Question

Water at \(15^{\circ} \mathrm{C}\) is flowing through a \(5-\mathrm{cm}\)-diameter smooth tube with a length of \(200 \mathrm{~m}\). Determine the Darcy friction factor and pressure loss associated with the tube for (a) mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) and (b) mass flow rate of $0.5 \mathrm{~kg} / \mathrm{s}$.

Short answer

Question: Determine the Darcy friction factors and pressure losses in a smooth tube for two different mass flow rates. Data: - Water temperature: 15°C - Tube diameter: 5 cm - Tube length: 200 m - Mass flow rates: (a) 0.02 kg/s, (b) 0.5 kg/s Solution: 1. Calculate the fluid properties and tube dimensions. 2. Calculate the volumetric flow rates for both cases. 3. Calculate the average fluid velocity for both cases. 4. Calculate the Reynolds Number for both cases. 5. Determine the Darcy Friction Factor for both cases. 6. Calculate the pressure losses for both cases. Result: (a) Darcy friction factor (f1) = 0.0302, and pressure loss (∆P1) = 4.835 x 10⁴ Pa (b) Darcy friction factor (f2) = 0.0175, and pressure loss (∆P2) = 2.670 x 10⁶ Pa

Step-by-step solution

Determine the fluid properties and tube dimensions

Calculate the properties of water at the given temperature (15°C). The properties can be taken from standard tables or online resources. The required properties are density (ρ) and dynamic viscosity (μ). For water at 15°C, we have ρ = 999 kg/m³ and μ = 1.139 x 10⁻³ Pa·s. Also, convert the diameter and length of the tube to meters: D = 0.05m and L = 200m.

Calculate the volumetric flow rate

Use the given mass flow rates (m) and the fluid properties to calculate the volumetric flow rate (Q) for both cases: (a) Q1 = m1/ρ = (0.02 kg/s)/(999 kg/m³) = 2.002 x 10⁻⁵ m³/s (b) Q2 = m2/ρ = (0.5 kg/s)/(999 kg/m³) = 5.005 x 10⁻⁴ m³/s

Calculate the average fluid velocity

We will now calculate the average fluid velocity (u) for both cases using the volumetric flow rate and the cross-sectional area of the tube, A = (πD²)/4: (a) u1 = Q1/A = (2.002 x 10⁻⁵ m³/s)/[π (0.05² m²)/4] = 0.636 m/s (b) u2 = Q2/A = (5.005 x 10⁻⁴ m³/s)/[π (0.05² m²)/4] = 15.92 m/s

Calculate the Reynolds Number

Now, we are going to calculate the Reynolds Number (Re) for both cases using the fluid properties and average fluid velocity, Re = (ρuD)/μ: (a) Re1 = (999 kg/m³)(0.636 m/s)(0.05 m)/(1.139 x 10⁻³ N·s/m²) = 2.80x10^⁴ (b) Re2 = (999 kg/m³)(15.92 m/s)(0.05 m)/(1.139 x 10⁻³ N·s/m²) = 6.994x10^⁵

Determine Darcy Friction Factor

For a smooth tube, the Darcy friction factor (f) is given by the Blasius correlation, f = 0.0791 Re^(-1/4): (a) f1 = 0.0791 (2.80x10⁴)^(-1/4) = 0.0302 (b) f2 = 0.0791 (6.994x10^⁵)^(-1/4) = 0.0175

Calculate pressure loss

Now, we can use the Darcy-Weisbach equation to calculate the pressure loss (∆P) for both cases, ∆P = f (L/D)(ρu²/2): (a) ∆P1 = (0.0302)(200m/0.05m)(999 kg/m³)(0.636 m/s)²/2 = 4.835 x 10⁴ Pa (b) ∆P2 = (0.0175)(200m/0.05m)(999 kg/m³)(15.92 m/s)²/2 = 2.670 x 10⁶Pa Thus, the Darcy friction factors and pressure losses are: (a) f1=0.0302 and ∆P1= 4.835 x 10⁴ Pa (b) f2=0.0175 and ∆P2= 2.670 x 10⁶ Pa