A General Guide to Two-Phase Flow Pressure Drop - WittyWriter
A General Guide to Two-Phase Flow Pressure Drop
1. Introduction and Application
This guide provides a universal philosophy for calculating pressure drop in in-plant piping systems involving two-phase flow (the simultaneous flow of gas and liquid). It covers the fundamental definitions, flow regime analysis, and general calculation principles for basic and detailed engineering.
These guidelines are intended for in-plant piping. They do not cover specialized applications such as cross-country pipelines, wells, or offshore platforms, which require more complex crude characterization.
Why is Two-Phase Flow Different from Single-Phase?
Two-phase flow calculations are significantly more complex than single-phase (all-liquid or all-gas) calculations for several reasons:
Slip: The gas phase and liquid phase often travel at different velocities (known as "slip").
Flow Regimes: The phases can arrange themselves in various patterns (e.g., bubbles, slugs, annular) that drastically change friction and density.
Phase Change: As pressure drops along the pipe, more liquid may flash into gas, changing the flow rate, velocity, and density of each phase.
Elevation Effects: In a vertical-upward pipe, the potential energy (static head) lost is often not fully regained in a downward-flowing section. This is because the "liquid hold-up" (and thus mixture density) is much lower in the downward section.
Counter-intuitive Sizing: Unlike single-phase flow, increasing the pipe diameter can sometimes *increase* the pressure drop if it causes a shift into an inefficient flow regime (like slug flow).
2. Key Definitions
Superficial Velocity (Vs)
The velocity a phase would have if it were the *only* fluid flowing through the *entire* pipe cross-section (A).
Vsl (Liquid) = Ql / A
Vsg (Gas) = Qg / A
Mixture Velocity (Vm)
The combined velocity of the total mixture, calculated as the sum of the superficial velocities.
Vm = Vsl + Vsg
Hold-Up (H)
The fraction of the pipe's cross-sectional volume that is occupied by a specific phase at any given moment. This is a critical value, as it is *not* the same as the volumetric flow ratio due to "slip."
Liquid Hold-Up (Hl): The fraction of the pipe occupied by liquid.
Gas Hold-Up (Hg): The fraction of the pipe occupied by gas. (Hg = 1 - Hl)
Actual Velocity (V)
The true velocity of each phase, found by dividing its superficial velocity by its hold-up. Since Hl and Hg are fractions, the actual velocities are always higher than the superficial velocities.
The effective density of the two-phase mixture in the pipe, calculated using the liquid and gas densities (Οl, Οg) weighted by their respective hold-ups.
Οm = (Οl Γ Hl) + (Οg Γ Hg)
Mixture Viscosity (ΞΌm)
The effective viscosity of the two-phase mixture. This is typically calculated using empirical correlations that are also weighted by the liquid hold-up.
Critical (Choked) Flow
At very high pressure drops, the two-phase mixture can reach a maximum velocity, analogous to the sonic velocity of a gas. A key difference is that the critical velocity for a two-phase mixture is often *much lower* than the critical velocity for the gas phase alone. This "choked" condition can occur at restrictions (like valves) or at the end of a pipe.
3. Understanding Two-Phase Flow Regimes
Flow regime describes the physical distribution of the gas and liquid phases inside the pipe. This pattern is the single most important factor in determining pressure drop, hold-up, and potential operational problems. The regime is determined by the fluid velocities, properties (density, viscosity), and the pipe's orientation (horizontal, vertical, or inclined).
Flow Regimes in Vertical (Upward) Flow
Bubble Flow: Dispersed bubbles of gas move upward through a continuous liquid phase. Occurs at low gas velocities.
Slug Flow: Bubbles coalesce into large "Taylor" bubbles that are separated by slugs of liquid. This regime is highly unstable and causes severe vibration and pressure fluctuations.
Froth (or Churn) Flow: A highly turbulent and chaotic regime where the slugs begin to break down. Often grouped with slug flow.
Annular Flow: The liquid flows as a thin film on the pipe wall, while the gas flows at high velocity as a continuous core in the center.
Mist Flow: At very high gas velocities, the liquid film is stripped from the wall and is carried as fine droplets within the continuous gas phase.
Flow Regimes in Horizontal Flow
Bubble Flow: Bubbles flow along the top of the pipe within the continuous liquid.
Plug Flow: Elongated bubbles (plugs) flow along the top of the pipe, separated by liquid.
Stratified Flow: At low liquid and gas rates, the gas and liquid separate completely, with the liquid flowing along the bottom of the pipe and the gas flowing above it.
Wavy Flow: As gas velocity increases, waves form on the surface of the stratified liquid.
Slug Flow: The waves grow large enough to touch the top of the pipe, creating large slugs of liquid that are pushed forward by high-pressure gas. This is a severe, high-vibration condition.
Annular Flow: The liquid forms a film around the entire pipe wall (thicker at the bottom), with a high-velocity gas core.
