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Chapter 13 Multi–Phase Flow
13.1 Introduction
Traditionally, the topic of multi–phase flow is ignored in an introductory class on fluid mechanics.
For many engineers, this class will be the only opportunity to be exposed to this topic.
The knowledge in this topic without any doubts, is required for many engineering problems.
Calculations of many kinds of flow deals with more than
one phase or material flow.
The author believes that the trends and effects of multiphase flow could
and should be introduced and considered by engineers.
In the past, books on multiphase flow were written more as
a literature review or heavy on the mathematics.
It is recognized that multiphase flow is still evolving.
In fact, there is not a consensus to the exact map of many flow regimes.
This book attempts to describe these issues as a fundamentals of
physical aspects and less as a literature review.
This chapter provides information that is more or less in
Additionally, the nature of multiphase flow requires solving many equations.
Thus, in many books the representations is by writing the whole set governing equations.
Here, it is believed that the interactions/calculations requires a full
year class and hence, only the trends and simple calculations are described.
13.2 History
The study of multi–phase flow started for practical purposes after World War II.
Initially the models were using simple assumptions.
For simple models,there are two possibilities
(1) the fluids/materials are flowing in well homogeneous mixed (where the main problem to find the viscosity),
(2) the fluids/materials are flowing separately where the actual total loss pressure can be correlated based
on the separate pressure loss of each of the material.
If the pressure loss was linear then the total loss will be the summation
of the two pressure losses (of the lighter liquid (gas) and the heavy liquid).
Under this assumption the total is not linear and experimental correlation was made.
This was suggested by Lockhart and Martinelli who use a model where the
flow of the two fluids are independent of each other.
They postulate that there is a relationship between the
pressure loss of a single phase and combine phases pressure loss
as a function of the pressure loss of the other phase.
It turned out this idea provides a good crude results in some cases.
Researchers that followed Lockhart and Martinelli looked for a different map for different combination of phases.
When it became apparent that specific models were needed for different
situations, researchers started to look for different flow regimes and provided different models.
Also the researchers looked at the situation when the different regimes are applicable.
Which leads to the concept of flow regime maps.
Taitle and Duckler suggested a map based on five dimensionaless groups which are considered as the most useful today.
However, Taitle and Duckler's map is not universal and it is only applied to certain liquid–gas conditions.
For example, Taitle–Duckler's map is not applicable for microgravity.
13.3 What to Expect From This Chapter
As oppose to the tradition of the other chapters in this book and all other Potto project books,
a description of what to expect in this chapter is provided.
It is an attempt to explain and convince all the readers that the multi–phase flow must be included
in introductory class on fluid mechanics.
Hence, this chapter will explain the core concepts of the multiphase flow
and their relationship, and importance to real world.
This chapter will provide: a category of combination of phases, the concept of flow regimes,
multi–phase flow parameters definitions, flow parameters effects on the flow regimes, partial discussion
on speed of sound of different regimes, double choking phenomenon
(hopefully), and calculation of pressure drop of simple homogeneous model.
This chapter will introduce these concepts so that the engineer not only be able to understand a conversation
on multiphase but also, and more importantly, will know and understand the trends.
However, this chapter will not provide a discussion of transient problems, phase change or transfer
processes during flow, and actual calculation of pressure of the different regimes.
13.4 Kind of MultiPhase Flow
Fig. 13.1 Different fields of multi phase flow.
All the flows are a form of multiphase flow.
The discussion in the previous chapters is only as approximation
when multiphase can be ``reduced'' into a single phase flow.
For example, consider air flow that was discussed and presented earlier as a single phase flow.
Air is not a pure material but a mixture of many gases.
In fact, many proprieties of air are calculated as if the air is made of well mixed gases of Nitrogen and Oxygen.
The results of the calculations of a mixture do not change much if it is assumed that the air flow
as stratified flow
of many concentration layers (thus, many layers (infinite) of different materials).
Practically for many cases, the homogeneous assumption is enough and suitable.
However, this assumption will not be appropriate when the air is stratified because of large body forces,
or a large acceleration. Adopting this assumption might lead to a larger error.
Hence, there are situations when air flow has to be considered as multiphase
flow and this effect has to be taken into account.
In our calculation, it is assumed that air is made of only gases.
The creation of clean room is a proof that air contains small particles.
In almost all situations, the cleanness of the air or the fact that air is a mixture is ignored.
The engineering accuracy is enough to totally ignore it.
Yet, there are situations where cleanness of the air can affect the flow.
For example, the cleanness of air can reduce the speed of sound.
In the past, the breaks in long trains were activated by reduction of the compressed line (a patent no. 360070 issued to
In a four (4) miles long train, the breaks would started to work after about 20 seconds in the last wagon.
Thus, a 10% change of the speed of sound due to dust particles in air could reduce the stopping time by 2 seconds (50 meter
difference in stopping) and can cause an accident.
One way to categorize the multiphase is by the materials flows.
For example, the flow of oil and water in one pipe is a multiphase flow.
This flow is used by engineers to reduce the cost of moving crude oil through a long pipes system.
The ``average'' viscosity is meaningless since in many cases the water follows around the oil.
The water flow is the source of the friction.
However, it is more common to categorize the flow by the distinct phases that flow in the tube.
Since there are three phases, they can be solid–liquid,
solid–gas, liquid–gas and solid–liquid–gas flow.
This notion eliminates many other flow categories that can and should be included in multiphase flow.
This category should include any distinction of phase/material.
There are many more categories,
for example, sand and grain (which are ``solids'') flow with rocks
and is referred to solid–solid flow.
The category of liquid–gas should be really viewed as the extreme case of
liquidliquid where the density ratio is extremely large.
For the gas, the density is a strong function of the temperature and pressure.
Open Channel flow is, although important, is only an extreme case of
liquidgas flow and is a sub category of the multiphase flow.
The multiphase is an important part of many processes.
The multiphase can be found in nature, living bodies (bio–fluids), and industries.
Gas–solid can be found in sand storms, and avalanches.
The body inhales solid particle with breathing air.
Many industries are involved with this flow category such as dust collection, fluidized bed, solid propellant rocket,
paint spray, spray casting, plasma and river flow with live creatures (small organisms to large fish) flow
of ice berg, mud flow etc.
The liquid–solid, in nature can be blood flow, and river flow.
This flow also appears in any industrial process that are involved in
solidification (for example die casting) and in moving solid particles.
Liquid–liquid flow is probably the most common flow in the nature.
Flow of air is actually the flow of several light liquids (gases).
Many natural phenomenon are multiphase flow, for an example, rain.
Many industrial process also include liquidliquid such as painting,
hydraulic with two or more kind of liquids.
