Answer:
[tex]{V_m}=26.66\ mph[/tex]
[tex]\omega_m=133.33\ rpm[/tex]
Explanation:
Given that
[tex]V_P=40\ mph[/tex]
We know that
1 mph = 0.44 m/s
[tex]V_P=17.88\ m/s[/tex]
[tex]\omega_p=300\ rpm[/tex]
[tex]D_P=7 \ in[/tex]
1 in = 0.0254 m
[tex]D_P=0.1778 \ in[/tex]
[tex]\dfrac{d_m}{d_p}=1.5[/tex]
For Reynolds similarity
[tex]Re=\dfrac{Vd}{\nu }[/tex]
[tex]\left(\dfrac{Vd}{\nu }\right)_m=\left(\dfrac{Vd}{\nu }\right)_p[/tex]
ν is same for model and prototype .
[tex]\left({Vd}\right)_m=\left({Vd}\right)_p[/tex]
[tex]{V_m}=V_p\dfrac{d_p}{d_m}[/tex]
[tex]{V_m}=\dfrac{40}{1.5}[/tex]
[tex]{V_m}=26.66\ mph[/tex]
[tex]St=\dfrac{\omega d}{V}[/tex]
[tex]\left(\dfrac{\omega d}{V}\right)_m=\left(\dfrac{\omega d}{V}\right)_p[/tex]
[tex]\omega_m=\omega_p\dfrac{d_p}{d_m}\dfrac{V_m}{V_p}[/tex]
[tex]\omega_m=300\times \dfrac{1}{1.5}\times \dfrac{1}{1.5}[/tex]
[tex]\omega_m=133.33\ rpm[/tex]