

Any zero to the ieft of nonzero digits is not significant unless it is also covered by Rule 2. 

Any zero to the right of nonzero digits and also to the right of a decimal point is significant. 

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. 

When we have a complete understanding of the physics of a problem and have no difficulty with the formulation but are mathematically stuck on the solution, we refer to dimensional analysis for a functional (implicit) form of the solution. Three distinct methods exist for dimensional analysis  

1) Formulation (nondimensionalized) Whenever a formulation is readily available, a term-by-term nondimensionalization of this formulation leads directly to the related dimensionless numbers. The procedure is not suitable to problems which cannot be readily formulated. 

2) II -Theorem 10 K a formulation is not readily accessible but all physical and geometric quantities which characterize a physical situation are clearly known, we write an implicit relation among these quantities, 

Expressing these quantities in terms of appropriate fundamental units, and making Eq. (5.120) independent of these fundamental units by an appropriate combination of Q s, yields the dimensionless numbers. 

Dimensionless numbers can be derived by making the appropriate balance equations dimensionless when the problem can be fully described. (See the example on heat transfer in Newtonian fluid between two plates later in this section.) Dimensionless numbers can also be derived from dimensional analysis this approach is 

Even though the actual form of the equation is not known, the equation has to be dimensionally homogeneous. This is also true if the force is written in the following form  

The original number of variables has now been reduced from 5 to 2 dimensionless numbers. The dimensionless number on the right-hand side of Eq. 5.52 is the well-known Reynolds number  

The solution is carried out by two methods. Because of the modular nature of twin-screw extruders, experiments were carried out in advance to determine what length of devolatilization section was required to reduce the level of MMA to 0.1%, which corresponds to a fractional separation, Fs, of about 0.85. Because the single-screw extmder is cheaper to build, it is desirable to determine the length of the DV section and the processing conditions (i.e., pQ, N, and f) required to accomplish the same reduction of MMA in PMMA as done in the SWCOR extruder. The first approach is based on dimensional analysis. In the second approach we use the penetration or diffusion theory summarized in Section 8.5.2. 

The starting point is to make sure that sufficient vacuum is available such that the equilibrium weight fraction, We, is less than the desired final weight fraction. From Henry s law and the data given in Table 8.2, we calculate at 250 °C to be 

Dimensional analysis requires that both extruders be geometrically and dynamically similar. To ensure geometric 

The ratio titp represents the number of devolatilization stages (Ns) available during the extrusion process. 

We first use the condition of geometric similarity to find the length of the DV section  

To add or subtract exponential numbers without a calculator, you need to align digit values (hundreds, tenths, units, and so on) vertically. This is done by adjusting coefficients and exponents so all exponentials are 10 raised to the same power. The coefficients are then added or subtraeted in the usual way. This adjustment is automatic on calculators. 

You probably did not round off the addition answer as we did, which is fine at this time. The reason for our round-off appears in Section 3.5. 

3 In a problem, identify given and wanted quantities that are related by a Per expression. Set up and solve the problem by dimensional analysis. 

You have probably figured out the answer already 21 days. How did you get it You probably reasoned that there are 7 days in 1 week, so there must be 3 X 7 days in 3 weeks 3 X 7 = 21. In doing this you used the problem-solving method called dimensional analysis. 

Let s examine this basic days-in-3-weeks problem in detail. The 7 days in each week relationship between these two units can be stated as 7 days per week. We call this a Per expression. This Per expression can also be written as a fraction or a ratio 7 days/week, or an equality, 7 days = 1 week. Similarly, there are 24 hours Per day (24 hours/day or 24 hours = 1 day) and 60 minutes Per hour (60 minutes/hour or 60 minutes = 1 hour). Any Per expression can be written as a fraction or an equality. We will identify Per expressions in examples by the symbol Per. 

From the physical point of view it is possible to suggest that the rate of bubble growth in micro-channel is determined by the following parameters  

In accordance with (6.40) one can present the functional equation for rate of bubble growth as follows 

Using dimensions of length L, mass M, time t, temperature T, and energy/, one can obtain dimensions of parameters on the right-hand side of Eq. (6.41)  

Among the dimensional variables of the problem, five parameters have independent dimensions and Eq. (6.41) may be written in dimensionless form. Choosing parameters Pl, CpL, U, ATs, and taking into account r-theorem (Sedov 1993), Eq. 

Data by Lee et al. (2004) and Li et al. (2004) contain the results related to bubble dynamics in a single micro-channel and two parallel ones. The experimen- 

Your objectives in studying this section are to be able to  

Explain the concept and importance of dimensional analysis in correlating experimental data on convective mass-transfer coefficients. 

And his answer trickled through my head Like water through a sieve. 

The use of dimensionless numbers improves the efficiency of experimental biology. With the use of dimensionless numbers, the number of experiments performed with all the combinations of parameter values may be reduced considerably because only the dimensionless numbers themselves must vary, not all components of the dimensionless numbers. This greatly reduces the effort needed to derive valid experimental data and to relate many experimental results together in the inductive process of forming predictive equations. 

