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Chapter 7: Problem 31

A screw propeller has the following relevant dimensional parameters: axial thrust, \(F\), propeller diameter, \(D\), fluid kinematic viscosity, \(v,\) fluid density, \(\rho,\) gravitational acceleration, \(g,\) advance velocity, \(V,\) and rotational speed, \(N .\) Find appropriate dimensionless parameters to present the test data.

Short Answer

Expert verified
The appropriate dimensionless parameters to present the test data are \(\pi_1 = F/(\rho V^2 D^2), \pi_2 = V/(ND), \pi_3 = (\rho V D)/v, \pi_4 = g/V^2\)

Step by step solution

01

Identify the Variables

The dimensional variables in the problem are F (force), D (length), v (viscosity), ρ (density), g (acceleration due to gravity), V (velocity) and N (frequency).
02

Find the Dimensional Matrix

As per Buckingham π theorem, we need to express each variable as a product of fundamental dimensions such as mass, length, and time. We can then form a dimensional matrix A that represents these quantities. The dimensions of force, length, viscosity, density, acceleration, velocity, and frequency with respect to M (mass), L (length), and T (time) are respectively: [MLT^-2], [L], [ML^2 T^-1], [ML^-3], [LT^-2], [LT^-1], and [T^-1].
03

Compute the rank of the Dimensional Matrix

The rank of a matrix is the maximum number of linearly independent rows or columns. The rank of our dimensional matrix is 3, since we have three basic dimensions of M, L, and T.
04

Use Buckingham π theorem

According to the Buckingham π theorem, if we have n-dimensional variables and r fundamental dimensions, the number of dimensionless groups or pi terms should be n - r. Here, we have 7 dimensional parameters and 3 fundamental dimensions. So the number of pi terms would be 7 - 3 = 4.
05

Form the Dimensionless Groups

Choose r quantities from the n quantities that have not yet occurred in the group and construct 4 π groups which are dimensionless. Here, we are going to form four π groups: \(\pi_1 = F/(\rho V^2 D^2), \pi_2 = V/(ND), \pi_3 = (\rho V D)/v, \pi_4 = g/V^2\) These π groups are dimensionless parameters and they are independent of each other.

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Most popular questions from this chapter

A thin elastic wire is placed between rigid supports. A fluid flows past the wire, and it is desired to study the static deflection. \(\delta,\) at the center of the wire due to the fluid drag. Assume that $$\delta=f(\ell, d, \rho, \mu, V, E)$$ where \(\ell\) is the wire length, \(d\) the wire diameter, \(\rho\) the fluid density, \(\mu\) the fluid viscosity, \(V\) the fluid velocity, ind \(E\) the modulus of elasticity of the wire material. Develop a suitable set of pi terms for this problem.

The following dimensionless groups are often used to present data on centrifugal pumps: flow coefficient \(\varphi=\frac{Q}{\omega D^{3}},\) head coefficient \(\psi=\frac{g H}{\omega^{2} D^{2}},\) power coefficient \(\xi=\frac{\dot{W}}{\omega^{3} D^{5}},\) efficiency \(r_{i}=\) \frac{\rho g Q H}{\dot{W}}, \text { specific speed } N_{s}=\frac{\omega \sqrt{Q}}{(g h)^{3 / 4}}, \text { specific diameter } D_{s}=\frac{\omega \sqrt{Q}}{(g h)^{1 / 4}} Show that the last three groups can be formed from combinations of the first three groups.

A \(\frac{1}{10}\) -scale model of an airplane is tested in a wind tunnel at \(70^{\circ} \mathrm{F}\) and 14.40 psia. The model test results are: $$\begin{array}{l|c|c|c|c|c} \text { Velocity (mph) } & 0 & 50 & 100 & 150 & 200 \\ \hline \text { Drag (lb) } & 0 & 5 & 21 & 46 & 85 . \end{array}$$ Find the corresponding airplane velocities and drags if only fluid compressibility is important and the airplane is flying in the U.S. Standard Atmosphere at 30,000 ft. Assume that the air is an ideal gas.

A dam spillway is \(40 \mathrm{ft}\) long and has fluid velocity of \(10 \mathrm{ft} / \mathrm{s}\) Considering Weber number effects as minor, calculate the corresponding model fluid velocity for a model length of \(5 \mathrm{ft}\).

A coach has been trying to evaluate the accuracy of a baseball pitcher. After two years of studying, he proposes a function that can be presented as the accuracy of any pitcher: \\[ \mathrm{Acc}=f(V, a, m, \rho, p, z) \\] where Acc is the dimensionless accuracy, \(V\) is the velocity of the ball, \(a\) is the age of the pitcher, \(m\) is the mass of the pitcher, \(\rho\) is the density of air where the game is played, \(p\) is the pressure where the game is played (which varies with elevation), and \(z\) is the elevation above sea level. Find the dimensionless groups for this function.
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