13.5 Classification of LiquidLiquid Flow Regimes
The general discussion on liquid–liquid will be provided and
the gas–liquid flow will be discussed as a special case.
Generally, there are two possibilities for two different materials
to flow (it is also correct for solid–liquid and any other combination).
The materials can flow in the same direction and it is referred as co–current flow.
When the materials flow in the opposite direction, it is referred as counter–current.
In general, the cocurrent is the more common.
Additionally, the counter–current flow must have special configurations of long length of flow.
Generally, the counter–current flow has a limited length window of possibility
in a vertical flow in conduits with the exception of magnetohydrodynamics.
The flow regimes are referred to the arrangement of the fluids.
The main difference between the liquid–liquid flow to gasliquid
flow is that gas density is extremely lighter than the liquid
density. For example, water and air flow as oppose to water and oil flow.
The other characteristic that is different between the gas flow and the
liquid flow is the variation of the density.
For example, a reduction of the pressure by half will double
the gas volumetric flow rate while the change in the liquid is negligible.
Thus, the flow of gas–liquid can have several flow regimes
in one situation while the flow of liquid–liquid will (probably) have only one flow regime.
13.5.1 Co–Current Flow
In Co–Current flow, two liquids can have three main categories:
vertical, horizontal, and what ever between them.
The vertical configuration has two cases, up or down.
It is common to differentiate between the vertical (and near vertical)
and horizontal (and near horizontal).
There is no exact meaning to the word ``near vertical'' or ``near horizontal''
and there is no consensus on the limiting angles
(not to mention to have limits as a function with any parameter that determine the limiting angle).
The flow in inclined angle (that not covered by the word ``near'')
exhibits flow regimes not much different from the other two.
Yet, the limits between the flow regimes are considerably different.
This issue of incline flow will not be covered in this chapter.
13.5.1.1 Horizontal Flow
Fig. 13.2 Stratified flow in horizontal tubes when the liquids flow is very slow.
The typical regimes for horizontal flow are stratified flow
(open channel flow, and non open channel flow),
dispersed bubble flow, plug flow, and annular flow.
For low velocity (low flow rate) of the two liquids, the heavy liquid
flows on the bottom and lighter liquid flows on the top
as depicted in Figure 13.2.
This kind of flow regime is referred to as horizontal flow.
When the flow rate of the lighter liquid is almost zero, the flow is referred to as open channel flow.
This definition (open channel flow) continues for small amount of lighter
liquid as long as the heavier flow can be calculated as open channel flow (ignoring the lighter liquid).
The geometries (even the boundaries) of open channel flow are very diverse.
Open channel flow appears in many nature (river) as well in industrial
process such as the die casting process where liquid metal is injected into a cylinder (tube) shape.
The channel flow will be discussed in a greater detail in Open Channel Flow chapter.
As the lighter liquid (or the gas phase) flow rate increases (superficial velocity), the friction between the phases increase.
The superficial velocity is referred to as the velocity that any phase will have if the other phase was not exist.
This friction is one of the cause for the instability which manifested itself as waves and changing the surface from
straight line to a different configuration (see Figure 13.3).
The wave shape is created to keep the gas and the liquid velocity equal
and at the same time to have shear stress to be balance by surface tension.
The configuration of the cross section not only depend on the surface tension,
and other physical properties of the fluids but also on the material of the conduit.
Fig. 13.3 Kind of Stratified flow in horizontal tubes.
As the lighter liquid velocity increases two things can happen (1) wave size increase, and (2) the shape of cross section continue to deform.
Some referred to this regime as wavy stratified flow but this definition is not accepted by all as a category by itself.
In fact, all the two phase flow are categorized by wavy flow which will proven later.
There are two paths that can occur on the heavier liquid flow rate.
If the heavier flow rate is small, then the wave cannot reach to the crown
and the shape is deformed to the point that all the heavier liquid is around the periphery.
This kind of flow regime is referred to as annular flow.
than the distance, for the wave to reach the conduit crown is smaller.
At some point, when the lighter liquid
flow increases, the heavier liquid wave reaches to the crown of the pipe.
At this stage, the flow pattern is referred to as slug flow or plug flow.
Plug flow is characterized by regions of lighter liquid
filled with drops of the heavier liquid with Plug (or Slug)
of the heavier liquid (with bubble of the lighter liquid).
These plugs are separated by large ``chunks'' that almost fill the entire tube.
The plugs are flowing in a succession (see Figure 13.4).
The pressure drop of this kind of regime is significantly larger than the stratified flow.
The slug flow cannot be assumed to be as homogeneous flow nor it can exhibit some average viscosity.
The ``average'' viscosity depends on the flow and thus making it
as insignificant way to do the calculations.
Further increase of the lighter liquid flow rate move the flow regime into annular flow.
Thus, the possibility to go through slug flow regime depends on if there is enough liquid flow rate.
Fig. 13.4 Plug flow in horizontal tubes when the liquids flow is faster.
Choking occurs in compressible flow when the flow rate is above a certain point.
All liquids are compressible to some degree.
For liquid which the density is a strong and primary
function of the pressure, choking occurs relatively closer/sooner.
Thus, the flow that starts as a stratified flow will turned into a slug
flow or stratified wavy flow after a certain distance depends on the
heavy flow rate (if this category is accepted).
After a certain distance, the flow become annular or the flow will choke.
The choking can occur before the annular flow regime is obtained
depending on the velocity and compressibility of the lighter liquid.
Hence, as in compressible flow, liquid–liquid flow has a maximum
combined of the flow rate (both phases).
This maximum is known as double choking phenomenon.
The reverse way is referred to the process where the starting point is
high flow rate and the flow rate is decreasing.
As in many fluid mechanics and magnetic fields, the return path is not move the exact same way.
There is even a possibility to return on different flow regime.
For example, flow that had slug flow in its path can be returned as stratified wavy flow.
This phenomenon is refer to as hysteresis.
Flow that is under small angle from the horizontal
will be similar to the horizontal flow.
However, there is no consensus how far is the ``near'' means.
Qualitatively, the ``near'' angle depends on the length of the pipe.
The angle decreases with the length of the pipe.
Besides the length, other parameters can affect the ``near.''
Fig. 13.5 Modified Mandhane map for flow regime in horizontal tubes.
The results of the above discussion are depicted in Figure 13.5.
As many things in multiphase, this map is only characteristics of the ``normal'' conditions, e.g.
in normal gravitation, weak to strong surface tension effects (air/water in ``normal'' gravity), etc.
13.5.1.2 Vertical Flow
Fig. 13.6 Gas and liquid in Flow in verstical tube against the gravity.
The vertical flow has two possibilities, with the gravity or against it.
