Example Dimensional Analysis

The distance that an accelerating object covers in a certain time is given by Equation (2), s = ut + ar. So, four variables, s, w, a and t, are needed to describe this system. If we are interested in the distance travelled, we can write 

To represent the effects of all the possible values of a, we require an infinite number of curves on this figure as a can take an infinite number of values. Now, this is just for one value of u, and u may itself take an infinite number of values. Therefore to represent s for all the possible values of t, a and u, we need an infinite number of figures (all possible u values), each with an infinite number of curves (all possible a values). This, clearly, is not very convenient. 

We saw earlier that Equation (2), s = ut + jat2, was dimensionally homogeneous each term has the same dimensions. Ratios of the terms in this equation will therefore be dimensionless. So we can make the original expression dimensionless by, for example, dividing each term 

1 Alternate Form. If instead we were interested in distance travelled 5, as a function of initial velocity u, we would wish 5 and u to appear only once each and in separate groups. So, taking s = ut + ar and dividing each term by at2 gives 

With dimensional analysis an equation can take several equally valid forms, e.g. sjut = f(at/u) or sjat2 = f(u/at). 

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On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

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. 

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... 

There are few problems of praetleal interest that ean be adequately approximated by one-dimensional simulations. As an example of sueh, eertain explosive blast problems are eoneerned with shoek attenuation and residual material stresses in nominally homogeneous media, and these ean be modeled as one-dimensional spherieally symmetrie problems. Simulations of planar impaet experiments, designed to produee uniaxial strain loading eonditions on a material sample, are also appropriately modeled with one-dimensional analysis teehniques. In faet, the prineipal use of one-dimensional eodes for the eomputational analyst is in the simulation of planar Impaet experiments for... 

Dimensional analysis techniques are especially useful for manufacturers that make families of products that vary in size and performance specifications. Often it is not economic to make full-scale prototypes of a final product (e.g., dams, bridges, communication antennas, etc.). Thus, the solution to many of these design problems is to create small scale physical models that can be tested in similar operational environments. The dimensional analysis terms combined with results of physical modeling form the basis for interpreting data and development of full-scale prototype devices or systems. Use of dimensional analysis in fluid mechanics is given in the following example. 

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

In the Rayleigh method of carrying out a dimensional analysis the dependent variable is assumed to be proportional to the product of the independent variables raised to different powers. By equating dimensions, the number of independent dimensionless groups and one set of their possible forms can be obtained. By way of illustration two examples may be considered. 

The procedure for performing a dimensional analysis will be illustrated by means of an example concerning the flow of a liquid through a circular pipe. In this example we will determine an appropriate set of dimensionless groups that can be used to represent the relationship between the flow rate of an incompressible fluid in a pipeline, the properties of the fluid, the dimensions of the pipeline, and the driving force for moving the fluid, as illustrated in Fig. 2-1. The procedure is as follows. 

As an example of the application of dimensional analysis to experimental design and scale-up, consider the following example. 

This manipulation of units is sometimes called dimensional analysis. Strictly speaking, though, dimensional analysis is independent of the units used. For example, the units of speed may be in metres per second, miles per hour, etc., but the dimensions of speed are always a length [L] divided by a time [T] ... 

Via Dimensional Analysis. Again, let the rate constant be k. In this example, imagine the value of k is 3.2 year-1. 

Dimensional analysis, sometimes called the factor label (unit conversion) method, is a method for setting up mathematical problems. Mathematical operations are conducted with the units associated with the numbers, and these units are cancelled until only the unit of the desired answer is left. This results in a setup for the problem. Then the mathematical operations can efficiently be conducted and the final answer calculated and rounded off to the correct number of significant figures. For example, to determine the number of centimeters in 2.3 miles ... 

You may be wondering Why bother with dimensional analysis at all There is a very good reason to know this method. It is the method scientists prefer because it allows you to do several conversions at once. Study the next example. Both methods of converting units will be used. 

When making these conversions in the field, it is often helpful to make approximations or estimates. One pound is a little more then 2 kilograms, so divide the persons weight in pounds by 2 and subtract an appropriate approximation for the decimal amount. For example, 176 pounds is 80 kilograms 176/2 = 88. You can approximate to deduct about 8 kilograms for the 0.2 decimal and subtract it to get the result, which is 80. Refer back to the section on dimensional analysis. This technique is very helpful in many of these problems. 

The elegant solution of this first example should not tempt the reader to believe that dimensional analysis can be used to solve every problem. To treat this example by dimensional analysis, the physics of unsteady-state heat conduction had to be understood. Bridgman s (2) comment on this situation is particularly appropriate ... 

This transparent and easy example clearly shows how dimensional analysis deals with specific problems and what conclusions it allows. It should now be easier to understand Lord Rayleigh s sarcastic comment with which he began his short essay on The Principle of Similitude (3) ... 

Pertinent examples of the value of dimensional analysis have been reported in a series of papers by Maa and Hsu (19,37,63). In their first report, they successfully established the scale-up requirements for microspheres produced by an emulsification process in continuously stirred tank reactors (CSTRs) (63). Their initial assumption was that the diameter of the microspheres, <7ms, is a function of phase quantities, physical properties of the dispersion and dispersed phases, and processing equipment parameters ... 

Although most physical properties (e.g., viscosity, density, heat conductivity and capacity, surface tension) must be regarded as variable, it is particularly the value of viscosity that can be varied by many orders of magnitude under certain process conditions [5,11]. In the following, dimensional analysis will be applied, via examples, to describe the temperature dependency of the density und viscosity of non-Newtonian fluids as influenced by the shear stress. 

It should be mentioned that, as is often the case in a first draft, the author based his treatment of the turbulent convection on certain assumptions which in fact were not necessary, in particular the semi-empirical concept of L. Prandtl. Moreover, even in the analysis of laminar convection, the author [cf., for example the transition from equation (9) to equation (10)], to derive the asymptotic laws, resorts to simplifications of the equations which are really not necessary. Actually, it is possible to manage without these assumptions so that Zeldovich s asymptotic laws (8), (8a), (11), and (11a) may be obtained by simple dimensional analysis under the most general assumptions. 

Other examples can be discussed in the same way. We note that all these results follow from simple dimensional analysis in terms of coil radius Rg, chain concentration cp, and (if present) momentum variables. The method is based on the assumptions that... 

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