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-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: Calculate the pressure drop and pumping power requirement for water flowing through a horizontal pipe. Answer: The pressure drop along the pipe is approximately 34,560 Pa, and the pumping power requirement to overcome this pressure drop is approximately 242 W.

Step-by-step solution

Calculate the average velocity of the fluid flow.

Using the provided flow rate \(Q=\frac{7 L}{s}=0.007\,m^3/s\), we will now calculate the average velocity \(V\) in the pipe using the formula: \( Q = V \cdot A \), where A is the cross-sectional area of the pipe. The diameter \(D\) of the pipe is \(4\,cm = 0.04\,m\). The area can be calculated using \(A = \pi(\frac{D}{2})^2\). \( V = \frac{Q}{A} = \frac{0.007\, m^3/s}{\pi (\frac{0.04\,m}{2})^2} = \frac{0.007}{\pi (0.02)^2} \approx 5.559\, m/s \)

Determine the Reynolds number

The Reynolds number \(Re\) is calculated using the formula: \( \textit{Re} = \frac{\rho V D}{\mu}\). Given the fluid properties, we can plug in the numbers and find the Reynolds number: \( \textit{Re} = \frac{(999.1 \, \frac{kg}{m^3}) (5.559\, m/s) (0.04\,m)}{1.138\times 10^{-3} \,\frac{kg}{m\cdot s}} \approx 196613 \)

Calculate the friction factor

Since the Moody chart is not available for us here, we can use the Darcy-Weisbach equation to estimate the friction factor \(f\) for turbulent flow in smooth pipes: \( f \approx 0.079\, \textit{Re}^{-1/4} = 0.079\, (196613)^{-1/4} \approx 0.0191\)

Calculate the pressure drop

To calculate the pressure drop \(\Delta P\), we can use the Darcy-Weisbach equation for pipe flow: \( \Delta P = f\, \frac{L\,\rho V^2}{2D} \) \( \Delta P = (0.0191) \frac{(25\,m)(999.1 \, \frac{kg}{m^3}) (5.559\,m/s)^2}{2(0.04\,m)} \approx 34560\, Pa\)

Calculate the pumping power requirement

The pumping power requirement \(P\) to overcome the pressure drop can be calculated using the formula: \(P = Q \cdot \Delta P\) \( P = (0.007\, m^3/s)(34560\, Pa) = 242\, W \)

Determine if the assumption of fully developed flow is valid

Based on the calculated Reynolds number, we have turbulent flow (\(\textit{Re} > 4000\)) in the pipe. Since it is a long pipe, we can assume that the flow is fully developed after a certain entrance length. So, the assumption of a fully developed flow is valid. To summarize, the pressure drop along the pipe is \(\Delta P\approx34560\,Pa\), and the pumping power requirement to overcome this pressure drop is \(P\approx242\,W\). The assumption of fully developed flow is valid.

Key concepts

Understanding Fluid Dynamics in Pipe Flow

Fluid dynamics is a subfield of fluid mechanics that deals with fluid flow—the science of liquids and gases in motion. It's crucial for predicting how fluids behave and interact with their environment, whether in natural settings or engineered systems like pipelines. In the context of the given exercise, the main concern is how water behaves as it flows through a pipe.

In a pipeline system, several factors influence the behavior of the fluid: the fluid properties (density \(\rho\) and viscosity \(\mu\)), the flow rate, and the pipe's characteristics (length, diameter, and surface roughness). Fluid flow can be laminar or turbulent, and this behavior directly affects the pressure drop, which is the reduction in the fluid's pressure from the pipe's inlet to its outlet.

The pressure drop is particularly important in engineering applications as it determines the energy required to pump the fluid through the pipe -- too high a pressure drop signals inefficiency and demands more energy, as reflected by the pumping power requirement in the solution.

Reynolds Number: Predicting Flow Patterns

The Reynolds number (\(Re\)) plays a pivotal role in determining the behavior of fluid flow. It is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. The Reynolds number compares inertial effects to viscous effects within the flow, serving as an indicator of whether the flow will be laminar or turbulent. A low Reynolds number, typically less than 2000, corresponds to laminar flow—steady and streamlined. Conversely, a high Reynolds number, typically greater than 4000, indicates turbulent flow—erratic and swirling.

In the exercise solution, we calculated the Reynolds number using the formula \(Re = \frac{\rho V D}{\mu}\), where \(V\) is the velocity of the fluid, \(D\) the diameter of the pipe, \(\rho\) the density of the fluid, and \(\mu\) the fluid's dynamic viscosity. With a calculated \(Re\) of approximately 196,613, we see the flow is well into the turbulent regime. This finding justifies the assumption of turbulent flow in subsequent calculations, such as determining the friction factor for the Darcy-Weisbach equation.

Darcy-Weisbach Equation: Estimating Pressure Drop

The Darcy-Weisbach equation is a fundamental expression in fluid dynamics, providing a means to calculate the pressure drop due to friction along a given length of pipe with a steady flow. The equation is given as \(\Delta P = f \frac{L \rho V^2}{2D}\), where \(\Delta P\) is the pressure drop, \(f\) the friction factor, \(L\) the length of the pipe, \(\rho\) the fluid density, \(V\) the average velocity of the fluid, and \(D\) the diameter of the pipe.

In cases where the flow is turbulent, as in this exercise, the friction factor can be estimated using empirical correlations or approximations like the one used in the solution step: \(f \approx 0.079\, Re^{-1/4}\). The Darcy-Weisbach equation then enables the calculation of pressure drop, integrating both the effect of the pipe's characteristics and fluid dynamics. The pumping power requirement can subsequently be derived from the pressure drop, which denotes the energy needed to move the fluid against resistance offered by friction. Understanding and applying the Darcy-Weisbach equation is essential for designing efficient piping systems and for the accurate assessment of operational costs.