(a) Evaluate the model scale when both viscous and gravity forces are necessary to secure similitude.
(b) What should be the model scale if oil of viscosity 9.29 x 10^–5 m²/s is used in the model tests and the prototype liquid has a viscosity of 74.3 x 10^–5 m²/s ?
(c) What would be the velocity and flow ratios for these liquids for the 1:4 model-prototype scale?

Question

Accepted Answer

(a) For this situation, both the Reynolds number and the Froude number must be satisfied simultaneously. We shall equate the velocity ratios for each model law. Using information from Problems 6.19 and 6.20,

Reynolds number [latex]V_{r} [/latex] = Froude number [latex]V_{r} [/latex]

[latex]( u /L)_{r}=\sqrt{L_{r}g_{r}} [/latex]

Since [latex]g_{r}=1 [/latex] , we obtain [latex]L_{r}= u ^{2/3}_{r} [/latex] .

(b) Using the above length ratio, [latex]L_{r}=\left(\frac{9.29 imes 10^{-5}}{74.3 imes 10^{-5}} ight)^{2/3}=\frac{1}{4} [/latex] . Model scale is 1:4.

(c) Using the Froude model laws (see Problems 6.18 and 6.19),

[latex]V_{r}=\sqrt{L_{r}g_{r}}=\sqrt{L_{r}}=\sqrt{\frac{1}{4} }=\frac{1}{2} [/latex] and [latex]Q_{r}=L^{5/2}_{r}=\left(\frac{1}{4} ight)^{5/2}=\frac{1}{32} [/latex]

Or, using the Reynolds model laws (see Problem 6.20),

[latex]V_{r}=\frac{ u _{r}}{L_{r}}=\frac{9.29/74.3}{1/4} =\frac{1}{2} [/latex] and [latex]Q_{r}=A_{r}V_{r}=L^2_{r} imes \frac{ u _{r}}{L_{r}}=L_{r} u _{r}=\left(\frac{1}{4} ight)\left(\frac{9.29}{74.3} ight)=\frac{1}{32} [/latex]

Question 6.34: (a) Evaluate the model scale when both viscous and gravity f...

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