In engineering application, the vertical flow against the gravity is more common used.
There is a difference between flowing with the gravity and flowing against the gravity.
The buoyancy is acting in two different directions for these two flow regimes.
For the flow against gravity, the lighter liquid has a buoyancy that acts
as an ``extra force'' to move it faster and this effect is opposite for the heavier liquid.
The opposite is for the flow with gravity.
Thus, there are different flow regimes for these two situations.
The main reason that causes the difference is that the heavier liquid is more dominated by gravity
(body forces) while the lighter liquid is dominated by the pressure driving forces.
Flow Against Gravity
For vertical flow against gravity, the flow cannot start as a stratified flow.
The heavier liquid has to occupy almost the entire cross section before it
can flow because of the gravity forces.
Thus, the flow starts as a bubble flow.
The increase of the lighter liquid flow rate will increase the number of bubbles
until some bubbles start to collide.
When many bubbles collide, they create a large bubble and the flow is referred
to as slug flow or plug flow (see Figure 13.6).
Notice, the different mechanism in creating the plug flow
in horizontal flow compared to the vertical flow.
Further increase of lighter liquid flow rate will increase the slug size as more bubbles collide to
create ``super slug''; the flow regime is referred as elongated bubble flow.
The flow is less stable as more turbulent flow and several ``super slug''
or churn flow appears in more chaotic way, see Figure 13.6.
After additional increase of ``super slug'' , all these ``elongated slug'' unite to become an annular flow.
Again, it can be noted the difference in the mechanism that create annular flow for vertical and horizontal flow.
Any further increase transforms the outer liquid layer into bubbles in the inner liquid.
Flow of near vertical against the gravity in two–phase does not deviate from vertical.
The choking can occur at any point depends on the fluids and temperature and pressure.
13.5.1.3 Vertical Flow Under Micro Gravity
Fig. 13.7 A dimensional vertical flow map under very low gravity against the gravity.
The above discussion mostly explained the flow in a vertical
configuration when the surface tension can be neglected.
In cases where the surface tension is very important.
For example, out in space between gas and liquid (large density difference) the situation is different.
The flow starts as dispersed bubble (some call it as
``gas continuous'') because the gas phase occupies most of column.
The liquid flows through a trickle or channeled flow that only partially wets part of the tube.
The interaction between the phases is minimal and can be
considered as the ``open channel flow'' of the vertical configuration.
As the gas flow increases, the liquid becomes more turbulent
and some parts enter into the gas phase as drops.
When the flow rate of the gas increases further, all the gas phase
change into tiny drops of liquid and this kind of regime referred to as mist flow.
At a higher rate of liquid flow and a low flow rate of
gas, the regime liquid fills the entire void and the gas is
in small bubble and this flow referred to as bubbly flow.
In the medium range of the flow rate of gas and liquid, there
is pulse flow in which liquid is moving in frequent pulses.
The common map is based on dimensionless parameters.
Here, it is presented in a dimension form to explain the trends (see Figure 13.7).
In the literature, Figure 13.7 presented
in dimensionless coordinates.
The abscissa is a function of combination of Froude ,Reynolds, and Weber numbers.
The ordinate is a combination of flow rate ratio and density ratio.
Flow With The Gravity
As opposed to the flow against gravity, this flow can starts with stratified flow.
A good example for this flow regime is a water fall.
The initial part for this flow is more significant.
Since the heavy liquid can be supplied from the ``wrong'' point/side,
the initial part has a larger section compared to the flow against the gravity flow.
After the flow has settled, the flow continues in a stratified configuration.
The transitions between the flow regimes is similar to stratified flow.
However, the points where these transitions occur are different from the horizontal flow.
While this author is not aware of an actual model, it must be possible to
construct a model that connects this configuration with the stratified
flow where the transitions will be dependent on the angle of inclinations.
13.6 Multi–Phase Flow Variables Definitions
Since the gas–liquid system is a specific case of the liquid–liquid system,
both will be united in this discussion.
However, for the convenience of the terms ``gas and liquid'' will be
used to signify the lighter and heavier liquid, respectively.
The liquid–liquid (also gas–liquid) flow is an extremely complex
three–dimensional transient problem since the flow conditions in a pipe
may vary along its length, over its cross section, and with time.
To simplify the descriptions of the problem and yet to retain the important
features of the flow, some variables are defined so that the
flow can be described as a onedimensional flow.
This method is the most common and important
to analyze twophase flow pressure drop and other parameters.
Perhaps, the only serious missing point in this discussion is the change
of the flow along the distance of the tube.
13.6.1 Multi–Phase Averaged Variables Definitions
The total mass flow rate through the tube is the sum of the mass
flow rates of the two phases
\begin{align}
\dot{m} = \dot{m}_G + \dot{m}_L
\label{phase:eq:massFrac}
\end{align}
It is common to define the mass velocity instead of the regular velocity
because the ``regular'' velocity changes along the length of the pipe.
The gas mass velocity is
\begin{align}
G_G = \dfrac{\dot{m}_G}{A}
\label{phase:eq:GmasV}
\end{align}
Where $A$ is the entire area of the tube.
It has to be noted that this mass velocity does not exist in reality.
The liquid mass velocity is
\begin{align}
G_L = \dfrac{\dot{m}_L}{A}
\label{phase:eq:LmasV}
\end{align}
The mass flow of the tube is then
\begin{align}
G = \dfrac{\dot{m}}{A}
\label{phase:eq:TmasV}
\end{align}
It has to be emphasized that this mass velocity is the actual velocity.
The volumetric flow rate is not constant (since the density is not constant)
along the flow rate and it is defined as
\begin{align}
Q_G = \dfrac{G_G}{\rho_G} = U_{sG}
\label{phase:eq:GqRate}
\end{align}
and for the liquid
\begin{align}
Q_L = \dfrac{G_L}{\rho_L}
\label{phase:eq:LqRate}
\end{align}
For liquid with very high bulk modulus (almost constant density),
the volumetric flow rate can be considered as constant.
The total volumetric volume vary along the tube length and is
\begin{align}
Q = Q_L + Q_G
\label{phase:eq:TqRate}
\end{align}
Ratio of the gas flow rate to the total flow rate is called the 'quality' or the ``dryness fraction'' and is given by
\begin{align}
X = \dfrac{\dot{m}_G}{\dot{m}} = \dfrac{G_G}{G}
\label{phase:eq:X}
\end{align}
In a similar fashion, the value of $(1  X)$ is referred to as the ``wetness fraction.''
The last two factions remain constant along the tube length as long the gas and liquid masses remain constant.
