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Air bubbles discharge from the end of a submerged tubeas shown in Fig. P7.57. The bubble
Chapter 7, Problem 7.57(choose chapter or problem)
Air bubbles discharge from the end of a submerged tube as shown in Fig. P7.57. The bubble diameter, D, is assumed tobe a function of the air flowrate, Q, the tube diameter, d, the acceleration of gravity, g, the density of the liquid, \(\rho\), and the surface tension of the liquid, \(\sigma\).
(a) Determine a suitable set of dimensionless variables for this problem.
(b) Model tests are to be run on the Earth for a prototype that is to be operated on a planet where the acceleration of gravity is 10 times greater than that on Earth. The model and prototype are to use the same fluid, and the prototype tube diameter is 0.25 in. Determine the tube diameter for the model and the required model flowrate if the prototype flowrate is to be \(0.001 \mathrm{ft}^{3} / \mathrm{s} \text {. }\)
Questions & Answers
QUESTION:
Air bubbles discharge from the end of a submerged tube as shown in Fig. P7.57. The bubble diameter, D, is assumed tobe a function of the air flowrate, Q, the tube diameter, d, the acceleration of gravity, g, the density of the liquid, \(\rho\), and the surface tension of the liquid, \(\sigma\).
(a) Determine a suitable set of dimensionless variables for this problem.
(b) Model tests are to be run on the Earth for a prototype that is to be operated on a planet where the acceleration of gravity is 10 times greater than that on Earth. The model and prototype are to use the same fluid, and the prototype tube diameter is 0.25 in. Determine the tube diameter for the model and the required model flowrate if the prototype flowrate is to be \(0.001 \mathrm{ft}^{3} / \mathrm{s} \text {. }\)
Step 1 of 6
(a) Express the bubble diameter as a function of the variables it depend on:
\(D=D(\Delta V,\ \sigma,\ \rho,\ g,\ d)\)
Next, express all the variables in their dimensional terms, using the MLT unit system:
\(\begin{array}{l} D=L \\ \Delta V=L^{3} T^{-1} \\ g=L T^{-2} \\ \rho=M L^{-3} \\ \sigma=M T^{-2} \\ d=L \end{array}\)
Calculate the number of dimensions for the pi term, using the next formula:
n = k - r
Where k is the number of variables, and r is the number of fundamental dimensions. Substitute k = 6 and r = 3 to obtain:
n = 2
Step 2 of 6
This implies that we have to find three pi terms. Calculate the first pi term from the next equation:
\(\Pi_{1}=D(d)^{a}(g)^{b}(\rho)^{c}\)
Express the pi term equation in its dimensional terms:
\(M^{0} L^{0} T^{0}=(L)(L)^{a}\left(L T^{-2}\right)^{b}\left(M L^{-3}\right)^{c}\)
Compare the powers between the left-hand side and the right-hand side, to obtain a solvable system of equations, and solve for the unknown constants:
c = 0 (1)
1 + a + b - 3c = 0 (2)
-2b = 0 (3)
Using the method of elimination we obtain the values for the unknown constants:
a = -1
b = 0
c = 0
Substitute the obtained values into the equation for the first pi term:
\(\begin{array}{l} \Pi_{1}=D(d)^{-1}(g)(\rho)^{0} \\ =\frac{D}{d} \end{array}\)
Step 3 of 6
Calculate the