Fluid Flow in Pipelines
A comprehensive technical reference on the principles, calculations, and applications of fluid dynamics in pipeline systems, including the role of the hydraulic flow control valve in managing flow characteristics.
Since all real fluids possess viscosity, energy losses are inevitable when fluids flow through pipelines. These losses are critical considerations in system design, affecting efficiency, performance, and the selection of appropriate components such as the hydraulic flow control valve. Understanding the nature of these losses requires knowledge of flow regimes, fundamental parameters, and resistance calculations.
This technical resource explores the fundamental principles governing fluid flow in pipelines, from flow regimes to pressure loss calculations, with specific attention to how devices like the hydraulic flow control valve influence these dynamics.
Two Regimes of Fluid Flow in Pipelines
Laminar Flow
In laminar flow, fluid particles move smoothly along the axis of the pipe without transverse movement. The fluid flows in distinct layers, with each layer sliding past adjacent layers with minimal mixing. This orderly flow pattern is characteristic of low-velocity conditions and is often observed in systems where a hydraulic flow control valve maintains precise, low-flow rates.
Velocity profile in laminar flow: parabolic distribution with maximum velocity at center
Turbulent Flow
Turbulent flow is characterized by fluid particles moving with both longitudinal and transverse motion, resulting in an irregular, chaotic pattern. This mixing action creates greater resistance to flow and higher energy losses. Turbulent conditions frequently occur in industrial systems, where a hydraulic flow control valve must regulate varying flow rates while accounting for the increased pressure drops associated with turbulence.
Velocity profile in turbulent flow: flatter distribution due to mixing
Fundamental Concepts
Critical Velocities
| Term | Explanation |
|---|---|
| Upper Critical Velocity | The velocity at which laminar flow transitions to turbulent flow. This parameter is important when selecting a hydraulic flow control valve to ensure proper operation across flow regimes. |
| Lower Critical Velocity | The velocity at which turbulent flow transitions back to laminar flow. Systems incorporating a hydraulic flow control valve must account for this threshold to maintain stable flow conditions. |
Through extensive experimental research, Reynolds discovered that for different pipe diameters (d), and fluids with different properties (density ρ, kinematic viscosity ν), the dimensionless number formed at critical velocities, Re = (dρv)/μ or Re = (vd)/ν, remains essentially constant. This Reynolds number is a fundamental parameter in fluid dynamics and critical for properly sizing a hydraulic flow control valve.
Rec is known as the critical Reynolds number. The value corresponding to the lower critical velocity is the lower critical Reynolds number Rec, while that corresponding to the upper critical velocity is the upper critical Reynolds number Re'c. Experimental measurements have determined Rec = 2320 and Re'c = 13800.
Reynolds Number
The Reynolds number expression corresponding to the average velocity (v) over the cross-section is:
Re = (vd)/ν = (ρvd)/μ
When Re ≤ Rec = 2320, the flow regime in the pipeline is laminar. When Re > Re'c = 13800, the flow regime is turbulent. For Reynolds numbers between these values, the flow may be either laminar or turbulent. Due to the instability of this transition state, even minor disturbances can cause laminar flow to become turbulent. Therefore, in engineering practice, the transition state is generally classified as turbulent, with the lower critical Reynolds number Rec = 2320 serving as the criterion: Re ≤ 2320 for laminar flow, and Re > 2320 for turbulent flow. This distinction is crucial when specifying a hydraulic flow control valve for a particular application.
The physical significance of the Reynolds number is the ratio of inertial forces to viscous forces acting on the fluid. A smaller Re indicates greater influence of viscous forces and more stable flow, which is often desirable in systems controlled by a precision hydraulic flow control valve. A larger Re indicates greater influence of inertial forces and more turbulent flow, which affects the performance characteristics of a hydraulic flow control valve.
Practical Application Note
When selecting a hydraulic flow control valve, engineers must calculate the Reynolds number for the expected operating conditions to ensure the valve performs optimally. Valves sized without considering Reynolds number characteristics may exhibit excessive pressure drops or inadequate flow control.
Non-circular Pipe Cross-sections
The parameter d in the Reynolds number expression represents the characteristic length of the pipeline. For circular cross-sections, d is simply the pipe diameter. For non-circular cross-sections, the hydraulic radius (R) or equivalent diameter (de) may be used.
