The air velocity in the duct of a heating system is to be measured by a Pitot-static probe inserted into the duct parallel to the flow. The differential height between the water columns connected to the two outlets of the probe is 67 mm. Take the density of water to be 1000 kg/m3. The gas constant of air is R = 0.287 kJ/kg-K. The air temperature and pressure in the duct are 580C and 96 kPa, respectively. Determine the flow velocity in m/s.
Question
The air velocity in the duct of a heating system is to be measured by a Pitot-static probe inserted into the duct parallel to the flow. The differential height between the water columns connected to the two outlets of the probe is 67 mm. Take the density of water to be 1000 kg/m3. The gas constant of air is R = 0.287 kJ/kg-K. The air temperature and pressure in the duct are 580C and 96 kPa, respectively. Determine the flow velocity in m/s.
Solution
To solve this problem, we will use Bernoulli's equation for fluid flow and the ideal gas law.
Step 1: Convert all the given quantities to SI units.
- The differential height between the water columns is 67 mm, which is 0.067 m.
- The density of water is given as 1000 kg/m^3.
- The gas constant of air is R = 0.287 kJ/kg-K, which is 0.287 * 10^3 J/kg-K.
- The air temperature is given as 58°C, which is 58 + 273.15 = 331.15 K in absolute temperature.
- The air pressure is given as 96 kPa, which is 96 * 10^3 Pa.
Step 2: Use the ideal gas law to find the density of air. The ideal gas law is P = ρRT, where P is the pressure, ρ is the density, R is the gas constant, and T is the temperature. Solving for ρ, we get ρ = P / RT. Substituting the given values, we get ρ = (96 * 10^3) / (0.287 * 10^3 * 331.15) = 1.11 kg/m^3.
Step 3: Use Bernoulli's equation to find the flow velocity. Bernoulli's equation for fluid flow is v = sqrt(2gh), where v is the velocity, g is the acceleration due to gravity, and h is the height difference. Substituting the given values, we get v = sqrt(2 * 9.81 * 0.067) = 3.65 m/s.
Therefore, the flow velocity in the duct is approximately 3.65 m/s.
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