Air at 45°C is flowing steadily in a 25 cm-diameter horizontal pipe made of plastic at
a rate of 900 l/min. Determine the head loss, and the required pumping power
input for flow over a 175m-long section of the pipe.
To calculate the head loss and the required pumping power input, we need to use the Darcy-Weisbach equation for head loss and the power equation for pumping power. Let's break down the steps to solve this problem:
Step 1: Convert the given flow rate from liters per minute to cubic meters per second.
- 900 L/min = 900/60 m^3/s = 15 m^3/s
Step 2: Calculate the average velocity of the air flow.
- The average velocity (V) can be determined using the flow rate (Q) and the pipe diameter (D).
- The formula for average velocity is: V = Q / (π * (D/2)^2)
- V = 15 / (π * (0.25/2)^2) = 15 / (π * 0.125^2) = 190.99 m/s
Step 3: Calculate the Reynolds number (Re) to determine the flow regime.
- The Reynolds number (Re) can be calculated using the average velocity (V), the pipe diameter (D), and the air properties (density and viscosity).
- The formula for Re is: Re = (ρ * V * D) / μ
- For air at 45°C, the air density (ρ) is 1.184 kg/m^3, and the dynamic viscosity (μ) is 1.8710^-5 kg/m·s.
- Re = (1.184 * 190.99 * 0.25) / 1.8710^-5 = 1.5887 * 10^7
Step 4: Determine the flow regime based on the Reynolds number.
- For pipe flow, there are three flow regimes: laminar, transitional, and turbulent.
- In this case, the Reynolds number (1.5887 * 10^7) suggests turbulent flow since it is well above the transitional flow range.
Step 5: Calculate the friction factor (f) using the Moody chart or an empirical equation.
- For turbulent flow in a smooth plastic pipe, we can use the empirical equation: f = 0.316 / Re^(1/4)
- f = 0.316 / (1.5887 * 10^7)^(1/4) = 0.016
Step 6: Calculate the head loss (hL) using the Darcy-Weisbach equation.
- The Darcy-Weisbach equation is: hL = (f * (L/D) * V^2) / (2 * g)
- In this case, the length of the pipe section (L) is 175 m, and the acceleration due to gravity (g) is approximately 9.81 m/s^2.
- hL = (0.016 * 175 / 0.25 * 190.99^2) / (2 * 9.81) = 6.743 m
Step 7: Calculate the required pumping power input.
- The equation for pumping power (P) is: P = (Q * hL * ρ * g) / η
- In this case, the pump efficiency (η) is assumed to be 100% (1).
- P = (15 * 6.743 * 1.184 * 9.81) / 1 = 1199.33 W or 1.2 kW
Therefore, the head loss for flow over a 175 m-long section of the pipe is 6.743 m, and the required pumping power input is approximately 1.2 kW.