Respuesta :
Answer:
[tex]\frac{Q}{ND_{pi}^{3}}=(\frac{Vis}{pND_{pi}^{2}} )(\frac{D_{pr}}{D_{pi}} )[/tex]
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= [tex]D_{pr}[/tex]
6. Pipe diameter= [tex]D_{pi}[/tex]
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:
[tex]Q=(N^{a})(p^{b})(Vis^{c})(D_{pi} ^{d})(D_{pr} ^{e})[/tex]
We have to express the variables with the fundamental dimensions:
[tex](L^{-3}T^{-1})=(T^{-1} )^{a}(ML^{-3} )^{b}(ML^{-1}T^{-1} )^{c}(L)^{d}(L)^{e}[/tex]
Fourth, developing and agrupating the similar terms, we have:
[tex]L^{3}=L^{(-3b-c+d+e)}[/tex]
[tex]T^{-1}=T^{(-a-c)}[/tex]
[tex]0=M^{(b+c)}[/tex]
From the previous equations we deduce:
[tex]a=1-c[/tex]
[tex]b=-c[/tex]
[tex]d=3-2c-e[/tex]
Now, we have to substitute the found exponents into the first equation that we wrote:
[tex]Q=(N^{a})(p^{b})(Vis^{c})(D_{pi} ^{d})(D_{pr} ^{e})[/tex]
[tex]Q=(N^{1-c})(p^{-c})(Vis^{c})(D_{pi} ^{3-2c-e})(D_{pr} ^{e})[/tex]
Developing and Agrupating the terms with the same exponent we get:
[tex]Q=(\frac{Vis}{pND_{pi}^2} )^c(ND_{pi}^3)(\frac{D_{pr}}{D_{pi}} )^e[/tex]
Finally, the three non-dimensional group terms which describe the volume flow rate in terms of the relevant parameters of the system are:
[tex]\frac{Q}{(ND_{pi}^3)} =(\frac{Vis}{pND_{pi}^2} )^c(\frac{D_{pr}}{D_{pi}} )^e[/tex]
