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# Fenômentos de Transporte

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corresponding points experience forces in a constant ratio (which may or may not have the same value as the ratio for lengths). Two bodies are thermally Chapter 5: Application of Dimensional Analysis 26.5 similar if at geometrically corresponding points the ratio of the temperatures is constant. Points in the model and prototype that correspond are called homologous points. Phenomena occurring in the model and prototype2* at homologous points are said to occur at homologous times, even though these events may not occur at the same real time. An equivalent definition of similarity is that two systems are similar if they behave in the Same manner at homologous points and times. The classical principle similarity from Buckingham n-theorem is (5.5-1) Two systems in which all relevant physical quantities are in a constant ratio will satisfy Equation (5.5-1). The z\u2019s - dimensionless groups - are the criteria for geometric, dynamic, thermal, etc., similarity. If we arbitrarily restrict the f of Equation (5.5-1) to a power function (a product of powers) then II;, = c (x2)x2 (n3)x3 * * . (n\u201c-JXn-r (5.5-2) where C is called the shape factor since it has been found to depend primarily on shape, and XI, x2 ... yn - r are empirical exponents. In this relationship, which is called the extended principle (really, it is a restricted principle) of similarity, the shape factor depends on geometry; therefore, this equation is limited to systems that have the same shape. The shape factor can be eliminated if two systems of similar geometric form are compared (5.5-3) 28 One dictionary definition of prococype is \u201cthe original on which a thing is modeled.\u201d This is a little ambiguous for common engineering usage, because w e often use data from our model to design an object, in which case the object is modeled on the model as rhe original. We usually mean the laboratory or mathematical representation when we say \u201cmodel,\u201d and the real world object under investigation or to be created when we say \u201cprototype.\u201d 266 Chapter 5: Application of Dimensional Anulysis where the prime distinguishes between the systems. Use of the analysis in this manner is called extrapolation since the values of the exponents x2, x3, etc., are extrapolated from one system to the other. It is possible to eliminate the effect of the exponents by comparing geometrically similar systems which have the same values of the corresponding dimensionless groups on the right-hand side. This makes all ratios on the right- hand side equal to unity, and therefore the right-hand side is equal to unity independent of the values of the exponents, thus (5.5-4) In general, when two physical systems are similar, knowledge about one system provides knowledge about the other. Equation (5.5-4) is called the principle of corresponding states. Often it is not possible to match the values for every pair of corresponding groups as required above. In such cases we must run our experimental models so as to minimize the unmatched effects. If it is not possible to do this, we get a scale effect. For this reason, when running models, one has to be careful that values of the groups are appropriate to both the model and the prototype For example, if in testing a model of a breakwater in the laboratory, one uses a very small scale (shallow depth), in the data obtained the surface tension of the water may have a large effect on the behavior; however, at the scale of an actual breakwater the surface tension of the water would usually have no perceptible effect. We can define scale factors as the ratio of a parameter in the model, Xmdel, to that in the prototype, Xproto. (5.5-5) le 5.5-1 Drap on immetsed body Consider scaling the drag on a body immersed in the stream of incompressible fluid. The following equation describes the flow, where 9 (. .) is functional notation. Chapter 5: Application of Dimensional Analysis 267 (5.5-6) If we wish to measure the same force in the model as in the prototype, and wish to use the Same fluid in each, determine the parameters within which we must construct the model. Solution For (Pmdcl to be the same as (Ppoto, the Reynolds numbers of the model and prototype must be equal. In terms of the scale factors = 1 If the same fluid is used in model and prototype K, = 1 K, = 1 Then 2 1 KD K v K p - KD K v ( l ) KP - (1) K,K, = 1 (5.5-7) (5.5-8) (5.5-9) (5.5- 10) so the product of D and v needs to correspond in model and prototype. Then 268 Chapter 5: Application of Dimensional Analysis F = 1 F- - 1 K Kt Kk which implies (5.5-1 1) (5.5-12) (5.5- 13) (5.5- 14) (5.5- 15) Therefore, so long as the same fluid is used, KF will remain equal to one for the same values of D and v in the model and prototype; that is, the force in the model and that in the prototype will be the same. Dle 5.5-2 Scale effech We wish to estimate the pressure drop per unit length for laminar flow of ambient atmospheric air at 0.1 ft/s in a long duct. Because of space limitations, the duct must be shaped like a right triangle with sides of 3,4, and 5 feet. We do not have a blower with sufficient capacity to test a short section of the full-size duct, so it has been proposed that we build a small-scale version of the duct, and Chapter 5: Application of Dimensional Analysis 269 further, that we use water as the fluid medium in order to achieve larger and therefore more accurately measurable pressure drops in the model? Discuss how to design and build the laboratory model. Solution Since there is no reason to create more scaling problems than we already have, we will make the laboratory model geometrically similar to the full-size duct. We will therefore be able to characterize it by a single length dimension, which we will designate as D. In the laminar region, we can expect the friction factor, f, which is a vehicle (to be discussed in detail in Chapter 6) for predicting pressure drops in internal flows, to be a function of only the Reynolds number, the same dimensionless group as used in the previous example. friction factor = f = f (Re) = f -F ("" (5.5- 16) In the laminar region the friction factor is not a function of relative roughness of the surface; therefore, we are free to choose the most easily fabricated material to build the laboratory model. The pressure drop is related to the friction factor for this case as (5.5-17) where C is a constant. If we choose to make the friction factors of the model and prototype the same, the scaling law follows as in the immediately preceding example = 1 ~ ~ ~ 29 This problem can be solved quite nicely by numerical methods if the necessary computing equipment is available. We ignore this alternative for pedagogical reasons in order to obtain a simple illustration. 270 Chapter 5: Application of Dimensional Analysis (5.5-18) This time, however, we have prescribed the scale factors for the fluids used. Approximate values at room temperature are 1 CP = 5 . 5 6 ~ 103 = K, 1.8 x 104 cP Therefore, to keep the Same friction factor will require KDK, = 7.2 Examining the scaling for pressure drop However, we intend to make Kf = 1.0, so (1) (772) K2, KD K(?) = K(?) = ( 7 7 2 ) z (5.5- 19) (5.5-20) (5 521) (5.5-22) (5 5 2 3 ) (5.5-24) (5 5 2 5 ) Chapter 5: Application of Dimensional Analysis 271 We now have considerable flexibility in choosing the size of our model and the velocity at which we pump the water. For example, suppose that we decide it will be easy to fabricate a model duct with hypotenuse of 2 inches. This will set the scale factor