The ratio of the gas flow cross sectional area to the total cross sectional
area is referred as the void fraction and defined as
\begin{align}
\alpha = \dfrac{A_G}{A}
\label{phase:eq:alpha}
\end{align}
This fraction is vary along tube length since the gas density is not constant along the tube length.
The liquid fraction or liquid holdup is
\begin{align}
L_H = 1  \alpha = \dfrac{A_L}{A}
\label{phase:eq:1ninusAlpha}
\end{align}
It must be noted that Liquid holdup, $L_H$ is not constant for the same reasons the void fraction is not constant.
The actual velocities depend on the other phase since the actual cross section the phase flows is dependent on the other phase.
Thus, a superficial velocity is commonly defined in which if only one phase is using the entire tube.
The gas superficial velocity is therefore defined as
\begin{align}
U_{sG} = \dfrac{G_G}{\rho_G} =
\dfrac{X\,\dot{m}\,}{\rho_G\,A} = Q_G
\label{phase:eq:Usg}
\end{align}
The liquid superficial velocity is
\begin{align}
U_{sL} = \dfrac{G_L}{\rho_L} =
\dfrac{\left(1X\right)\,\dot{m}\,}{\rho_L\,A} = Q_L
\label{phase:eq:Usl}
\end{align}
Since $U_{sL} = Q_L$ and similarly for the gas then
\begin{align}
U_m = U_{sG} + U_{sL}
\label{phase:eq:QlQg}
\end{align}
Where $U_m$ is the averaged velocity.
It can be noticed that $U_m$ is not constant along the tube.
The average superficial velocity of the gas and liquid are different.
Thus, the ratio of these velocities is referred to as the slip
velocity and is defined as the following
\begin{align}
SLP = \dfrac{U_G}{U_L}
\label{phase:eq:skip}
\end{align}
Slip ratio is usually greater than unity.
Also, it can be noted that the slip velocity is not constant along the tube.
For the same velocity of phases ($SLP = 1$), the mixture density is defined as
\begin{align}
\rho_m = \alpha\, \rho_G + (1\alpha)\, \rho_L
\label{phase:eq:RhoMix}
\end{align}
This density represents the density taken at the ``frozen'' cross section
(assume the volume is the cross section times infinitesimal thickness of $dx$).
The average density of the material flowing in the tube can
be evaluated by looking at the definition of density.
The density of any material is defined as $\rho = m/V$
and thus, for the flowing material it is
\begin{align}
\rho = \dfrac{\dot{m} } {Q}
\label{phase:eq:rhoAvreI}
\end{align}
Where $Q$ is the volumetric flow rate.
Substituting equations \eqref{phase:eq:massFrac} and \eqref{phase:eq:TqRate}
into equation \eqref{phase:eq:rhoAvreI} results in
\begin{align}
\rho_{average} = \dfrac {\overbrace{X\,\dot{m}}^{\dot{m}_G} + \overbrace{(1X)\,\dot{m} }^{\dot{m}_L}}
{Q_G + Q_L} \, = \,
\dfrac{X\,\dot{m} + (1X)\,\dot{m} }
{\underbrace{\dfrac{X\,\ddot{m}}{\rho_G }} _{Q_G} + \underbrace{\dfrac{(1X)\,\dot{m}} {\rho_L}} _{Q_L} }
\label{phase:eq:rhoAvreM}
\end{align}
Equation \eqref{phase:eq:rhoAvreM} can be simplified by canceling the
$\dot{m}$ and noticing the $(1X) +X = 1$ to become
Averaged Density
\begin{align}
\label {phase:eq:rhoAvreF}
\rho_{\text{average}} = \dfrac{1}{\dfrac{X}{\rho_G} + \dfrac{(1X)}{\rho_L} }
\end{align}
The average specific volume of the flow is then
\begin{align}
v_{\text{average}} = \dfrac{1}{\rho_{\text{average}}} =
\dfrac{X}{\rho_G} + \dfrac{(1X)}{\rho_L} =
X\,v_G + (1X)\,v_L
\label{phase:eq:spesificVolume}
\end{align}
The relationship between $X$ and $\alpha$ is
\begin{align}
X = \dfrac{\dot{m}_G}{\dot{m}_G + \dot{m}_L } =
\dfrac{\rho_G\, U_G\, \overbrace{A\, \alpha}^{A_G} }
{\rho_L U_L \underbrace{A (1\alpha) }_{A_L} +
\rho_G\, U_G\, A\, \alpha } =
\dfrac{\rho_G\, U_G\, \alpha}
{\rho_L\, U_L\, (1\alpha) + \rho_G\, U_G\, \alpha }
\label{phase:eq:Xalpha}
\end{align}
If the slip is one $SLP=1$, thus equation \eqref{phase:eq:Xalpha}
becomes
\begin{align}
X = \dfrac{\rho_G\, \, \alpha} {\rho_L\,(1\alpha) + \rho_G\, \alpha }
\label{phase:eq:XalphaUG=UL}
\end{align}
13.7 Homogeneous Models
Before discussing the homogeneous models, it is worthwhile to appreciate the complexity of the flow.
For the construction of fluid basic equations, it was assumed that the flow is continuous.
Now, this assumption has to be broken, and the flow is continuous only in many chunks (small segments).
Furthermore, these segments are not defined but results of the conditions imposed on the flow.
In fact, the different flow regimes are examples of typical configuration
of segments of continuous flow.
Initially, it was assumed that the different flow regimes can be
neglected at least for the pressure loss (not correct for the heat transfer).
The single phase was studied earlier in this book and there is a considerable
amount of information about it.
Thus, the simplest is to used it for approximation.
The average velocity (see also equation \eqref{phase:eq:QlQg}) is
\begin{align}
U_m = \dfrac{Q_L + Q_G} {A} = U_{sL} + U_{sG} = U_m
\label{phase:eq:Uavarge}
\end{align}
It can be noted that the continuity equation is satisfied as
Averaged Mass Rate
\begin{align}
\label{phase:eq:m}
\dot{m} = \rho_m \,U_m \, A
\end{align}
Under what conditions equation \eqref{phase:eq:m} is correct?
Solution
Under construction
The governing momentum equation can be approximated as
\begin{align}
\dot{m}\, \dfrac{dU_m}{dx} =  A\, \dfrac{dP}{dx}  S\, \tau_w 
A\,\rho_m\,g\,\sin\theta
\label{phase:eq:momentum}
\end{align}
or modifying equation \eqref{phase:eq:momentum} as
Averaged Momentum
\begin{align}
\label{phase:eq:Pmomentum}
 \dfrac{dP}{dx} =  \dfrac{S}{A} \, \tau_w 
\dfrac{\dot{m}}{A} \, \dfrac{dU_m}{dx} + \rho_m\,g\,\sin\theta
\end{align}
The energy equation can be approximated as
Averaged Energy
\begin{align}
\label{phase:eq:energy}
\dfrac{dq}{dx}  \dfrac{dw}{dx} = \dot{m}\, \dfrac{d}{dx}
\left( h_m + \dfrac{{U_m}^2}{2} + g\,x\,\sin\theta \right)
\end{align}
13.7.1 Pressure Loss Components
In a tube flowing upward in incline angle $\theta$, the pressure loss
is affected by friction loss, acceleration, and body force(gravitation).