For instance, dimensional analysis applied to the heart yields two pi (dimensionless) terms  

There have been very few attempts to apply dimensional analysis to biological systems. However, as quantitative biological research becomes ever more sophisticated, dimensional analysis will help to discover similarities and produce new scaling factors. 

It takes 10.5 s for a sprinter to run 100.00 m. Calculate her average speed in meters per second, and express the result to the correct number of significant figures. 

To calculate the density, we must know both the mass and the volume of the gas. The mass of the gas is just the difference in the masses of the full and empty container  

In subtracting numbers, we determine the number of significant figures in our result by counting decimal places in each quantity. In this case each quantity has one decimal place. Thus, the mass of the gas, 1.4 g, has one decimal place. 

Using the volume given in the question, 1.05 X 10 cm, and the definition of density, we have 

To how many significant figures should the mass of the container be measured (with and without the gas) in Sample Exercise 1.8 for the density to be calculated to three significant figures  

Unit Abbreviation Meter equivalent Exponential equivalent 

The nanometer (10 m) is used extensively in expressing the wavelength of light,  

Many chemical principles can be illustrated mathematically. Learning how to set up and solve numerical problems in a systematic fashion is essential in the study of chemistry. 

Usually a problem can be solved by several methods. But in all methods it is best to use a systematic, orderly approach. The dimensional analysis method is emphasized 

Dimensional analysis converts one unit to another unit by using conversion factors, uniti X conversion factor = unit2 

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Feiste, K.L. Stegemann, D. Reimehe, W. Three dimensional analysis of growing casting defects. International Symposium on Computerized Tomography for Industrial Applieatlons, 8.10. Junl 1994, Berlin... 

Doering, E.R. Basart,. I.P. Gray,. I.N. Three-dimensional flaw reconstruction and dimensional analysis using a real-time X-ray imaging system. NDT-I-E International, Vol. 26(1), 1993, pp. 7-17. 

Resonant timesteps can be estimated on the basis of one-dimensional analysis [65, 62] from the propagating rotation matrices in phase-space for... 

The number of terms of a complete polynomial of any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that corresponds to complete polynomials in terms of three spatial variables can also be constructed for three-dimensional analysis. 

Entrance andExit SpanXireas. The thermal design methods presented assume that the temperature of the sheUside fluid at the entrance end of aU tubes is uniform and the same as the inlet temperature, except for cross-flow heat exchangers. This phenomenon results from the one-dimensional analysis method used in the development of the design equations. In reaUty, the temperature of the sheUside fluid away from the bundle entrance is different from the inlet temperature because heat transfer takes place between the sheUside and tubeside fluids, as the sheUside fluid flows over the tubes to reach the region away from the bundle entrance in the entrance span of the tube bundle. A similar effect takes place in the exit span of the tube bundle (12). 

Static mixing of immiscible Hquids can provide exceUent enhancement of the interphase area for increasing mass-transfer rate. The drop size distribution is relatively narrow compared to agitated tanks. Three forces are known to influence the formation of drops in a static mixer shear stress, surface tension, and viscous stress in the dispersed phase. Dimensional analysis shows that the drop size of the dispersed phase is controUed by the Weber number. The average drop size, in a Kenics mixer is a function of Weber number We = df /a, and the ratio of dispersed to continuous-phase viscosities (Eig. 32). 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

Dimensional analysis (qv) shows that is generally a function of the particle Reynolds number ... 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

Other dimensional systems have been developed for special appHcations which can be found in the technical Hterature. In fact, to increase the power of dimensional analysis, it is advantageous to differentiate between the lengths in radial and tangential directions (13). In doing so, ambiguities for the concepts of energy and torque, as well as for normal stress and shear stress, are eliminated (see Ref. 13). 

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) ... 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

In applying dimensional analysis, it is first necessary to be able to identify the variables that govern a particular physical phenomenon. The naming of the governing variables requites some prior knowledge of a particular branch of physics involved. This may include analytical studies, experimental observations, or both. Whatever the source, there must be some prior knowledge upon which a selection can be made. 

Suppose that an experiment were set up to determine the values of drag for various combinations of O, p, and ]1. If each variable is to be tested at ten values, then it would require lO" = 10, 000 tests for all combinations of these values. On the other hand, as a result of dimensional analysis the drag can be calculated by means of the drag coefficient, which, being a function of the Reynolds number Ke, can be uniquely determined by the values of Ke. Thus, for data of equal accuracy, it now requires only 10 tests at ten different values of Ke instead of 10,000, a remarkable saving in experiments. 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

The functional relation ia equation 53 or 54 cannot be determined by dimensional analysis alone it must be suppHed by experiments. The significance is that the mean-free-path problem is reduced from an original relation involving seven variables to an equation involving only three dimensionless products, a considerable saving ia terms of the number of experiments required ia determining the governing equation. 

J. F. Douglas, Introduction to Dimensional Analysis forEngineers Sir Isaac Pitman Sons, London, 1969. 

H. L. Langhaar, Dimensional Analysis and Theory ofModelSs oEn Wiley Sons, Inc., New York, 1951. 

A. D. Sloan and W. W. Happ, "Literature Search Dimensional Analysis," NMSM Rept. ERClCQD 68-631 (Aug. 1968). 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

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