Question

Water at $15^{\circ} \mathrm{C}\left(\rho=999.1 \mathrm{~kg} / \mathrm{m}^{3}\right.\( and \)\mu=1.138 \times 10^{-3}\( \)\mathrm{kg} / \mathrm{m} \cdot \mathrm{s}\( ) is flowing in a \)4-\mathrm{cm}$-diameter and \(25-\mathrm{m}\)-long horizontal pipe made of stainless steel steadily at a rate of \(7 \mathrm{~L} / \mathrm{s}\). Determine \((a)\) the pressure drop and \((b)\) the pumping power requirement to overcome this pressure drop. Assume flow is fully developed. Is this a good assumption?

Short answer

Question: Assuming fully developed flow in a horizontal pipe, calculate (a) the pressure drop across the pipe, and (b) the pumping power required to overcome this pressure drop for water with a flow rate of 7 L/s, pipe diameter of 4 cm, and pipe length of 25 m. Use the following properties for water: density = 999.1 kg/m³ and dynamic viscosity = 0.001138 kg/m·s. Answer: (a) The pressure drop across the pipe is approximately 285.536 kPa. (b) The pumping power required to overcome the pressure drop is approximately 2.036 kW. Note that the assumption of fully developed flow might not be accurate in this case due to the calculated Reynolds number, indicating transition or turbulent flow.

Step-by-step solution

Convert flow rate to volumetric flow rate and calculate average velocity

First, convert the flow rate from liters per second to cubic meters per second. Then, using the pipe's cross-sectional area, calculate the average velocity of the water flow. The flow rate is given in liters per second (L/s). To convert it to cubic meters per second (m³/s), note that 1 L = 0.001 m³. So, Q = 7 L/s * 0.001 m³/L = 0.007 m³/s The diameter of the pipe (D) is 4 cm, so the radius (r) is 2 cm or 0.02 m. The cross-sectional area (A) of the pipe can be calculated using the formula A = π * r² = π * (0.02 m)² = 0.001257 m² Now we can find the average velocity (v) of the water flow using the formula v = Q / A = 0.007 m³/s / 0.001257 m² ≈ 5.566 m/s

Calculate the Reynolds number

The Reynolds number (Re) is a dimensionless number that describes the flow regime in the pipe. It can be calculated using the formula Re = (ρ * v * D) / μ where ρ is the density of water (999.1 kg/m³), v is the average velocity, D is the diameter of the pipe, and μ is the dynamic viscosity (0.001138 kg/m·s). Re = (999.1 kg/m³ * 5.566 m/s * 0.04 m) / 0.001138 kg/m·s ≈ 19561

Decide on the appropriate flow model

Since Re > 2000, the flow is considered transition or turbulent which means we should use Darcy-Weisbach equation instead of Hagen-Poiseuille equation for pressure drop calculation. However, as mentioned in the question, we're asked to assume the flow is fully developed (laminar) which might not be accurate, but it allows us to use the Hagen-Poiseuille equation for calculating the pressure drop.

Calculate the pressure drop using Hagen-Poiseuille equation

The Hagen-Poiseuille equation relates pressure drop (ΔP) to the flow rate (Q), pipe length (L), fluid viscosity (μ), and pipe radius (r): ΔP = (8 * μ * L * Q) / (π * r⁴) Substituting the given values into the equation: ΔP = (8 * 0.001138 kg/m·s * 25 m * 0.007 m³/s) / (π * (0.02 m)⁴) ≈ 285536 Pa or 285.536 kPa So the pressure drop across the pipe is approximately 285.536 kPa.

Calculate the pumping power requirement

The pumping power (P) required to overcome the pressure drop can be calculated using the formula P = Q * ΔP / (ρ * g_eff) where g_eff is the effective gravitational acceleration, which equals 9.81 m/s² for Earth. Here, we can assume g_eff is approximately the same as the gravitational constant "g" because the flow is horizontal, and the vertical effect due to gravity is negligible. P = (0.007 m³/s * 285536 Pa) / (999.1 kg/m³ * 9.81 m/s²) ≈ 2035.49 W or 2.036 kW The pumping power required to overcome the pressure drop is approximately 2.036 kW.

Evaluate the assumption of fully developed flow

We calculated the Reynolds number to be approximately 19561, which indicates that the flow is likely in the transition or turbulent regime. The assumption of fully developed (laminar) flow might not be accurate in this case. The Hagen-Poiseuille equation is valid for laminar flow only; for a more accurate calculation of pressure drop and pumping power, the Darcy-Weisbach equation should be used for transition or turbulent flows. However, it's essential to consider the given input conditions and assumptions as mentioned in the question.