These losses are nonlinear and depend on each other.
For example, the gravitation pressure loss reduce the pressure and thus the density must change
and hence, acceleration must occur.
However, for small distances ($dx$) and some situations, this dependency can be neglected.
In that case, from equation \eqref{phase:eq:Pmomentum}, the total pressure loss can be written as
Pressure Loss
\begin{align}
\label{phase:eq:dP}
\dfrac{dP}{dx} =
\overbrace{\left.\dfrac{dP}{dx} \right_{f}}^{friction} +
\overbrace{\left.\dfrac{dP}{dx} \right_{a}}^{acceleration} +
\overbrace{\left.\dfrac{dP}{dx} \right_{g}}^{gravity}
\end{align}
Every part of the total pressure loss will be discussed in the following section.
13.7.1.1 Friction Pressure Loss
The frictional pressure loss for a conduit can be calculated as
\begin{align}
 \left. \dfrac{dP}{dx}\right_f = \dfrac{S}{A} \tau_w
\label{phase:eq:PLfriction}
\end{align}
Where $S$ is the perimeter of the fluid.
For calculating the frictional pressure loss in the pipe is
\begin{align}
 \left. \dfrac{dP}{dx} \right_f = \dfrac{ 4\,\tau_w}{D}
\label{phase:eq:PLfPipe}
\end{align}
The wall shear stress can be estimated by
\begin{align}
\tau_w = f \dfrac {\rho_m\, {U_m}^2 } {2}
\label{phase:eq:f}
\end{align}
The friction factor is measured for a single phase flow
where the average velocity is directly related to the wall shear stress.
There is not available experimental data for the relationship of the averaged
velocity of the two (or more) phases and wall shear stress.
In fact, this friction factor was not measured for the ``averaged''
viscosity of the two phase flow.
Yet, since there isn't anything better, the experimental data that was
developed and measured for single flow is used.
The friction factor is obtained by using the correlation
\begin{align}
f = C \,\left( \dfrac{\rho_m\,U_m\,D}{\mu_m}\right)^ {n}
\label{phase:eq:fFurmula}
\end{align}
Where $C$ and $n$ are constants which depend on the flow regimes (turbulent or laminar flow).
For laminar flow $C=16$ and $n=1$.
For turbulent flow $C=0.079$ and $n=0.25$.
There are several suggestions for the average viscosity.
\begin{align}
\mu_m = \dfrac{\mu_G \, Q_G}{ Q_G + Q_L} + \dfrac{\mu_L \, Q_L}{ Q_G + Q_L}
\label{phase:eq:dukler}
\end{align}
suggest similar to equation \eqref{phase:eq:rhoAvreF} average viscosity as
\begin{align}
\mu_{average} = \dfrac{1}{\dfrac{X}{\mu_G} + \dfrac{(1X)}{\mu_L} }
\label{phase:eq:muAvreF}
\end{align}
Or simply make the average viscosity depends on the mass fraction as
\begin{align}
\mu_m = X\, \mu_G + \left( 1  X \right) \mu_L
\label{phase:eq:muMass}
\end{align}
Using this formula, the friction loss can be estimated.
13.7.1.2 Acceleration Pressure Loss
The acceleration pressure loss can be estimated by
\begin{align}
\left. \dfrac{dP}{dx} \right_a = \dot{m} \, \dfrac{dU_m} {dx}
\label{phase:eq:accPL}
\end{align}
The acceleration pressure loss (can be positive or negative) results from
change of density and the change of cross section.
Equation \eqref{phase:eq:accPL} can be written as
\begin{align}
\left. \dfrac{dP}{dx} \right_a =
\dot{m}\, \dfrac{d} {dx} \left( \cfrac{\dot{m}} {A\,\rho_m} \right)
\label{phase:eq:accPLa}
\end{align}
Or in an explicit way equation \eqref{phase:eq:accPLa} becomes
\begin{align}
\left. \dfrac{dP}{dx} \right_a =
{\dot{m}}^2 \left[
\overbrace{\dfrac{1}{A} \, \dfrac{d} {dx} \left( \dfrac{1} {\rho_m} \right)}
^{\text{pressure loss due to density change}} +
\overbrace{\dfrac{1}{\rho_m\,A^2} \dfrac{dA} {dx}}
^{\text{pressure loss due to area change}}
\right]
\label{phase:eq:accPLae}
\end{align}
There are several special cases.
The first case where the cross section is constant, $\left. dA \right/ dx = 0$.
In second case is where the mass flow rates of gas and liquid is constant
in which the derivative of $X$ is zero, $\left. dX \right/ dx = 0$.
The third special case is for constant density of one phase only,
$\left. d\rho_L \right/ dx = 0$.
For the last point, the private case is where densities are constant for both phases.
13.7.1.3 Gravity Pressure Loss
Gravity was discussed in Chapter 4 and is
\begin{align}
\left.\dfrac{dP}{dx} \right_{g} = g\,\rho_m\, \sin\theta
\label{phase:eq:Pgravity}
\end{align}
The density change during the flow can be represented as a function of density.
The density in equation \eqref{phase:eq:Pgravity} is the density without the ``movement'' (the ``static'' density).
13.7.1.4 Total Pressure Loss
The total pressure between two points, ($a$ and $b$) can be calculated with integration as
\begin{align}
\Delta P_{ab} = \int_a^b \dfrac{dP}{dx} dx
\label{phase:eq:TdeltaP}
\end{align}
and therefore
\begin{align}
\Delta P_{ab} = \overbrace{{\Delta P_{ab}}_{
ule[4pt]{0pt}{2pt}f}}
^{friction} +
\overbrace{{\Delta P_{ab}}_{
ule[4pt]{0pt}{2pt}a}}^{acceleration} +
\overbrace{{\Delta P_{ab}}_{
ule[4pt]{0pt}{2pt}g}}^{gravity}
\label{phase:eq:TdeltaPpart}
\end{align}
13.7.2 Lockhart Martinelli Model
The second method is by assumption that every phase flow separately
One such popular model by Lockhart and Martinelli.
