Scale-model experiments dimensionless numbers

It is only po.ssible to obtain similar solutions in situations where the governing equations (Eqs. (12.40) to (12.44)) are identical in the full scale and in the model. This tequirement will be met in situations where the same dimensionless numbers are used in the full scale and in the model and when the constants P(i, p, fj.Q,.. . have only a small variation within the applied temperature and velocity level. A practical problem when water is used as fluid in the model is the variation of p, which is very different in air and in water see Fig. 12.27. Therefore, it is necessary to restrict the temperature differences used in model experiments based on water. 

Is one model scale sufficient or should tests be carried out in models of different sizes One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called process point in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. 

Dimensionless numbers are commonly used in diffusion and advection models because they simplify the scaling of the models from laboratory experiments to practical dimensions. This approach also takes advantage of the Buckingham n theorem (Barenblatt, 2003), which states that n independent variables with k independent dimensions can be expressed in terms of p independent dimensionless numbers. 

An offshoot of dimensional analysis is the theory of similarities, which sets out a way to model a physical system at a different scale without altering its nature. This theory is commonly used in mechanics, as well as in chemical engineering, whenever a pilot experiment is involved. Similarity theory is based on defining dimensionless numbers to describe a physical phenomenon. 

It can be concluded that none of the simple theories give a completely general correlation that can be applied to any combination of the relative cyclone proportions. The vast majority of them, however, show that for a family of geometrically similar cyclones, there is a dimensionless group (we shall call it the cyclone number ) which should be a constant. This constant can be obtained from experiment, rather than from the correlations given by the various theories, and this approach leads to much more reliable performance prediction. This is the approach adopted in the chemical engineering model for hydrocyclone scale-up. 

In fully turbulent flow, viscous forces become negligible relative to turbulent stresses and can be neglected (except for their action at the dissipative scales of motion). This has an important implication above a certain Reynolds number, all velocities will scale with the tip speed of the impeller, and the flow characteristics can be reduced to a single set of dimensionless information, regardless of the fluid viscosity. One experiment in the fiiUy turbulent regime can be applied for all tanks that are exactly geometrically similar to the model, at all Reynolds numbers... 

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