This document discusses two-phase flow line sizing for liquid-gas flows in piping systems. It describes the different flow regimes that can occur using Baker's flow regime map. The key steps outlined are: 1) determining the flow regime based on fluid properties and flow rates, 2) calculating pressure drops for the liquid and gas phases separately using correlations, 3) using a multiplier to determine the two-phase pressure drop based on the flow regime, and 4) summing pressure drops from friction, elevation changes, and fittings to obtain the total pressure drop. Care must be taken to size each pipe segment separately as properties and regimes can change along the line.

1. PIPING SIZING EXERCISE: TWO-PHASE FLOW LINE SIZING
Vikram Sharma
Piping is an integral part of a chemical process and chemical plant where fluids are conveyed from one
point to another point of the plant. The nature of fluid flow in a piping can be in the form of single or
multiphase fluid flow. Single-phase fluid flow can be described as the fluid flow in a single state such as
liquid-only or gas-only flow. Multiphase fluid flow is described as a simultaneous flow of several fluid
phases. The common multiphase fluid flow encountered in the process plants are gas-liquid, liquid-liquid
or liquid-solid flow. This article shall focus on liquid-gas two-phase flow.
The existence of two-phase fluid flow shall be taken seriously by process engineers due to the severity of
pressure drop problems that may result to operational problems in a process. Coker (2007) highlighted
that the fluid flow patterns may vary significantly along a pipeline. In the case for cross country pipeline,
an engineer shall consider the effect of terrain profile in predicting the pressure drop across the pipeline.
Besides this, the pressure drop in a two-phase flow can be up to 10 times higher than the single-phase
flow (Coker, 2007).
The first step is to gather important fluid properties for both phases. For liquid and gas phases, a process
engineer shall require the flowrates (kg/h), density (kg/m3
), viscosity (cP) and surface tension (N/cm).
The information is then used to establish the two-phase flow regime. Besides this, the pipe layout such as
pipe size, length and fittings shall be considered in calculating the pressure drop. Before that, we must
have some understanding on the flow regimes. There are seven flow regimes that are bubble flow, plug
flow, stratified flow, slug flow, annular flow and dispersed flow. Each flow regime is identified using an
old and widely used map called Baker’s map. Refer to Figure 1 for the Baker’s map.
Figure 1. Baker's map for two-phase fluid flow
(Source: Szilas, 1975)
Szilas (1975) provided the equations to determine the horizontal and vertical axis. The equations are
provided below.
̇
Eq.1(a)
̇
Eq.1(b)
( )
⁄
Eq.1(c)
Bx =
By =

2. [ ( ) ]
⁄
Eq.1(d)
where:
vml: Superficial liquid velocity [m/s]
vmg: Superficial gas velocity [m/s]
̇ : Volumetric liquid flow rate [m3
/s]
̇ : Volumetric gas flow rate [m3
/s]
ρl: Liquid density [kg/m3
]
ρG: Gas density [kg/m3
]
di: Internal pipe diameter [m]
µl: Liquid viscosity [Ns/m2
]
σl: Surface tension of liquid [N/m]
The bubble flow is characterized as bubbles of gas that are dispersed throughout the liquid (Coker, 2007)
and moves along the upper part of the pipe due to their buoyancy (Thome, n.d.). The velocity of the
bubbles of gas moves at a velocity approximately the same as the liquid. This flow occurs when the gas
content is less than about 0.3 weight fraction of the total volumetric flow and at high mass flow rates. The
linear velocities of the liquid phase are in the region of 1.5-4.8 m/s and 0.15-0.61 m/s for the vapour
phase (Coker, 2007).
Plug flow, as seen in Figure 2, is a form of intermittent two-phase fluid flow. The fluid flow is seen as
alternate plugs of liquid and gas where the gas portion moves along the upper part of the pipe. The liquid
portion moves along the bottom part of the pipe. This flow is expected to occur when the liquid phase is
at 0.61 m/s with the vapour phase less than 1.22 m/s (Coker, 2007).
Stratified flow is another common two-phase flow seen in horizontal piping. It occurs when the two
phases are separated from each other by a continuous interface and the liquid phase is stratified at the
bottom due to gravity (Coker, 2007 ; Alain & Fabre, 2011). Such flow regime is noticed for both
horizontal and slightly inclined pipelines. The structure of the fluid interface may be smooth or wavy in
nature dependent on the gas flow rate. Low gas flowrate results smooth fluid interface or possible rippling
by small capillary waves of a few millimeter lengths. As the gas flow rate increases, waves of small
amplitude appear and eventually droplets can be entrained and deposited at the wall or at the interface
(Alain & Fabre, 2011). Stratified flow occurs when the liquid superficial velocity is less than 0.15 m/s
with gas superficial velocities between 0.15 to 3.05 m/s (Coker, 2007).
Wave flow is another commonly seen two-phase flow regime in horizontal piping. It is similar to
stratified flow and the gas-flow rate is moving at a higher velocity (Coker, 2007). At low gas velocities,
the gas-liquid interface is flat. As the velocity increases, the interface becomes unstable due to small
disturbances and waves are seen on it. The shape and size of waves are dependent on the pipeline
geometry and fluids flowrate (Sreenivas, 2011).