For a non-circular pipe with cross-sectional area A and wetted perimeter (length of contact between fluid and pipe) l, the equivalent radius is:
R = A / l
The equivalent diameter is:
de = 4R = 4A / l
The Reynolds number expression applicable to non-circular pipes is therefore:
Re = (vde)/ν = (ρvde)/μ
Common Cross-sectional Shapes and Hydraulic Diameters
| Cross-sectional Shape | Hydraulic Diameter de | Critical Re Value |
|---|---|---|
| Circular pipe | d | 2300 |
| Square | a | 2100 |
| Concentric annular gap | D - d | 1100 |
| Eccentric gap | 2δ | 1000 |
| Parallel plates | 2δ | 1000 |
| Spool valve opening | 2δ | 260 |
Note: When designing systems with non-circular passages and selecting components like a hydraulic flow control valve, engineers must use the appropriate hydraulic diameter to accurately calculate Reynolds numbers and predict flow behavior.
Pressure Losses in Pipes
1. Frictional (沿程) Pressure Loss
When fluid flows through a pipeline, adhesive forces between the fluid and pipe wall, along with internal friction between fluid particles, create resistance to flow along the entire length. This resistance is known as frictional resistance. Overcoming this resistance consumes energy, typically manifested as a pressure drop known as frictional pressure loss (Δpf). This loss can be calculated using the Darcy formula, which is essential for properly sizing a hydraulic flow control valve and designing efficient pipeline systems.
Δpf = λ × (l/d) × (ρv²/2)
or in terms of head loss:
hf = λ × (l/d) × (v²/(2g))
Where:
- λ = frictional resistance coefficient, which is a function of Reynolds number (Re) and relative roughness (ε/d)
- l = length of the pipe section, in meters
- d = inner diameter of the pipe, in meters
- v = average fluid velocity, in m/s
- ρ = fluid density, in kg/m³
- g = acceleration due to gravity (9.81 m/s²)
Absolute Roughness of Pipe Surfaces (ε)
| Pipe Material | Absolute Roughness ε (mm) |
|---|---|
| Aluminum | 0.0015 ~ 0.01 |
| Cold-drawn copper, brass tubes | 0.0015 ~ 0.06 |
| Cold-drawn aluminum, aluminum alloy tubes | 0.01 ~ 0.03 |
| Cold-drawn seamless steel tubes | 0.05 ~ 0.1 |
| Hot-rolled seamless steel tubes | 0.05 ~ 0.1 |
| Galvanized steel pipes | 0.12 ~ 0.15 |
| Asphalt-coated steel pipes | 0.03 ~ 0.05 |
| Corrugated steel pipes | 0.75 ~ 7.5 |
| Cast iron pipes | 0.05 |
| Smooth plastic pipes | 0.0015 ~ 0.01 |
Calculation Formulas for Frictional Resistance Coefficient (λ)
| Flow Regime | Reynolds Number Range | Calculation Formula |
|---|---|---|
| Laminar | Re < 2320 | λ = 64/Re |
| Hydraulically smooth pipes | Re < 2×105 | λ = 0.3164/Re0.25 |
| Hydraulically rough pipes | 3000 < Re < 105 | λ = 0.308/(0.842 - lgRe)2 |
| Resistance square region | 105 < Re < 108 | 1/√λ = 2lg(d/ε) + 1.74 |
Engineering Application: Accurate calculation of frictional pressure losses is essential when specifying a hydraulic flow control valve, as these losses directly impact the valve's performance characteristics and the overall system efficiency. A properly sized hydraulic flow control valve will account for these losses to maintain desired flow rates across varying operating conditions.
2. Local Pressure Loss
When fluid flows through pipelines and encounters fittings such as bends, sudden expansions or contractions, valves, and tees, the velocity magnitude and direction change abruptly. This causes fluid particle collisions, vortices, secondary flows, and flow separation and reattachment. Due to viscous effects, intense friction and momentum exchange occur between particles, creating resistance to flow. This resistance at local disturbances is known as local resistance. Overcoming local resistance consumes energy, typically manifested as a pressure drop known as local pressure loss (Δpζ). Components like the hydraulic flow control valve introduce specific local pressure losses that must be accounted for in system design.
Δpζ = ζ × (ρv²/2)
or in terms of head loss:
hζ = ζ × (v²/(2g))
Where:
- ζ = local resistance coefficient, which depends on the fitting geometry and Reynolds number
- v = average fluid velocity, in m/s (unless specified otherwise, refers to the average velocity in the cross-section downstream of the fitting)
Local Resistance Coefficients
Except for sudden expansion fittings, most local resistance coefficients (ζ) are determined experimentally or calculated using empirical formulas. Most available data for local resistance coefficients apply to turbulent flow conditions. Data for laminar flow is limited, which is important when selecting a hydraulic flow control valve for low-flow applications.
a. Sudden Expansion Local Resistance Coefficients
The configuration of a sudden pipe expansion is shown in Figure 1-16.
