5.5 Pipe flow and pressure drop

Pipe flow and pressure drop describe how fluids move through piping systems and how much energy they lose along the way. These concepts drive two of the most common decisions in chemical engineering: what size pipe to use and how powerful a pump to install.

Laminar vs Turbulent Flow

Reynolds Number and Flow Regimes

The Reynolds number (Re) is a dimensionless ratio of inertial forces to viscous forces. It tells you which flow regime you're dealing with:

$Re = \frac{\rho v D}{\mu}$

where $\rho$ is fluid density, $v$ is average velocity, $D$ is pipe diameter, and $\mu$ is dynamic viscosity.

• Laminar flow ($Re < 2300$): Fluid moves in smooth, parallel layers with no mixing. The velocity profile is parabolic, meaning the fluid at the center moves fastest while fluid near the wall barely moves. Think slow-moving oil in a small-diameter tube.
• Turbulent flow ($Re > 4000$): Fluid motion is chaotic, with eddies and mixing throughout the cross-section. The velocity profile is much flatter because all that mixing evens out the speeds. Think fast-moving water in a large pipe.

Transition Region and Flow Instability

Between $Re = 2300$ and $Re = 4000$, flow is unstable. Small disturbances (a pipe joint, a vibration) can flip the flow between laminar and turbulent behavior, causing unpredictable pressure fluctuations.

The exact transition point shifts depending on pipe roughness, entrance conditions, and fluid properties. For design purposes, you generally want to know clearly whether you're laminar or turbulent, so operating in this transition zone is something engineers try to avoid.

Pressure Drop and Head Loss

Darcy-Weisbach Equation

The Darcy-Weisbach equation calculates the pressure drop due to friction along a straight pipe:

$\Delta p = \frac{f L \rho v^2}{2D}$

where $f$ is the Darcy friction factor, $L$ is pipe length, $\rho$ is fluid density, $v$ is average velocity, and $D$ is pipe diameter.

The friction factor $f$ depends on the Reynolds number and the relative roughness ($\varepsilon / D$), where $\varepsilon$ is the absolute roughness of the pipe wall.

How you find $f$ depends on the flow regime:

• Laminar flow ($Re < 2300$): Use the Hagen-Poiseuille result directly:

$f = \frac{64}{Re}$

Notice that in laminar flow, roughness doesn't matter. The friction factor depends only on $Re$.

• Turbulent flow ($Re > 4000$): Use the Colebrook-White equation:

$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$

This equation is implicit in $f$ ($f$ appears on both sides), so it requires iteration or a chart to solve.

Head Loss and the Moody Diagram

Head loss ($h_L$) expresses the same friction loss as an equivalent height of fluid column:

$h_L = \frac{\Delta p}{\rho g}$

This form is convenient when working with pumps, since pump performance is typically given in units of head (meters or feet of fluid).

The Moody diagram is a graph that plots $f$ versus $Re$ for various values of $\varepsilon / D$. It covers both laminar and turbulent regimes on a single chart. Instead of iterating through the Colebrook-White equation, you can read $f$ directly from the diagram once you know $Re$ and $\varepsilon / D$. You'll use this chart constantly in pipe flow problems.

Pipe Roughness Effects

Surface Irregularities and Friction

Pipe roughness ($\varepsilon$) is the average height of surface bumps on the pipe wall. Common values range from about 0.0015 mm for drawn tubing to 0.26 mm for cast iron.

In turbulent flow, rougher pipes create more friction and a larger pressure drop. The parameter that matters is the relative roughness ($\varepsilon / D$), which compares bump height to pipe diameter. A small scratch matters a lot more in a 1 cm tube than in a 1 m pipe.

In laminar flow, roughness has negligible effect because the viscous sublayer near the wall is thick enough to smooth over the bumps.

Pipe Diameter and Fluid Properties

Pipe diameter has an outsized effect on pressure drop. For a given volumetric flow rate, the pressure drop scales as:

$\Delta p \propto \frac{1}{D^5}$

This means doubling the pipe diameter reduces the pressure drop by a factor of $2^5 = 32$. That's why even small increases in pipe size can dramatically cut pumping costs.

Fluid properties also play a direct role:

• Higher density ($\rho$) increases pressure drop because heavier fluid requires more force to push through.
• Higher viscosity ($\mu$) lowers the Reynolds number, pushing flow toward the laminar regime. Laminar flow in viscous fluids (like honey) can actually produce lower pressure drops than turbulent flow in low-viscosity fluids (like water) at comparable conditions, though this depends on the specific velocities and geometry involved.

Pipe Network Design

Network Components and Flow Considerations

Real systems aren't just single straight pipes. Pipe networks include interconnected pipe segments, fittings (elbows, tees), valves, and other components.

The continuity equation ensures mass conservation at every junction:

$Q = Av$

where $Q$ is volumetric flow rate, $A$ is cross-sectional area, and $v$ is average velocity. Whatever total flow enters a junction must leave it.

Pressure Drop Calculations and Minor Losses

For each straight pipe segment, you calculate friction losses using the Darcy-Weisbach equation as described above.

Fittings and valves cause additional minor losses. These are calculated using loss coefficients:

$h_{L,\text{minor}} = K \frac{v^2}{2g}$

where $K$ is a dimensionless loss coefficient specific to each fitting type (for example, a standard 90° elbow has $K \approx 0.9$, while a fully open gate valve has $K \approx 0.2$).

The total head loss in the system is the sum of all friction losses plus all minor losses. A network with long, narrow pipes and many fittings will have a much higher total head loss than a short, wide, straight run.

Pump Selection and Economic Considerations

Once you know the required flow rate and total system head loss, you can select a pump. Pumps are characterized by performance curves that plot head, power, and efficiency against flow rate. You want to operate near the pump's best efficiency point.

There's always a trade-off in pipe network design: larger pipes cost more upfront but reduce pressure drop and long-term energy costs. Smaller pipes save on materials but require more powerful pumps and consume more electricity over the system's lifetime. A good design balances these initial and operating costs.