# Similarity Principles and Conditions for Model Experiments

The general conditions for scale-model experiments with flow in a room are

1. Identical dimensionless sets of boundary conditions, including geometry, in the room and in the model;

2. Identical dimensionless numbers, Eqs. (12.56) to (12,60), in the governing equations for the flow in the room and in the model;

3. The constants p0, /3, /u0,… in the governing equations (Eqs. (12.40) to (12.44)) should have only a small variation within the applied temperature and velocity levels.

The requirement of identical dimensionless boundary conditions is met when the model is geometrically similar to full scale in all details that are im­portant for the volume flow, the energy flow and the contaminant flow; see Fig. 12.24. "

The dimensionless supply profile from the diffuser in an experiment with general ventilation given by

U’,i>w’=f(x, y,z, t) (12.63)

Must thus be identical for both room and model. This can be obtained by us­ing a diffuser in the model that is geometrically similar to the expected diffuser in the room, but it is an expensive solution because it is difficult to reproduce all the details in the diffuser when the model, for example, is one-tenth of full scale. The problem can be simplified if it is possible to replace the diffuser by a simple opening able to generate a similar flow in the model. The problem is somewhat similar to the use of the inlet-box method or the prescribed velocity method in computational fluid dynamics; see Section 11.2. O
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 FIGURE 12.26 Velocity decay in a wall jet along the ceiling in a room and in a model. Is possible to obtain an identical dimensionless temperature distribution T*(x, y, z) in the full scale and in the model. The conditions for model experiments can be explained in the following way. The governing equations are made nondimensional in the full scale and in the reduced scale used in the model experiments. For example, the velocity in the room is divided by the diffuser velocity in the room, and the velocity in the model is divided by the supply velocity in the model in order to normalize all velocities. The two sets of equations are identical and they describe the same solution provided that requirements 1, 2, and 3 mentioned at the begin­ning of this section have been met. It is difficult to carry out a model experiment on a reduced scale if all the dimensionless numbers must be kept constant. If, for example, the scale is re­duced by a factor of 10, then the velocity also has to be increased by a factor of 10 due to the Reynolds number, which will give an increase in the temperature difference by a factor of 1000 in order to keep the Archimedes number. The Prandtl number is, on the other hand, unchanged when air is used as the fluid in the model experiments. The problem with the temperature level can be slightly reduced when water is used as fluid in the model experiments and, as shown in the next section, the problem can also be reduced when the flow is a fully developed turbulent flow. It is only possible 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 requirement 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(), /3, Fi(),… 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 /3, 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. 