Spray (or Mist) Flow: At very high gas velocities, all liquid is entrained as droplets in the gas core.
4. Critical Challenge: Identifying and Avoiding Slug Flow
DANGER: Slug Flow
Slug flow is the most common and destructive flow regime in plant piping. It is an unstable, oscillating flow that can cause severe mechanical vibration, pressure surges, and inconsistent instrument readings. This can lead to pipe support failure, equipment damage, and process upsets.
Slug Formation Mechanisms:
Wave Growth: In horizontal or inclined pipes, fast-moving gas creates waves that grow large enough to bridge the pipe.
Terrain Slugging: Liquid accumulates in low points (sags) in the piping. The gas pressure builds up behind the liquid until it has enough force to push the entire liquid "slug" out at high velocity.
Transient Flow: Sudden changes in flow rate (e.g., from a control valve) can cause liquid to build up and form a slug.
Design Strategies to Avoid Slug Flow:
Line Sizing: Reduce the pipe diameter to increase the gas velocity, pushing the flow into a more stable regime (like Annular or Mist flow).
Eliminate Low Points: Ensure horizontal lines are self-draining and avoid "pocketed" or sagged pipe runs where liquid can accumulate.
Valve Location: As shown in the example of a vacuum transfer line, placing the control valve at the *end* of the run (right at the vessel) can create a stable, low-pressure flow. Placing it at the *beginning* (near the heater) can cause flashing and slugging in the line after the valve.
Piping Layout: A shortened, self-draining pipeline is always preferred over a layout with low points, even if it impacts convenience.
5. Calculating Two-Phase Pressure Drop
The General Equation
The total pressure drop (ΞP) in a two-phase line is the sum of three components: static head (elevation), friction, and acceleration.
ΞPElevation (Static Head): The pressure loss or gain due to changes in elevation. This is calculated using the mixture density (Οm) and the angle of inclination (ΞΈ).
ΞPElevation β Οm Γ L Γ sin(ΞΈ)
ΞPFriction: The pressure loss due to friction against the pipe wall. This is calculated using a two-phase friction factor, the mixture density (Οm), and the mixture velocity (Vm).
ΞPFriction β fm Γ L Γ Οm Γ VmΒ² / D
ΞPAcceleration: The pressure loss that occurs as the gas phase expands due to dropping pressure. This expansion "accelerates" the flow and is only significant in high-velocity systems or lines with a very large pressure drop.
Calculation Approach
Because properties change as pressure drops, the calculation must be done in segments:
Calculate the superficial gas and liquid velocities (Vsg, Vsl) at the inlet conditions.
Determine the flow regime (e.g., slug, annular) using a standard flow map.
Select a correlation (see Table 1) appropriate for that flow regime and pipe orientation to calculate the liquid hold-up (Hl).
Calculate the mixture density (Οm), mixture viscosity (ΞΌm), and two-phase friction factor (fm).
Calculate the total pressure drop (ΞP) for a short segment of the pipe.
Using the new, lower pressure at the end of the segment, flash the fluid to find new phase fractions and properties. Repeat steps 1-5 for the next segment.
Common Two-Phase Flow Correlations
No single correlation works for all conditions. The choice of method is critical for an accurate result. The table below, based on common hydraulic software packages, gives general recommendations. (β = Recommended, β = Not Recommended)
Correlation
Horizontal
Vertical Upward
Vertical Downward
Upward Inclined
Downward Inclined
Remarks
Begges & Brill
β
Under-predicts hold-up.
Begges & Brill - Moody
β
β
β
β
β
Recommended general method. Performs reasonably well for the widest range of flow conditions.
Begges & Brill - No Slip
β
β
β
β
β
To be used for low hold-up.
Eaton
β
β
β
β
β
Do not use for diameters < 2 in. Under-predicts hold-up for Hl < 0.1.
Eaton-Flannigan
β
β
β
β
β
Works well for 0.1 < Hl < 0.35.
Dukler
β
β
Good for horizontal flow. Tends to under-predict pressure drop & hold-up.
Dukler-Flannigan
β
β
Dukler-Eaton-Flannigan
β
β
Lockhart-Martinelli
β
β
β
β
β
Generally over-predicts pressure drop. Do not use for large pipes.
Mukherjee-Brill
β
β
β
β
β
Recommended for low liquid hold-up systems.
Begges & Brill - Moody-Eaton
β
β
β
β
β
Non-standard hybrid model.
Begges & Brill - Moody-Dukler
β
β
β
β
β
Non-standard hybrid model.
Mukherjee-Brill-Eaton
β
β
β
β
β
Non-standard hybrid model.
Begges & Brill High velocity
β
β
For high velocity (e.g., flare) systems.
Begges & Brill-Moody-High velocity
β
β
For high velocity (e.g., flare) systems.
Begges & Brill-No slip-High velocity
β
β
β
β
β
For high velocity (e.g., flare) systems.
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