Lockhart and Martinelli built model based on the assumption that the
separated pressure loss are independent from each other.
Lockhart Martinelli parameters are defined as the ratio of the pressure loss of two phases and pressure of a single phase.
Thus, there are two parameters as shown below.
\begin{align}
\phi_{G} =
\left. {
\sqrt{ { \left. {\left. \dfrac{dP}{dx} \right_{TP}}
\right/ }
\left.\dfrac{dP}{dx}\right_{SG}
}
} \;\right_f
\label{phase:eq:LM_G}
\end{align}
Where the $TP$ denotes the two phases and $SG$ denotes the pressure loss for the single gas phase.
Equivalent definition for the liquid side is
\begin{align}
\phi_{L} =
\left. {
\sqrt{ { \left. {\left. \dfrac{dP}{dx} \right_{TP}}
\right/ }
\left.\dfrac{dP}{dx}\right_{SL}
}
} \;\right_f
\label{phase:eq:LM_L}
\end{align}
Where the $SL$ denotes the pressure loss for the single liquid phase.
The ratio of the pressure loss for a single liquid phase and the pressure loss for a single gas phase is
\begin{align}
\Xi =
\left. {
\sqrt{ { \left. {\left. \dfrac{dP}{dx} \right_{SL}} \right/ }
\left.\dfrac{dP}{dx}\right_{SG}
}
} \;\right_f
\label{phase:eq:xi}
\end{align}
where $\Xi$ is Martinelli parameter.
It is assumed that the pressure loss for both phases are equal.
\begin{align}
\left.\dfrac{dP}{dx}\right_{SG} =
\left.\dfrac{dP}{dx}\right_{SL}
\label{phase:eq:SG=SL}
\end{align}
The pressure loss for the liquid phase is
\begin{align}
\left.\dfrac{dP}{dx}\right_{L} = \dfrac{2\,f_L\, {U_L}^2\, \rho_l}{D_L}
\label{phase:eq:GDP}
\end{align}
For the gas phase, the pressure loss is
\begin{align}
\left.\dfrac{dP}{dx}\right_{G} = \dfrac{2\,f_G\, {U_G}^2\, \rho_l}{D_G}
\label{phase:eq:LDP}
\end{align}
Simplified model is when there is no interaction between the two phases.
To insert the Diagram.
13.8 Solid–Liquid Flow
Solid–liquid system is simpler to analyze than the liquidliquid system.
In solid–liquid, the effect of the surface tension are very minimal and can be ignored.
Thus, in this discussion, it is assumed that the surface tension is
insignificant compared to the gravity forces.
The word ``solid'' is not really mean solid but a combination of many solid particles.
Different combination of solid particle creates different ``liquid.''
Therefor,there will be a discussion about different particle size and different geometry (round, cubic, etc).
The uniformity is categorizing the particle sizes, distribution, and geometry.
For example, analysis of small coal particles in water is different from large coal particles in water.
The density of the solid can be above or below the liquid.
Consider the case where the solid is heavier than the liquid phase.
It is also assumed that the ``liquids'' density does not change significantly and it is far from the choking point.
In that case there are four possibilities for vertical flow:
beginNormalEnumerate change startEnumerate=1

The flow with the gravity and lighter density solid particles.

The flow with the gravity and heavier density solid particles.

The flow against the gravity and lighter density solid particles.

The flow against the gravity and heavier density solid particles.
All these possibilities are different.
However, there are two sets of similar characteristics, possibility, 1 and 4 and the second set is 2 and 3.
The first set is similar because the solid particles are moving faster
than the liquid velocity and vice versa for the second set (slower than the liquid).
The discussion here is about the last case (4) because very little is
known about the other cases.
13.8.1 Solid Particles with Heavier Density $\rho_S>\rho_L$
Solid–liquid flow has several combination flow regimes.
When the liquid velocity is very small, the liquid cannot carry
the solid particles because there is not enough resistance to lift up the solid particles.
The force balance of spherical particle in field viscous fluid (creeping flow) is
\begin{align}
\overbrace{ \dfrac{\pi\, D^3 \,g\,(\rho_S  \rho_L)}{6}}
^{\text{ gravity and buoyancy forces} } =
\overbrace{ \dfrac{{C_D}_{\infty}\,\pi\, D^2 \,\rho_L\,{U_L}^2}{8}}
^{\text{ drag forces} }
\label{phase:eq:forceSP}
\end{align}
Where ${C_D}_{\infty}$ is the drag coefficient and is a function of
Reynolds number, $Re$, and $D$ is the equivalent radius of the particles.
The Reynolds number defined as
\begin{align}
Re = \dfrac{{U_L}\, D\,\rho_L}{\mu_L}
\label{phase:eq:Re}
\end{align}
Inserting equating \eqref{phase:eq:Re} into equation \eqref{phase:eq:forceSP} become
\begin{align}
\overbrace{f(Re)}^{{C_D}_{\infty}({U_L})} {U_L}^2 =
\dfrac{ 4 \, D\,g\,(\rho_S  \rho_L)}{3\,\rho_L}
\label{phase:eq:Uparticle}
\end{align}
Equation \eqref{phase:eq:Uparticle} relates the liquid velocity that
needed to maintain the particle ``floating'' to the liquid and particles properties.
The drag coefficient, ${C_D}_{\infty}$ is complicated function of the Reynolds number.
However, it can be approximated for several regimes.
The first regime is for $Re<1$ where Stokes' Law can be approximated as
\begin{align}
{C_D}_{\infty} = \dfrac{24}{Re}
\label{phase:eq:Stokes}
\end{align}
In transitional region 1$<$Re$<$1000
\begin{align}
{C_D}_{\infty} = \dfrac{24}{Re}
\left( 1+ \dfrac{1}{6}\,Re^{2/3} \right)
\label{phase:eq:transitionalCD}
\end{align}
For larger Reynolds numbers, the Newton's Law region, ${C_D}_{\infty}$, is nearly constant as
\begin{align}
{C_D}_{\infty} = 0.44
\label{phase:eq:last}
\end{align}
In most cases of solidliquid system, the Reynolds number is in
the second range.
For the first region, the velocity is small to lift the particle unless the
density difference is very small (that very small force can lift the
particles).
In very large range (especially for gas) the choking might be approached.
Thus, in many cases the middle region is applicable.
So far the discussion was about single particle.
When there are more than one particle in the cross section, then the actual
velocity that every particle experience depends on the void fraction.
The simplest assumption that the change of the cross section of the fluid
create a parameter that multiply the single particle as
\begin{align}
\left.{C_D}_{\infty}\right_{\alpha} =
{C_D}_{\infty} \, f(\alpha)
\label{phase:eq:CDepsilon}
\end{align}
When the subscript $\alpha$ is indicating the void,
the function $f(\alpha)$ is not a linear function.
