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

Answer:

$\frac{Q}{ND_{pi}^{3}}=(\frac{Vis}{pND_{pi}^{2}} )(\frac{D_{pr}}{D_{pi}} )$

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:

$Q=(\frac{Vis}{pND_{pi}^2} )^c(ND_{pi}^3)(\frac{D_{pr}}{D_{pi}} )^e$

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

$\frac{Q}{(ND_{pi}^3)} =(\frac{Vis}{pND_{pi}^2} )^c(\frac{D_{pr}}{D_{pi}} )^e$