Question

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.

Answer 1

`(Q)/(ND_(pi)^(3))=((Vis)/(pND_(pi)^(2)) )((D_(pr))/(D_(pi)) )`

Step-by-step 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=
`D_(pr)`

6. Pipe diameter=
`D_(pi)`

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:

`Q=(N^(a))(p^(b))(Vis^(c))(D_(pi) ^(d))(D_(pr) ^(e))`

We have to express the variables with the fundamental dimensions:

`(L^(-3)T^(-1))=(T^(-1) )^(a)(ML^(-3) )^(b)(ML^(-1)T^(-1) )^(c)(L)^(d)(L)^(e)`

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

`L^(3)=L^((-3b-c+d+e))`

`T^(-1)=T^((-a-c))`

`0=M^((b+c))`

From the previous equations we deduce:

`a=1-c`

`b=-c`

`d=3-2c-e`

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

`Q=(N^(a))(p^(b))(Vis^(c))(D_(pi) ^(d))(D_(pr) ^(e))`

`Q=(N^(1-c))(p^(-c))(Vis^(c))(D_(pi) ^(3-2c-e))(D_(pr) ^(e))`

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

`Q=((Vis)/(pND_(pi)^2) )^c(ND_(pi)^3)((D_(pr))/(D_(pi)) )^e`

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

`(Q)/((ND_(pi)^3)) =((Vis)/(pND_(pi)^2) )^c((D_(pr))/(D_(pi)) )^e`