In the literature there are many functions for various conditions.
Minimum velocity is the velocity when the particle is ``floating''.
If the velocity is larger, the particle will drift with the liquid.
When the velocity is lower, the particle will sink into the liquid.
When the velocity of liquid is higher than the minimum velocity many particles will be floating.
It has to remember that not all the particle are uniform in size or shape.
Consequently, the minimum velocity is a range of velocity rather than
a sharp transition point.
Fig. 13.8 The terminal velocity that left the solid particles.
As the solid particles are not pushed by a pump but moved by the forces the fluid applies to them.
Thus, the only velocity that can be applied is the fluid velocity.
Yet, the solid particles can be supplied at different rate.
Thus, the discussion will be focus on the fluid velocity.
For small gas/liquid velocity, the particles are what some call fixed fluidized bed.
Increasing the fluid velocity beyond a minimum will move the
particles and it is referred to as mix fluidized bed.
Additional increase of the fluid velocity will move all the particles and
this is referred to as fully fluidized bed.
For the case of liquid, further increase will create a slug flow.
This slug flow is when slug shape (domes) are almost empty of the solid particle.
For the case of gas, additional increase create ``tunnels'' of empty almost from solid particles.
Additional increase in the fluid velocity causes large turbulence and the
ordinary domes are replaced by churn type flow or large bubbles that are almost empty of the solid particles.
Further increase of the fluid flow increases the empty spots to the whole flow.
In that case, the sparse solid particles are dispersed all over.
This regimes is referred to as Pneumatic conveying (see Figure 13.9).
Fig. 13.9 The flow patterns in solidliquid flow.
One of the main difference between the liquid and gas flow
in this category is the speed of sound.
In the gas phase, the speed of sound is reduced dramatically with increase
of the solid particles concentration (further reading Fundamentals of
Compressible Flow'' chapter on Fanno Flow by this author is recommended).
Thus, the velocity of gas is limited when reaching the Mach somewhere
between $1/\sqrt{k}$ and $1$ since the gas will be choked (neglecting the double choking phenomenon).
Hence, the length of conduit is very limited.
The speed of sound of the liquid does not change much.
Hence, this limitation does not (effectively) exist for most cases of solid–liquid flow.
13.8.2 Solid With Lighter Density $\rho_S< \rho$ and With Gravity
This situation is minimal and very few cases exist.
However, it must be pointed out that even in solid–gas, the fluid density can
be higher than the solid (especially with micro gravity).
There was very little investigations and known about the solid–liquid flowing down (with the gravity).
Furthermore, there is very little knowledge about the solid–liquid when
the solid density is smaller than the liquid density.
There is no known flow map for this kind of flow that this author is aware of.
Nevertheless, several conclusions and/or expectations can be drawn.
The issue of minimum terminal velocity is not exist and therefor there is no fixed or mixed fluidized bed.
The flow is fully fluidized for any liquid flow rate.
The flow can have slug flow but more likely will be in fast Fluidization regime.
The forces that act on the spherical particle are the buoyancy force and drag force.
The buoyancy is accelerating the particle and drag force are reducing the speed as
\begin{align}
\dfrac{\pi\,D^3\,g(\rho_S  \rho_L)}{6} =
\dfrac{{C_D}_\infty\,\pi \, D^2 \rho_L \left( U_SU_L\right)^2 }{8}
\label{phase:eq:ligthSolid}
\end{align}
From equation 13.54, it can observed that increase
of the liquid velocity will increase the solid particle velocity at the same amount.
Thus, for large velocity of the fluid it can be observed that $U_L/U_S \rightarrow 1$.
However, for a small fluid velocity the velocity ratio is very large, $U_L/U_S \rightarrow 0$.
The affective body force ``seems'' by the particles can be in some cases larger than the gravity.
The flow regimes will be similar but the transition will be in different points.
The solid–liquid horizontal flow has some similarity to horizontal gas–liquid flow.
Initially the solid particles will be carried by the liquid to the top.
When the liquid velocity increase and became turbulent, some of the particles enter into the liquid core.
Further increase of the liquid velocity appear as somewhat similar to slug flow.
However, this author have not seen any evidence that show the annular flow does not appear in solid–liquid flow.
13.9 Counter–Current Flow
This discussion will be only on liquid–liquid systems (which also includes liquidgas systems).
This kind of flow is probably the most common to be realized by the masses.
For example, opening a can of milk or juice.
Typically if only one hole is opened on the top of the can, the liquid will flow in pulse regime.
Most people know that two holes are needed to empty the can easily and continuously.
Otherwise, the flow will be in a pulse regime.
Fig. 13.10 Counter–flow in vertical tubes map.}
In most cases, the possibility to have counter–current flow is limited to having short length of tubes.
In only certain configurations of the infinite long pipes the counter–current flow can exist.
In that case, the pressure difference and gravity (body forces) dominates the flow.
The inertia components of the flow, for long tubes, cannot compensate for the pressure gradient.
In short tube, the pressure difference in one phase can be positive while
the pressure difference in the other phase can be negative.
The pressure difference in the interface must be finite.
Hence, the counter–current flow can have opposite pressure gradient for short conduit.
But in most cases, the heavy phase (liquid) is pushed by the gravity and
lighter phase (gas) is driven by the pressure difference.
woImg{cont/phase/canOne}
{cont/phase/can}
{Counter Flow In Can}
{0.15}
{phase:fig:can}
{Counter–current flow in a can}
{Counter–current flow in a can (the left figure) has only one hole thus
pulse flow and a flow with two holes (right picture).}
The countercurrent flow occurs, for example, when cavity is filled or emptied with a liquid.
The two phase regimes ``occurs'' mainly in entrance to the cavity.
For example, Figure 13.11 depicts emptying of can filled with liquid.
The air is ``attempting'' to enter the cavity to fill the vacuum created thus forcing pulse flow.
If there are two holes, in some cases, liquid flows through one hole
and the air through the second hole and the flow will be continuous.
It also can be noticed that if there is one hole (orifice) and a long and narrow
tube, the liquid will stay in the cavity (neglecting other phenomena such as dripping flow.).
Fig. 13.12 Picture of Countercurrent flow in liquid–gas and solid–gas configurations.
The container is made of two compartments. The upper compartment is filled with the heavy phase (liquid, water solution, or small wood particles) by rotating the container. Even though the solid–gas ratio is smaller, it can be noticed that the solid–gas is faster than
There are three flow regimes that have been observed.
The first flow pattern is pulse flow regime.
In this flow regime, the phases flow turns into different direction (see Figure 13.12).
