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Chapter 7: Problem 11

Assume that the flowrate, \(Q\), of a gas from a smokestack is a function of the density of the ambicnt air, \(\rho_{\alpha}\), the density of the gas, \(\rho_{x},\) within the stack, the acceleration of gravity, \(g,\) and the height and diameter of the stack, \(h\) and \(d\), respectively. Use \(\rho_{c}, d,\) and \(g\) as repeating variables to develop a set of pi terms that could be used to describe this problem.

Short Answer

Expert verified
The three Pi terms that can describe the problem are \(\Pi_1 = \frac{Q}{{d^2 \sqrt{{\rho_x g}}}}\), \(\Pi_2 = \frac{h}{d}\), \(\Pi_3 = \frac{{\rho_a}}{{\rho_x}}\).

Step by step solution

01

Identify Variables and Their Dimensions

Identify the variables and their dimensions in the SI system. Here, \(Q\) - flowrate (m^3/s), \(\rho_{a}\) - density of ambient air (kg/m^3), \(\rho_{x}\) - density of gas (kg/m^3), \(g\) - acceleration due to gravity (m/s^2), \(h\) - height of stack (m), \(d\) - diameter of stack (m).
02

Choose Repeating Variables

Choose the repeating variables as mentioned in the question, which are \(\rho_{x}, d\) and \(g\).
03

Apply Buckingham Pi theorem

Apply the Buckingham Pi theorem which states that a problem that involves m physical variables can be rewritten into a problem of m-n dimensionless parameters. We can make formed term \(\Pi_1, \Pi_2, \ldots , \Pi_{m-n}\), where \(m=6\) and \(n=3\). Thus, there are \(6-3=3\) Pi terms to be formed.
04

Form Pi terms

Based on the Buckingham theorem, \(3\) pi terms can be formed as follows: \[\Pi_1 = \frac{Q}{{d^2 \sqrt{{\rho_x g}}}}\] \[\Pi_2 = \frac{h}{d}\] \[\Pi_3 = \frac{{\rho_a}}{{\rho_x}}\]. The dimensions of these Pi terms are dimensionless as per the theorem.

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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.

For a certain fluid flow problem it is known that both the Froude number and the Weber number are important dimensionless parameters. If the problem is to be studied by using a 1: 15 scale model, determine the required surface tension scale if the density scale is equal to \(1 .\) The model and prototype operate in the same gravitational field.

A student drops two spherical balls of different diameters and different densities. She has a stroboscopic photograph showing the positions of each ball as a function of time. However, she wants to express the velocity of each as a function of time in dimensionless form. Develop the dimensionless group. The equation of motion for each ball is $$m g-\frac{C_{D}}{2} \rho A V^{2}=m \frac{d V}{d t}$$ where \(m\) is ball mass, \(g\) is acceleration of gravity, \(C_{D}\) is a dimensionless and constant drag coefficient, \(\rho\) is air mass density, \(A\) is ball cross-sectional area \(\left(=\pi \mathrm{D}^{2} / 4\right)\) with \(D\) ball diameter, \(V\) is ball velocity, and \(t\) is time.

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.

The pressure rise, \(\Delta p=p_{2}-p_{1},\) across the abrupt expansion of Fig. \(\mathrm{P} 7.38\) through which a liquid is flowing can be expressed as $$\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)$$ where \(A_{1}\) and \(A_{2}\) are the upstream and downstream cross-sectional areas, respectively, \(\rho\) is the fluid density, and \(V_{1}\) is the upstream velocity. Some experimental data obtained with \(A_{2}=1.25 \mathrm{ft}^{2}\) \(V_{1}=5.00 \mathrm{ft} / \mathrm{s},\) and using water with \(\rho=1.94\) slugs/ft \(^{3}\) are given in the following table: $$\begin{array}{l|l|l|l|l|r} A_{1}\left(\mathrm{ft}^{2}\right) & 0.10 & 0.25 & 0.37 & 0.52 & 0.61 \\ \hline \Delta p\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 3.25 & 7.85 & 10.3 & 11.6 & 12.3 \end{array}$$ Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for \(\Delta p\) and use this equation to predict \(\Delta p\) for water flowing through an abrupt expansion with an area ratio \(A_{1} / A_{2}=0.35\) at a velocity \(V_{1}=3.75 \mathrm{ft} / \mathrm{s}\).
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