3. Figure 2. Gas-liquid flow regime
(Source: Coker, 2007)
Slug flow, an intermittent two-phase flow, described as liquid rich slugs that may or may not cover the
entire inner section of a pipe. The pattern is also observed when the rapidly moving gas creates waves and
form frothy slugs that travel along the pipeline at a velocity higher than average liquid velocity. Slug flow
is an undesirable two-phase in piping as it results to severe vibrations in equipment. The vibrations are
due to high velocity slugs travelling against fittings (Coker, 2007 ; McCready, n.d.). For slug flow to
occur, the liquid superficial velocity approximately 4.58 m/s and gas superficial velocities between 4.58
m/s to 15.24 m/s (Coker, 2007).
Annular flow is seen when the gas velocity in the pipe further increases resulting to gas flow through the
liquid flow. Due to gravity, the liquid film at the bottom section of the pipe is thicker compared to the top
(Hewitt & Taylor-Hall, 2013). This flow is noticed when the liquid superficial velocity is less than 0.15
m/s and the gas superficial velocity is greater than 6.1 m/s (Coker, 2007).
Dispersed flow occurs when the liquid is entrained as fine droplets by the gas phase especially in gas-
liquid flow. The dispersed phase in both gas-liquid and liquid-liquid is primarily determined by the flow
rates of both phases as the interface between both phases is deformable. The dispersed phase of this
dispersed flow coalesces and become continuous phase with increasing flow rate (Akiwi, 2010). This type
of two-phase flow occur when the gas content is greater than 30% of the total weight flow rate (Coker,
2007).
A brief description of each flow regime has been provided in the above paragraphs. The x-axis and y-axis
of the Baker’s plot are calculated with reference to Eq.1(a) to Eq.1(d). The meeting point of the values
from both x- and y-axis provides the two-phase flow regime (Szilas, 1975).
( ) [ ( ) ] Eq.2(a)
( ) Eq.2(b)
The calculation procedure proceeds with the pressure drop calculation for both liquid and gas phase. At
this stage, the pressure drop calculation shall treat the liquid and gas independently. The pressure drop for
liquid and gas phases are expressed in bar/100m (Coker, 2007).

4. For liquid-phase: Eq.3(a)
For gas-phase: Eq.3(b)
where:
fD: Darcy friction factor [-]
ΔPL,bar/100m: Pressure drop of liquid phase [bar/100m]
ΔPG,bar/100m: Pressure drop of gas phase [bar/100m]
The friction factors seen in Eq.3(a) and Eq.3(b) is the Darcy friction factor (fD). Point to note, process
engineers are advised not to be confused with Fanning friction factor (fF) as the former is 4 times the
Fanning friction factor. Churchill (1977) equation is referred to compute the Darcy friction factorfor both
laminar and turbulent flow (Coker, 2007).
[( ) ( )
] Eq.4
{ (
( )
)} Eq.4(a)
( ) Eq.4(b)
where:
ε: Pipe roughness [m]
di: Pipe internal diameter [m]
Re: Reynolds number [-]
The Lockhart-Martinelli parameter, X, is expressed as the ratio of liquid and gas pressure drops. This
parameter is a function of mass fluxes, densities, viscosities of the liquid and the gas, and diameter of the
pipe (Mekisso, 2004).
[ ] Eq.5
A generalized method, Lockhart-Martinelli (1949), was developed to determine the frictional pressure
drop multipliers for both liquid, , and gas phases, . (Refer Figure 3). These multipliers depend on
whether the liquid-phase only flow and gas-phase only flow are laminar (a.k.a viscous) or turbulent
(Thermal-FluidsCentral, 2010). Another point to consider is that the transitional flow is treated as
turbulent flow (Coker, 2007). A careful observation in Figure 3 showed that increases with increasing
X. The decreases with increasing X (Coker, 2007).

