Question

Determine the average velocity and hydrodynamic and thermal entry lengths for water, engine oil, and liquid mercury flowing through a standard 2 -in Schedule 40 pipe with a mass flow rate of \(0.1 \mathrm{lbm} / \mathrm{s}\) and a temperature of \(100^{\circ} \mathrm{F}\).

Short answer

Question: Determine the average velocity, hydrodynamic and thermal entry lengths for water, engine oil, and liquid mercury flowing through a standard 2-inch Schedule 40 pipe with a mass flow rate of 0.1 lbm/s and a temperature of 100°F.

Step-by-step solution

Fluid properties

For this problem, we will need the density (\(\rho\)), dynamic viscosity (\(\mu\)), specific heat capacity (\(c_{p}\)), and thermal conductivity (\(k\)) of the fluids at \(100^{\circ} F\). These properties can be found in engineering fluid mechanics textbooks, or online resources like Perry’s Chemical Engineer’s Handbook. Note that the properties may vary slightly based on the source, but should be close enough for this problem. Step 2: Calculating average velocity

Average velocity

The average velocity of a fluid can be found using the mass flow rate (\(\dot{m}\)), density (\(\rho\)), and cross-sectional area of the pipe (\(A\)). The formula for average velocity (\(V\)) is: \(V = \frac{\dot{m}}{\rho A}\). We will calculate the cross-sectional area, A, of the 2-inch Schedule 40 pipe using the inner diameter \(D\) found in pipe dimension charts: \(A=\pi (\frac{D}{2})^2\). The average velocity can then be calculated for each fluid. Step 3: Calculating Reynolds number

Reynolds number

The Reynolds number (Re) is a dimensionless quantity used to describe the flow regime of a fluid in a pipe. It can be calculated using the formula: \(\mathrm{Re} = \frac{VD\rho}{\mu}\), where V is the average velocity, D is the inner pipe diameter, \(\rho\) is the density, and \(\mu\) is the dynamic viscosity of the fluid. Calculate the Reynolds number for each fluid flowing through the 2-inch Schedule 40 pipe. Step 4: Determining hydrodynamic entry length

Hydrodynamic entry length

The hydrodynamic entry length is the length in the pipe where the flow transitions from a developing flow to a fully-developed flow. It depends on the flow regime and the Reynolds number. For laminar flow (\(\mathrm{Re} < 2300\)), the hydrodynamic entry length (L_h) can be estimated using the formula: \(L_{h} \approx 0.05 D \cdot \mathrm{Re}\). For turbulent flow (\(\mathrm{Re} > 4000\)), the formula is: \(L_{h} \approx 4.4 D \cdot \mathrm{Re}^{1/6}\). Calculate the hydrodynamic entry length for each fluid in the pipe. Step 5: Determining thermal entry length

Thermal entry length

The thermal entry length is the length required for the temperature profile to fully develop inside the pipe. It can be estimated using the Prandtl number (Pr) and Reynolds number (Re). The Prandtl number can be calculated using the formula: \(\mathrm{Pr} = \frac{c_{p}\mu}{k}\), where \(c_{p}\) is the specific heat capacity, \(\mu\) is the dynamic viscosity, and \(k\) is the thermal conductivity of the fluid. For laminar flow (\(\mathrm{Re} < 2300\)), the thermal entry length (L_t) can be estimated using the formula: \(L_{t} \approx 0.05 D \cdot \mathrm{Re} \cdot \mathrm{Pr}\). For turbulent flow (\(\mathrm{Re} > 4000\)), the formula is: \(L_{t} \approx 4.4 D \cdot (\mathrm{Re} \cdot \mathrm{Pr})^{1/3}\). Calculate the thermal entry length for water, engine oil, and liquid mercury flowing through the pipe. With these calculations, we can determine the average velocity, and hydrodynamic and thermal entry lengths for water, engine oil, and liquid mercury flowing through a standard 2-inch Schedule 40 pipe with a mass flow rate of \(0.1 \mathrm{lbm}/\mathrm{s}\) and a temperature of \(100^{\circ}F\).