Stahl67 bmp

170 WALTER R. STAHL

Consider a small-scale test model of a new type of turbinę, which would be too expensive to build full-size without assurance of proper operation and which is too complex for a definitive analysis by mathe-matical analysis. A geometrically similar scalę model, usually madę of the same materials, can be madę, but it is critically important to define how its operating parameters correspond with the larger prototype. For example, if fluid flow pattem is important, data from the model can be extrapolated to the larger unit if and only if the Reynolds number is invariant for both. The Reynolds number is dimensionless and has the composition

Re = vLp/ii    (1)

where L is the characteristic length of geometrically similar systems, e.g., turbinę diameter; v is some characteristic flow velocity, e.g., veloc-ity of turbinę blade tip or center stream of output flow streamline; p is the density of fluid; ij is the viscosity of fluid. If the same fluid is used in the model and prototype (p and rj are invariants), then invariance of the Reynolds number reąuires that (vL) = invariant and a decrease in length by a factor of ten reąuires an increase in velocity by the same factor.

There are also other different kinds of fluid flow models. Another major class is based on inyariance of the Euler number:

Eu = pv²/P    (2)

in which P is the characteristic pressure. This criterion governs flow models in which viscosity effects are smali and leaves velocity unchanged with size; it is incompatible with the Reynolds number unless a different fluid is used in the model and prototype.

Approximately a hundred twenty named dimensionless numbers, and thirty to fifty morę unnamed ones, are known and many of them dis-cussed elsewhere (Stahl, 1967). Any given physical model may be governed by several to over ten different similarity criteria simultane-ously. The entire model design problem is handled by defining specific numerical functions relating the pertinent similarity criteria, which is equivalent to definition of a physical system, such as a “rotary pump” or “internal combustion engine.”

The important point to be madę here is that a dimensionless number is an example of a numerical criterion of similarity whose inyariance indicates that ą model is physically similar to a prototype. This means