An engineering model is a physical or mathematical representation of a prototype engineering system design that is utilized to predict some aspects of the prototype behavior. Physical models of smaller size may be used in wind tunnels to obtain information on the prototype flow around airplanes, automobiles and skyscrapers. With the advent of supercomputers, flow over aircraft under conditions that cannot be simulated in a wind tunnel can now be calculated using mathematical models of such flow. Short range forecasts of weather conditions are aided by the exercising of mathematical models of the atmospheric circulation system. In this section, we will concentrate on the physical modeling of engineering systems as carried out in laboratory experiments.
The use of engineering models goes back at least to 1872 when William Froude built the first towing tank to test the resistance of ocean vessels. A small wind tunnel was utilized by the Wright brothers to develop their first aircraft. Nowadays wind tunnels, towing tanks and hydraulic models of dams, rivers and harbors are commonplace aids to the design of engineering systems. The usefulness of such models relies
upon the principles that underlie the relation, called similitude, between the model characteristics and those of the prototype.
Similitude
In geometry we learned how to prove the similarity of plane figures, such as triangles and rectangles. Similar figures had the same shape but different sizes. Corresponding angles must be equal and the ratio of the lengths of corresponding sides also must be equal. This illustrates the principle of geometric similitude. As applied to modeling, it requires that the shapes of the model and the prototype be identical, but the size may be different by a geometric scale ratio SR that equals the ratio of their characteristic lengths:where and are the same characteristic length (such as ship length or aircraft wing span) of the model and prototype, respectively.
If we were to take photographs of a model and its prototype, the photographs would be indistinguishable if taken at distances from the object that are in the same proportion to and .
If we are modeling an engineering structure that moves through a fluid, we must ensure that the model moves with time in a manner that is similar to that of the prototype. This will be achieved if the ratio of the acceleration of the structure to the acceleration of a fluid particle is the same for both model and prototype. For example, if we want to model a wind turbine of radius L rotating at an angular speed in a wind of speed , we must require that the ratio of the centrifugal acceleration of the turbine, , to a typical acceleration of a fluid element, , should be the same for both model and prototype:
which is equivalent to the equality of the Strouhal number for both model and prototype. Note that, in combination with equation 10.34, this implies a constraint on the value of the flow length :
This illustrates the principle of kinematic similitude, where the acceleration of the fluid and its boundaries are maintained in the same ratio for both model and prototype. Using the photographic analogy, kinematic similarity between model and prototype implies that moving pictures of the model and prototype flows would appear identical if the framing rates are proportional to and and the flow speeds satisfy the relation of equation 10.36.
Although geometric and kinematic similitude are necessary conditions for a model experiment to portray a prototype flow, they are not sufficient. We need to ensure that the entire flow fields of the model and prototype flows are similar. One way to assure ourselves that this will be so is to require that the solutions to the dimensionless conservation laws, as described in section 10.2.4, are identical for both model and prototype flows. For this to be true, the dimensionless variables such as etc., must be the same for both model and prototype since they appear as parameters in the dimensionless expression of the conservation laws. However, these dimensionless variables are the same as those 's that we derived in undertaking a dimensional analysis of a flow problem. We can then ensure a complete dynamical similitude between model and prototype flow by requiring that the model and prototype 's be equal:
The similarity of model and prototype flow fields is thus ensured if the length scales of all dimensions conform to geometric similarity (equation 10.34) and if the independent dimensionless 's are the same for both model and prototype (equation 10.37). These conditions enable us to relate the dimensional model and prototype variables to each other, thereby providing a prediction of the prototype variables from a measurement of the model values. This is the objective of the model experiments.
As an example of modeling, suppose that we construct a laboratory scale model of a complex piping system in order to investigate the pressure drops and flow rates in the system. To ensure similitude, we need a geometrically similar model of scale ratio SR (all pipe diameters, lengths and roughness heights are in proportion to SR) and all independent 's must be equal for both model and prototype. Using the dimensional analysis of equation 10.17 for pipe flow, the independent 's may be chosen to be the Reynolds number and the roughness ratio . Because we have a scale model, the roughness ratio equality is ensured:
by virtue of equation 10.34. The equality of the Reynolds numbers requires:
If we decide to use the same fluid in the model as is used in the prototype flow (although this is not necessary ), the model flow speed must be higher than the prototype speed by the factor .
Changes in pressure between any two points in the system will be proportional to the pressure gradient times the diameter D, but because the roughness ratio and Reynolds number of the model and prototype are each equal, so must be the friction factors (see equation 10.17):
where we have used equation 10.38 to eliminate the velocities in 10.39. If the fluids are the same for model and prototype, then the prototype differences in are smaller than the model values by the factor , for corresponding points in the piping system.
No comments:
Post a Comment