Air under standard conditions flows through a 5.0 mm-diameter drawn tubing with
an average velocity of V = 55 m/s. Assume that the tube is smooth, the air is
incompressible and no shear stress occur in the tube. Determine
i. the pressure drop in 0.1 m section of the tube using Colebrook formula,
ii. the pressure drop in 0.1 m section of the tube using Moody chart. Compare
the difference of both analyses.
To determine the pressure drop in a 0.1 m section of the tube using the Colebrook formula and the Moody chart, we need to calculate the Reynolds number first. The Reynolds number is a dimensionless quantity that relates the flow velocity, tube diameter, and fluid properties.
Reynolds number (Re) formula:
Re = (ρ * V * D) / μ
Where:
ρ = density of air
V = flow velocity of air
D = diameter of the tube
μ = dynamic viscosity of air
1. Calculating Reynolds Number:
To calculate the Reynolds number, we need to know the density (ρ) and dynamic viscosity (μ) of air. The density of air at standard conditions is approximately 1.225 kg/m³, and the dynamic viscosity of air is approximately 1.81 x 10^-5 Pas.
Re = (1.225 kg/m³ * 55 m/s * 5.0 mm) / (1.81 x 10^-5 Pas)
Now, calculate the Reynolds number using the given values.
2. Determining the Pressure Drop using Colebrook Formula:
The Colebrook formula is an equation used to calculate the friction factor in smooth pipes. The friction factor is then used to determine the pressure drop.
The Colebrook formula:
1 / √f = -2 * log10((ε / (3.7 * D)) + (2.51 / (Re * √f)))
Where:
f = friction factor
ε = roughness of the tube surface
To use the Colebrook formula, we need to know the roughness of the tube surface. Assuming the tube is smooth, we can consider the roughness as negligible (ε ≈ 0).
Now, using the Reynolds number calculated in step 1, we can use an iterative method (e.g., Newton-Raphson) to solve the Colebrook formula for the friction factor (f). Once we have the friction factor, we can calculate the pressure drop using the Darcy-Weisbach equation:
Pressure drop (ΔP) = (f * L * ρ * V^2) / (2 * D)
Where:
L = length of the pipe section (0.1 m in this case)
3. Determining the Pressure Drop using Moody Chart:
The Moody chart provides a graphical representation of the friction factor, Reynolds number, and relative roughness. To use the Moody chart, we need to know the relative roughness (ε / D) and the Reynolds number.
Using the values of relative roughness and Reynolds number obtained in step 1, find the corresponding point on the Moody chart. From that point, we can read the friction factor (f) directly.
Once we have the friction factor, we can calculate the pressure drop using the Darcy-Weisbach equation as described in step 2.
Comparison of both analyses:
After calculating the pressure drops using the Colebrook formula and the Moody chart, compare the results. The values should be approximately the same, confirming the validity of both methods. Any differences could be due to the limitations of the formula or inaccuracies in data.
To determine the pressure drop in a 0.1 m section of the tube using the Colebrook formula and the Moody chart, we will need to calculate the Reynolds number (Re) and the relative roughness (ε/D) first.
1. Reynolds Number:
The Reynolds number (Re) is calculated using the formula:
Re = (ρVD) / μ
Where ρ is the density of air, V is the velocity of air, D is the diameter of the tube, and μ is the dynamic viscosity of air.
2. Relative roughness:
The relative roughness (ε/D) is the ratio of the roughness height (ε) to the tube diameter (D).
Now, let's calculate both values step by step:
Step 1: Calculate Reynolds number (Re)
Given:
- Diameter of the tube (D) = 5.0 mm = 0.005 m
- Velocity of air (V) = 55 m/s
We need the density (ρ) and dynamic viscosity (μ) values for air under standard conditions. The density of air at room temperature and atmospheric pressure is approximately 1.225 kg/m³. The dynamic viscosity of air at room temperature is approximately 1.81 x 10^(-5) Pa.s.
Substituting the values into the Reynolds number formula:
Re = (ρVD) / μ
Re = (1.225 kg/m³ * 55 m/s * 0.005 m) / 1.81 x 10^(-5) Pa.s
Calculating the Reynolds number (Re) will give the dimensionless number that characterizes the flow regime in the tube.
Step 2: Calculate relative roughness (ε/D)
Given:
- The tube is smooth, so there is no roughness (ε) on its surface.
Since there is no roughness, the relative roughness (ε/D) will be zero.
Now, let's compare the calculations using the Colebrook formula and the Moody chart:
i. Pressure drop using Colebrook formula:
The Colebrook formula is used to determine the pressure drop in turbulent flow regimes and is given by the formula:
1 / √f = -2 * log10((ε/D) / 3.7 + 2.51 / (Re √f))
We'll need to solve this equation iteratively to determine the friction factor (f).
ii. Pressure drop using Moody chart:
The Moody chart is a graphical representation of the Colebrook formula. We can use the Moody chart to directly find the friction factor (f) and then calculate the pressure drop.
To make a comparison, we need the value of the friction factor (f) obtained from the Colebrook formula and the Moody chart for the same Reynolds number (Re) and relative roughness (ε/D) values.
Once we have the friction factor (f), we can use the following formula to calculate the pressure drop (ΔP) in a 0.1 m section of the tube:
ΔP = f * (L / D) * (ρV² / 2)
Comparing the results from both analyses will allow us to see if there are any differences between the Colebrook formula and the Moody chart in calculating the pressure drop.