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# Fenômentos de Transporte

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four fundamental dimensions) F P Cr D gc V F M L t 1 0 0 0 0 1 - 3 0 0 1 -1-1 0 0 1 0 0 0 1-1 -1 1 1 -2 (5.2.1-2) It can be shown that this matrix is of rank 4, so in a complete set we should be able to obtain 6 - 4 = 2 dimensionless products. Carrying out the dimensional analysis yields 228 W1 Chapter 5: Application of Dimensional Analysis [q.),i")] p v2 D2 = 0 (5.2.1-3) In principle, we can solve W I explicitly for (p:fb2) - to obtain a new function (*) = W2[(.?)] This relation may be rewritten as (5.2.1 -4) (5.2.1-5) where (p v2) is proportional to the kinetic energy per unit volume and D2 is proportional to a characteristic area. For internal flows in circular conduits this may be expressed a.. (5.2.1-6) where we have used the bulk velocity as the characteristic velocity, the product (R L) in place of the square of the characteristic length, and the conduit diameter as the characteristic length in the Reynolds number (for geometrically similar systems, defined below, these lengths remain in constant ratio to one another). The Fanning friction factor, f, is then defined by the relation F = (2xRL)($p(v)*)f [(D(v)p,l .p (5.2.1-7) where the first factor on the right-hand side is the lateral area of the conduit and the second factor on the right-hand side is the kinetic energy per unit volume. The function f is just the function ~2 divided by (ng,). Consistent with our practice throughout the text, we do not incorporate g, in the formula, because Chapter 5: Application of Dimensional Analysis 229 the definition of f is the same for all systems of units, and when g, is needed it is self-evident by dimensional homogeneity. The friction factor is a dimensionless function of Reynolds number. A friction factor (unfortunately also designated by f') four times the magnitude of the Fanning friction factor is sometimes also defined in the literature. This friction factor is called the Darcy or BIasius friction factor. Two different friction factors, both designated by f, cause no real difficulty as long as one is consistent in which friction factor one uses and observes the factor of four. The biggest danger of mistake is usually in using a value from a chart or table which incorporates one friction factor in a formula using the other. For external flows, Equation (5.2.1-5) may be written as F = ( q q w " ( y ) ] (5.2.1-8) where we now have a different function because we have a different flow situation. For flow around objects, we usually convert the square of the characteristic length to the cross-section of the object normal to flow. The reference velocity usually used in the kinetic energy term and in the Reynolds number is the freestream velocity (the velocity far enough away from the surface that there is no perturbation by the object). The characteristic length, if the cross- sectional area is circular, is the diameter of the circle - if not, a specific dimension must be specified. Then (5.2.1 -9) where CD, called the drag coefficient, is a dimensionless function of Reynolds number (CD and ~3 are different functional forms because of the constants introduced). Again, we do not incorporate g, because the definition is valid for any consistent system of units. 5.2.2 Shape factors Dimensional analysis can seldom be used to predict ways in which phenomena are affected by different shapes of object. Therefore, when applying dimensional analysis to a problem it is usually convenient to eliminate the 230 Chapter 5: Application of Dimensional Analysis consideration of shape by considering bodies of similar shape, that is, bodies that are geometrically similar. D r u force on ship hull Tbe drag force exerted on the hull of a ship depends on the shape of the hull. Hulls of similar shape may be specified by a single characteristic dimension such as keel17 length, L, and hulls of other lengths but similar shapes have their dimensions in the same ratio to this characteristic length - for example, the ratio of beam width (width across the hull) to length will be the same for all hulls of similar shape. If the hulls are considered to have the same relative water line (for example, with respect to freeboard, the distance of the top of the hull above the waterline), keel length will also be proportional to the draft (displacement). The drag force will then depend on the keel length, L, the speed of the ship, v, the viscosity of the liquid, p, the density of the liquid, p, and the acceleration of gravity (which affects the drag produced by waves). What are the dimensionless groups that describe this situation? So 1 u t ion Our function is (5.2.2- 1) The rank of the dimensional matrix may be shown to be 3 for a system with three fundamental dimensions. We will therefore have (6 - 3) = 3 dimensionless groups. If we carry out the dimensional analysis, we obtain (5.2.2-2) (the dimensionless group F/[pv2L2] is sometimes called the pressure coefficient, P) . l* l7 The keel is the longitudinal member reaching from stem to stern to which the ribs are attached. I * Langhaar, H. L. (1951). Dimensional Analysis and Theory of Models. New York, NY, John Wiley and Sons, p. 21. Chapter 5: Application of Dimensional Analysis 231 To run experiments, we would fearrange (5.2.2-1) as and would rearrange (5.2.2-2) as F= P = w2(Re, Fr) p v2 L2 (5.2.2-3) (5.2.2-4) In each case the experiments would consist of measuring the variable on the left- hand side of the equation as a function of the variables on the right-hand side of the equation, which would be varied from experiment to experiment. Notice the numerous advantages of (5.2.2-4) over (5.2.2-3). Using (5.2.2-3), to run three levels of each variable would require 35 = 243 experiments. Using (5.2.2-4), to run three levels of each variable would require only 32 = 9 experiments. . To use (5.2.2-4), we could easily vary both Reynolds and Froude numbers by varying velocity (which is easily accomplished in a towing tank) and hull length (which could be accomplished by using models of the same shape but different sizes). To use (5.2.2-3), we would also have to find a way to vary p (perhaps by vaqing temperature or fluid or adding a thickening agent to water), p (perhaps by varying fluid; or dissolving something in water to vary density; or varying temperature to vary density, even though this is not a very big effect), and g (perhaps by building two laboratories, one on a very high mountain and one in a mine, or building the experiment in a centrifuge). None of these alternatives is particularly appealing. To present the results of (5.2.2-3) in a compact form is difficult, as is extrapolating or interpolating the results. The results of (5.2.2-4), however, can be concisely presented on a two-dimensional graph, where the approximate results of interpolation or extrapolation can be easily visualized, as is illustrated in the following figure (the data is artificial and must not be used for design purposes). The figure shows the data from nine hypothetical experiments arranged in 232 Chapter 5: Application of Dimensional Analysis a factorial design (each of three levels of Re combined with each of three levels of Fr, and hypothetical curve fits which could be used for interpolation (or extrapolation, if the curves were to be extended beyond the range of the data - which would be dangerous). Fr = 6 F r = 4 Fr = 2 Reynolds number vle 5.2.2-2 Deceleration o f corUpressible -fh&i The important variables that determine the maximum pressure, pmax, resulting when flow of a compressible fluid is stopped instantly (for example, by shutting a valve