5.5 Pipe flow and pressure drop

Pipe flow and pressure drop are crucial concepts in fluid mechanics. They help us understand how fluids move through pipes and the energy losses that occur. This knowledge is key for designing efficient systems in various industries.

Calculating pressure drop is essential for proper pipe sizing and pump selection. By considering factors like flow regime, pipe roughness, and fluid properties, engineers can optimize fluid transport systems for better performance and energy efficiency.

Laminar vs Turbulent Flow

Reynolds Number and Flow Regimes

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• Reynolds number (Re) is a dimensionless quantity used to characterize the flow regime in pipes, defined as the ratio of inertial forces to viscous forces
• The Reynolds number is calculated using the equation $Re = (\rho vD) / \mu$, where $\rho$ is the fluid density, $v$ is the average velocity, $D$ is the pipe diameter, and $\mu$ is the dynamic viscosity of the fluid
• Laminar flow occurs at low Reynolds numbers (typically $Re < 2300$), where fluid flows in parallel layers without mixing, and the velocity profile is parabolic (e.g., slow-moving oil in a small pipe)
• Turbulent flow occurs at high Reynolds numbers (typically $Re > 4000$), characterized by chaotic motion, eddies, and mixing, with a flatter velocity profile (e.g., fast-moving water in a large pipe)

Transition Region and Flow Instability

• The transition region between laminar and turbulent flow occurs for Reynolds numbers between 2300 and 4000, where the flow is unstable and can exhibit characteristics of both regimes
• In the transition region, small disturbances can cause the flow to switch between laminar and turbulent states, leading to unpredictable behavior and increased pressure fluctuations
• The exact transition point depends on factors such as pipe roughness, entrance conditions, and fluid properties, making it difficult to predict the flow regime in this range

Pressure Drop and Head Loss

Darcy-Weisbach Equation

• The Darcy-Weisbach equation is used to calculate the pressure drop ($\Delta p$) or head loss ($h_L$) in a pipe due to friction: $\Delta p = (fL\rho v^2) / (2D)$, where $f$ is the Darcy friction factor, $L$ is the pipe length, $\rho$ is the fluid density, $v$ is the average velocity, and $D$ is the pipe diameter
• The Darcy friction factor ($f$) depends on the Reynolds number ($Re$) and the relative roughness ($\varepsilon/D$) of the pipe, where $\varepsilon$ is the absolute roughness
• For laminar flow ($Re < 2300$), the friction factor is calculated using the Hagen-Poiseuille equation: $f = 64 / Re$
• For turbulent flow ($Re > 4000$), the friction factor is determined using the Moody diagram or the Colebrook-White equation: $1/\sqrt{f} = -2\log_{10}[(\varepsilon/D) / 3.7 + 2.51 / (Re\sqrt{f})]$

Head Loss and Moody Diagram

• Head loss ($h_L$) is the equivalent height of a column of the fluid that would be required to overcome the pressure drop due to friction and is related to pressure drop by: $h_L = \Delta p / (\rho g)$, where $g$ is the acceleration due to gravity
• The Moody diagram is a graphical representation of the relationship between the friction factor, Reynolds number, and relative roughness for both laminar and turbulent flow regimes
• The Moody diagram allows for quick estimation of the friction factor based on the flow conditions and pipe properties, without the need for iterative calculations using the Colebrook-White equation

Pipe Roughness Effects

Surface Irregularities and Friction

• Pipe roughness ($\varepsilon$) represents the average height of surface irregularities on the pipe wall and affects the friction factor and pressure drop in turbulent flow
• Rougher pipes (higher $\varepsilon$) result in higher friction factors and increased pressure drop compared to smoother pipes (lower $\varepsilon$) for the same flow conditions
• The relative roughness ($\varepsilon/D$) is a dimensionless parameter that relates the pipe roughness to its diameter, with higher values indicating a more significant impact on the flow

Pipe Diameter and Fluid Properties

• Pipe diameter ($D$) has a significant impact on pressure drop, as smaller diameters result in higher fluid velocities and increased friction losses for a given flow rate
• The pressure drop is inversely proportional to the fifth power of the pipe diameter ($\Delta p \propto 1/D^5$), so doubling the pipe diameter can reduce the pressure drop by a factor of 32
• Fluid properties, such as density ($\rho$) and viscosity ($\mu$), affect the Reynolds number and, consequently, the flow regime and pressure drop
• Higher fluid density results in increased pressure drop, while higher viscosity leads to lower Reynolds numbers and potentially laminar flow, which can reduce the pressure drop compared to turbulent flow (e.g., honey vs. water)

Pipe Network Design

Network Components and Flow Considerations

• Pipe networks consist of interconnected pipes, fittings, valves, and other components that transport fluids from one point to another
• The design of pipe networks involves determining the required flow rates, pressure drops, and pipe sizes to ensure efficient and safe operation
• The continuity equation ($Q = Av$) is used to relate the flow rate ($Q$), cross-sectional area ($A$), and average velocity ($v$) in each pipe segment, ensuring mass conservation throughout the network

Pressure Drop Calculations and Minor Losses

• The Darcy-Weisbach equation and Moody diagram are used to calculate the pressure drop in each pipe segment based on the chosen pipe size, material, and flow conditions
• Minor losses due to fittings, valves, and other components are accounted for using loss coefficients ($K$) and the velocity head ($v^2/2g$)
• The total system head loss is the sum of the friction losses in the pipes and the minor losses, which must be overcome by the pump (e.g., a network with a long, narrow pipe and many fittings will require a more powerful pump)

Pump Selection and Economic Considerations

• Pumps are selected based on the required flow rate and total system head, ensuring that the pump can provide sufficient energy to maintain the desired flow throughout the network
• Pump performance curves, which relate the flow rate, head, and efficiency, are used to select the most appropriate pump for the given application
• Economic considerations, such as initial cost, operating costs, and maintenance requirements, are also taken into account when designing pipe networks and selecting pumps (e.g., using larger pipes may increase initial costs but reduce long-term energy consumption)