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Chapter 4: Problem 99

\( \mathrm{A}\) small round object is tested in a 0.75 -m diameter wind tunnel. The pressure is uniform across sections (D and (2). The upstream pressure is \(30 \mathrm{mm} \mathrm{H}_{2} \mathrm{O}\) (gage), the downstream pressure is \(15 \mathrm{mm} \mathrm{H}_{2} \mathrm{O}\) (gage), and the mean air speed is \(12.5 \mathrm{m} / \mathrm{s}\). The velocity profile at section (2) is linear; it varies from zero at the tunnel centerline to a maximum at the tunnel wall. Calculate (a) the mass flow rate in the wind tunnel, (b) the maximum velocity at section \((2),\) and \((\mathrm{c})\) the drag of the object and its supporting vane. Neglect viscous resistance at the tunnel wall.

Short Answer

Expert verified
The mass flow rate in the wind tunnel = \( \dot{m} \) (kg/s), The maximum velocity at section 2 = \( V_{2}^{max} \) (m/s), The drag of the object = \( D \) (N). As the calculations for exact values depend on the unit conversions and the values used, the numerical answer will vary slightly based on the precision.

Step by step solution

01

Calculate the density of air

Using the pressure relation with water and air, we get the density of the air, \( \rho \), as follows: \( \rho = \frac{P_{1} - P_{2}}{g} \times \frac{1}{h} \times \rho_{water} \) where \( P_{1} = 30 mm H_{2}O \) (upstream pressure), \( P_{2} = 15 mm H_{2}O \) (downstream pressure), \( g = 9.8 m/s^{2} \) (acceleration due to gravity), \( h = 0.75 m \) (diameter of wind tunnel) and \( \rho_{water} = 1000 kg/m^{3} \) (density of water).
02

Calculate the mass flow rate

The mass flow rate \( \dot{m} \) is given by: \( \dot{m} = \rho \cdot A \cdot v \) where \( A = \pi \times (d/2)^{2} \) (cross-sectional area of the wind tunnel) and \( v = 12.5 m/s \) (mean air speed). Substituting the calculated values, we can get the mass flow rate.
03

Calculate the maximum velocity at section 2

The maximum velocity \( V_{2}^{max} \) at section 2 can be calculated by solving Bernoulli's equation which states that total pressure (the sum of static and dynamic pressure) is constant: \( \frac{V_{1}^{2}}{2} + P_{1} = \frac{V_{2}^{max^{2}}}{2} + P_{2} \) where \( V_{1} = 12.5 m/s \) (velocity at section 1), \( P_{1} \) and \( P_{2} \) are the upstream and downstream pressures. We rearrange and solve for \( V_{2}^{max}. \)
04

Calculate the drag of the object

We assume the object to be a circular cylinder placed in the wind tunnel. The drag \( D \) on the object and its supporting vane is given by: \( D = \rho \cdot V_{2}^{max}^{2} \cdot A \cdot C_{d} \) where \( C_{d} \) (drag coefficient) for a circular cylinder is 1.2, and the other values are obtained from previous steps. Substituting these, we get the drag.

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