Piping Networks and Pressure Losses

When fluid flows through closed channels (e.g., pipes, tubing, blood vessels) it must overcome frictional resistance from the channel walls. This causes a continual loss of pressure in the flow direction as mechanical energy is converted to heat. This experiment focuses of measurement and modeling of such pressure losses in internal flow systems.

To measure pressure drop along channels, this experiment will use the principle of hydrostatic pressure variation. In stationary fluid, pressure only varies with depth due to fluid weight (Eqn. 1, Fig. 1a).

(1)

Here and are the pressures at two points, ρ is the fluid density, g is the gravitational acceleration, and h₁ and h₂ are the depths (measured in the direction of gravity) of the points from a reference level. At typical ambient conditions, the density of water is ρ_w = 998 kg m^-3 and the density of air is ρ_a = 1.15 kg m^-3. Because ρ_a << ρ_w, hydrostatic pressure variations in air can be neglected compared with liquid hydrostatic pressure variations, and the ambient atmospheric pressure can be assumed uniform (P_atm ~ 101 kPa). Following this principle, the pressure drop along a channel flow can be measured by the difference in fluid levels in vertical open-top tubes connected to the channel:  (Fig. 1b). Such liquid-level-based pressure measurement devices are called manometers.

The pressure loss along a length of a channel can be predicted with the Darcy friction factor formula (Eqn. 2). Here, is the pressure loss along a length (L) of channel with internal diameter D. U is the average channel velocity, defined as the volume flow rate of fluid (e.g., in m³ s^-1) divided by the channel cross-section area (e.g., in m², for circular channels). f is the Darcy friction factor, which follows different trends for different channel geometries and flow rates. In this experiment, friction factors will be measured experimentally for straight and helically coiled lengths of tube, and compared with previously published formulas.

  (2)

Channel-flow friction factor trends depend on the Reynolds number (Re), which measures the relative strength of effects from fluid inertia to effects from fluid viscosity (frictional effects). Re is defined as , where is fluid dynamic viscosity (~0.001 kg m^-1 s^-1 for water at ambient conditions). At low Re ( 2000 in straight channels), viscous effects are strong enough to damp out eddies in the flow, leading to smooth laminar flow. At higher Re (2000), random eddies can form in the flow, leading to turbulent behavior. Commonly used friction factor models for straight circular channel flows are presented in Eqn. 3.

(3)

When fluid flows through helical tube coils, secondary internal vortices form (Fig. 1c). As a result, the friction factor  also depends on the Dean number, which accounts for the relative influence of tube curvature: . Here R is the radius of the tube coil, measured from the central axis to halfway into the tubing. A common correlation for  is:

  (4)

Pipe fittings, valves, expansions/contractions, and other obstructions also cause pressure losses. One approach to model such minor losses is in terms of the equivalent length of plain channel required to yield the same pressure drop (L_e/D). Here, and  are the friction factor and flow velocity in the inlet / outlet channel lengths (Fig 1d).

(5)

Tables of representative equivalent channel lengths are reported in handbooks for common plumbing components (c.f., [1]). This experiment will measure the equivalent lengths for sharp 90°-bend fittings (elbows). Typical reported equivalent lengths for such fittings are L_e/D ~ 30.

Measured friction factor and equivalent length data are presented in Fig. 3a-c. For the straight tube section, a clear PVC tube with D = 6.4 mm and L = 284 mm is used. Measured flow rates (0.75 - 2.10 l min^-1) correspond to turbulent conditions (Re = 2600 - 7300). Friction factors match predictions from the analytic model to within experimental uncertainty. Relatively high f uncertainty is found at low flow rates due to the limited accuracy of the selected (low-cost) flow meter (± 0.15 l min^-1).

Friction factor results for the tube coil case also match the provided correlation (Eqn. 4) within experimental uncertainty (Fig. 3b). Five coil loops of radius R = 33 mm with tube inner diameter D = 6.4 mm are employed. Here, the Dean number is 500 - 5600, which corresponds to the laminar portion of Eqn. 4. Measured friction factors are significantly higher than for the straight section at equal flow rates. This stems from the stabilizing effect of the coil tube geometry, which delays the transition to turbulence to high Re.

For the elbow case, 4 elbow fittings (part number in materials list) are employed, connected by short lengths of D = 6.4 mm tubing. The equivalent frictional length of each elbow fitting approaches (L_e/D) ~ 30 - 40 at high Re (Fig. 3c). This is similar to a commonly reported value of 30. Note that the actual frictional resistance is specific to the fitting geometry, and reported L_e/D values should only be considered as guidelines.

Figure 1: a. Schematic of hydrostatic pressure variation in a stationary body of fluid. b. Pressure change along a straight length of tube, measured with open-top manometers. c. Schematic of coiled tube, with internal vortices indicated in cross-section view.

Figure 2: (a) Schematic and (b) photograph of pressure drop measurement facility. Please click here to view a larger version of this figure.

Figure 3: Friction factor and equivalent length measurements and model predictions for: a. Straight tube, b. Coiled tube, c. Elbow fittings.

Summary

This experiment demonstrates methods for measuring pressure-drop friction factors and equivalent lengths in internal flow networks. Modeling methods are presented for common flow configurations, including straight tubes, coiled tubes, and pipe fittings. These experimental and analysis techniques are key engineering tools for the design of fluid flow systems.

Applications

Internal flow networks arise in numerous applications including power generation plants, chemical processing, flow distribution inside heat exchangers, and blood circulation in organisms. In all cases, it is critical to be able to predict and model pressure losses and pumping requirements. Such flow systems can be decomposed into sections of straight and curved channels, connected by fittings or junctions. By applying friction factor and minor loss models to such components, whole network descriptions can be formulated.

Materials List

Name	Company	Catalog Number	Comments
Equipment
Submersible water pump	Uniclife	B018726M9K
Covered plastic container			Water reservoir, plastic food container used in this study.
Water flow meter	UXCell	LZM-15	Rotameter, 0.5 – 4.0 l min-1
Rigid clear PVC tube	McMaster	53945K13	For test sections and manometers, 1/4” ID, 3/8” OD
Flexible soft PVC tubing	McMaster	5233K63 5233K56	For tubing connections and coil test section
Plastic tube fitting tee	McMaster	5016K744	For test sections inlet and outlet connections/manometers
Plastic tube fitting elbow	McMaster	5016K133	For test section with elbows