The flow through a closed, circular sectioned pipe may be metered by measuring the speed of rotation of a propeller having its axis along the pipe central line. using dimensional analysis, derive a set of non-dimensional parameters describing the relationship between the volume flow rate and relevant parameters of the system.

Explanation:

To solve this problem we have to make a dimensional analysis:

First, we have to write the variables involved and their dimensions:

1. Volume flow rate = Q

2. Speed of rotation= N

3. Density =ρ

4. Viscosity = Vis

5. Propeller diameter=

6. Pipe diameter=

Second, we have to write the fundamental dimensions:

Lenght = L

Mass= M

Time =T

Third, we must express the variable we want to know as a product of the other variables and to each variable we have to assign a respectic exponent:

We have to express the variables with the fundamental dimensions:

Fourth, developing and agrupating the similar terms, we have:

From the previous equations we deduce:

Now, we have to substitute the found exponents into the first equation that we wrote:  

Developing and Agrupating the terms with the same exponent we get:

Finally, the three non-dimensional group terms which describe the volume flow rate in terms of the relevant parameters of the system are:

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