Figure 1-16: Schematic of Sudden Pipe Expansion
i. Laminar Flow
When Re < 2320, for the average velocity in the larger pipe, the local resistance coefficient for sudden expansion can be calculated using:
ζ = 2(1 - A1/A2) + 16/Re2
ii. Turbulent Flow
For the average velocity in the larger pipe, the local resistance coefficient for sudden expansion is:
ζ = (1 - A1/A2)2
Where A1 and A2 are the cross-sectional areas of the pipe before and after expansion, respectively.
Local resistance coefficients for sudden expansion can also be found in Table 1-22.
| Table 1-22: Local Resistance Coefficients for Sudden Expansion | |
|---|---|
| A1/A2 | ζ |
| 0.1 | 0.81 |
| 0.2 | 0.64 |
| 0.3 | 0.49 |
| 0.4 | 0.36 |
| 0.5 | 0.25 |
| 0.6 | 0.16 |
| 0.7 | 0.09 |
| 0.8 | 0.04 |
| 0.9 | 0.01 |
| 1.0 | 0 |
b. Local Resistance Coefficients for Pipe Inlets and Outlets
These coefficients are presented in Tables 1-23 and 1-24, which are critical references when integrating a hydraulic flow control valve into a pipeline system.
Table 1-23: Local Resistance Coefficients for Pipe Outlets
| Outlet Configuration | Resistance Coefficient ζ |
|---|---|
| Flow from straight pipe into large reservoir | ζ = 1.0 (turbulent flow) |
| Flow from straight pipe (laminar flow) | ζ = 2.0 |
Table 1-24: Local Resistance Coefficients for Pipe Inlets
| Inlet Configuration | Resistance Coefficient ζ |
|---|---|
| Sharp-edged inlet (Re > 104) | ζ = 0.5 |
| Well-rounded inlet (r/d ≥ 0.15) | ζ = 0.03 ~ 0.05 |
| Projecting pipe inlet | ζ = 1.0 ~ 1.5 |
c. Local Resistance Coefficients for Pipe Contractions
These coefficients are presented in Table 1-25, which is particularly useful when designing systems with a hydraulic flow control valve that modulates flow through variable orifice sizes.
| Table 1-25: Local Resistance Coefficients for Pipe Contractions (Re > 104) | |
|---|---|
| A2/A1 | ζ |
| 0.1 | 0.45 |
| 0.2 | 0.40 |
| 0.3 | 0.35 |
| 0.4 | 0.30 |
| 0.5 | 0.25 |
| 0.6 | 0.20 |
| 0.7 | 0.15 |
| 0.8 | 0.10 |
| 0.9 | 0.05 |
| 1.0 | 0 |
d. Local Resistance Coefficients for Pipe Bends
These coefficients are presented in Table 1-26, which is important when calculating pressure losses in systems where a hydraulic flow control valve is installed downstream of elbow fittings.
| Table 1-26: Local Resistance Coefficients for Pipe Bends | |
|---|---|
| Bend Angle α (°) | Resistance Coefficient ζ |
| 10 | 0.04 |
| 20 | 0.10 |
| 30 | 0.17 |
| 40 | 0.27 |
| 50 | 0.40 |
| 60 | 0.55 |
| 70 | 0.70 |
| 80 | 0.90 |
| 90 | 1.12 |
3. Total Energy Loss
Hydraulic systems are always composed of various hydraulic components and fittings. Therefore, the total pressure loss in a system is the arithmetic sum of all frictional pressure losses and local pressure losses in the pipeline. This comprehensive calculation is essential when specifying a hydraulic flow control valve to ensure it can operate effectively within the system's pressure constraints.
Δptotal = ΣΔpf + ΣΔpζ = ρ/2 (Σλ(l/d)v² + Σζv²)
Similarly, total head loss is expressed as:
htotal = Σhf + Σhζ = 1/(2g) (Σλ(l/d)v² + Σζv²)
System Design Consideration
When designing a hydraulic system, accurate calculation of total energy loss ensures proper component selection, including the hydraulic flow control valve. Undersized components or inadequate pressure ratings can lead to inefficient operation, excessive energy consumption, and premature failure. The hydraulic flow control valve must be rated to handle the calculated pressure losses while maintaining precise flow regulation across all operating conditions.