5. Figure 3. Pressure drop correlations for Lockhart-Martinelli (1946)
(Source: Coker, 2007)
The multipliers for both liquid and gas phases are a factor of the fluid flow regime. Noticed that there are
4 curves for and that corresponds to a combination of 2 flow regimes that are laminar (viscous) and
turbulent for liquid-only flow and vapour-only flow (Thermal-FluidsCentral, 2010). Tabulated data from
Figure 3 is presented in Table 1 (Coker, 2007).
Table 1.Two-phase multipliers with respect to Lockhart-Martinelli parameter, X
The subscripts seen in the top row of Table 1 indicate the fluid flow regime. For example, tt represents
turbulent flow for both liquid and gas phases (Coker, 2007).
The two-phase pressure drop can be estimated either from liquid or gas-phase pressure drop via a
multiplier which is specific to liquid and gas phases
Eq.6(a)
Eq.6(b)

6. The above approach in obtaining the multipliers is cumbersome as process engineers are required to
extract the information manually and could result to highly inaccurate data. Therefore, Chisholm (1967)
presented a correlation by incorporating the effect of interfacial shear forces in the Lockhart-Martinelli
(1949) equation. This new correlation allows process engineers to determine the hydraulic diameters of
the phases more accurately compared to Lockhart-Martinelli (1949). Besides this, it do not require the use
of graph (Figure 9) to determine the multipliers (Mekisso, 2004). Chisholm (1967) correlations in terms
of Lockhart-Martinelli (1949) are presented in Eq.7(a) and Eq.7(b) and the values of coefficient C is
presented in Table 2 (Thermal-FluidsCentral, 2010).
Eq.7(a)
Eq.7(b)
Table 2. Values of C for Eq.7(a) and Eq.7(b)
Liquid Vapour Subscripts C
Turbulent Turbulent tt 20
Viscous Turbulent vt 12
Turbulent Viscous tv 10
Viscous Viscous vv 5
The above correlations are focused on determining the frictional pressure drop in a horizontal piping. The
effect of elevation on the overall two-phase flow pressure drop shall not be ignored. An expression based
on homogeneous model for gas-liquid flow is presented below.
( ) Eq.8
̇ ̇
(
̇ ̇
)
Eq.8(a)
where:
ΔPstatic: Static pressure drop [bar]
ρM: Two-phase density [kg/m3
]
h: Pipe elevation [m]
̇ : Liquid mass flowrate [kg/s]
̇ : Gas mass flowrate [kg/s]
θ: Angle of piping from horizontal position [°]
The total pressure drop of a gas-liquid flow in a piping system is the sum of pressure drop due to friction,
static and fittings. For pipe fittings, the author recommends to use the equivalent length method to
calculate the pressure drop. This method approximates the pressure drop of fittings based on hypothetical
piping length.
Eq.9
( ) ( ) Eq.9(a)

7. (
̇ ̇
) Eq.9(b)
where:
A: Internal pipe area based on di [m2]
vm: Two-phase flow velocity [m/s]
g: Gravitation acceleration [m/s2
]
ΔPfittings: Pressure drop due to pipe fittings [bar]
The total pressure drop of a gas-liquid flow is the sum of 3 pressure drop components that are frictional,
elevation and pipe fittings.
Eq.10
where:
ΣPTP: Total two-phase pressure drop [bar]
ΔPf: Two-phase friction pressure drop [bar]
ΔPstatic: Two-phase static pressure drop [bar]
ΔPfittings: Two-phase static pressure drop [bar]
Process engineers must consider the impact of erosion-corrosion on process and utility piping in a two-
phase flow. In certain flow regime, the liquid velocities approach or exceed the gas velocities. In some
cases, the liquid velocities are higher than the required velocity and this leads to erosion-corrosion. An
expression was formulated to determine whether erosion-corrosion may occur at a particular velocity.
Eq.11
Point to note, two-phase fluid flow line sizing shall be carried out based on each pipe segment in a piping
system. This is important as the fluid properties such as densities, viscosities etc. are not constant and the
fluid flow regime changes throughout a piping length. Therefore, the line sizing approach outline above
shall be repeated on each pipe segment.
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