The name pulse flow is used to signify that the flow is flowing in pulses that occurs in a certain frequency.
This is opposed to counter–current solid–gas flow when almost no pulse was observed.
Initially, due to the gravity, the heavy liquid is leaving the can.
Then the pressure in the can is reduced compared to the outside and some lighter liquid (gas)entered into the can.
Then, the pressure in the can increase, and some heavy liquid will starts to flow.
This process continue until almost the liquid is evacuated (some liquid stay due the surface tension).
In many situations, the volume flow rate of the two phase is almost equal.
The duration the cycle depends on several factors.
The cycle duration can be replaced by frequency.
The analysis of the frequency is much more complex issue and will not be dealt here.
Annular Flow in Counter–current flow
Fig. 13.13 Flood in vertical pipe.
The other flow regime is annular flow in which the heavier phase
is on the periphery of the conduit (In the literature, there are someone who claims that heavy liquid will be inside).
The analysis is provided, but somehow it contradicts with the experimental evidence.
Probably, one or more of the assumptions that the analysis based is erroneous).
In very small diameters of tubes the counter–current flow is not possible because
of the surface tension (see section ).
The ratio of the diameter to the length with some combinations of the
physical properties (surface tension etc) determines the point where the counter flow can start.
At this point, the pulsing flow will start and larger diameter will
increase the flow and turn the flow into annular flow.
Additional increase of the diameter will change the flow regime into extended open channel flow.
Extended open channel flow retains the characteristic of open channel
that the lighter liquid (almost) does not effect the heavier liquid flow.
Example of such flow in the nature is water falls in which water flows down and air (wind) flows up.
The driving force is the second parameter which effects the flow existence.
When the driving (body) force is very small, no counter–current flow is possible.
Consider the can in zero gravity field, no counter–current flow possible.
However, if the can was on the sun (ignoring the heat transfer issue),
the flow regime in the can moves from pulse to annular flow.
Further increase of the body force will move the flow to be in the extended ``open channel flow.''
In the vertical co–current flow there are two possibilities, flow with gravity or against it.
As opposed to the co–current flow, the counter–current flow has no possibility for these two cases.
The heavy liquid will flow with the body forces (gravity).
Thus it should be considered as non existent flow.
13.9.1 Horizontal Counter–Current Flow
Up to this point, the discussion was focused on the vertical tubes.
In horizontal tubes, there is an additional flow regime which is stratified .
Horizontal flow is different from vertical flow from the stability issues.
A heavier liquid layer can flow above a lighter liquid.
This situation is unstable for large diameter but as in static
(see section page \pageref{static:eckertBarmeir})
it can be considered stable for small diameters.
A flow in a very narrow tube with heavy fluid above the lighter
fluid should be considered as a separate issue.
Fig. 13.14 A flow map to explain the horizontal counter–current flow.
When the flow rate of both fluids is very small, the flow will be stratified counter–current flow.
The flow will change to pulse flow when the heavy liquid flow rate increases.
Further increase of the flow will result in a single phase flow regime.
Thus, closing the window of this kind of flow.
Thus, this increase terminates the two phase flow possibility.
The flow map of the horizontal flow is different from the vertical flow
and is shown in Figure 13.14.
A flow in an angle of inclination is closer to vertical flow unless the angle of inclination is very small.
The stratified counter flow has a lower pressure loss (for the liquid side).
The change to pulse flow increases the pressure loss dramatically.
13.9.2 Flooding and Reversal Flow
The limits of one kind the counter–current flow regimes, that is stratified flow are discussed here.
This problem appears in nuclear engineering (or boiler engineering) where
there is a need to make sure that liquid (water) inserted into the pipe reaching the heating zone.
When there is no water (in liquid phase), the fire could melt or damage the boiler.
In some situations, the fire can be too large or/and the water supply
failed below a critical value the water turn into steam.
The steam will flow in the opposite direction.
To analyze this situation consider a two dimensional conduit with a
liquid inserted in the left side as depicted in Figure 13.13.
The liquid velocity at very low gas velocity is constant but not uniform.
Further increase of the gas velocity will reduce the average liquid velocity.
Additional increase of the gas velocity will bring it to a point where the
liquid will flow in a reverse direction and/or disappear (dried out).
Fig. 13.15 A diagram to explain the flood in a two dimension geometry.
A simplified model for this situation is for a two dimensional configuration
where the liquid is flowing down and the gas is flowing up as shown in Figure 13.15.
It is assumed that both fluids are flowing in a laminar regime and steady state.
Additionally, it is assumed that the entrance effects can be neglected.
The liquid flow rate, $Q_L$, is unknown.
However, the pressure difference in the ($x$ direction) is known and equal to zero.
The boundary conditions for the liquid is that velocity at the wall is zero
and the velocity at the interface is the same for both phases
$U_G = U_L$ or $\left.\tau_i\right_G = \left.\tau_i\right_L$.
As it will be shown later, both conditions cannot coexist.
The model can be improved by considering turbulence, mass transfer,
wavy interface, etc.
This model is presented to exhibits the trends and the special features of countercurrent flow.
Assuming the pressure difference in the flow direction for the gas is constant and uniform.
It is assumed that the last assumption does not contribute or change significantly the results.
The underline rational for this assumption is that gas density does not change
significantly for short pipes (for more information look for the book
``Fundamentals of Compressible Flow'' in Potto book series in the Fanno flow chapter.).
The liquid film thickness is unknown and can be expressed as a function of the above boundary conditions.
Thus, the liquid flow rate is a function of the boundary conditions.
On the liquid side, the gravitational force has to be balanced by the shear forces as
\begin{align}
\dfrac{d\tau_{xy}} {dx} = \rho_L\,g
\label{phase:eq:dTauDx}
\end{align}
The integration of equation \eqref{phase:eq:dTauDx} results in
\begin{align}
\tau_{xy} = \rho_L\,g\,x + C_1
\label{phase:eq:Taufx}
\end{align}
The integration constant, $C_1$, can be found from the boundary condition where $\tau_{xy} (x=h) = \tau_i$.
Hence,
\begin{align}
\tau_i = \rho_L\,g\,h + C_1
\label{phase:eq:tauIini}
\end{align}
The integration constant is then $C_i = \tau_i  \rho_L\,g\,h$ which leads to
\begin{align}
\tau_{xy} = \rho_L\,g\,(xh) + \tau_i
\label{phase:eq:tauL}
\end{align}
Substituting the newtonian fluid relationship into equation \eqref{phase:eq:tauL} to obtained
\begin{align}
\mu_L \dfrac{dU_y}{dx} = \rho_L\,g\,(xh) + \tau_i
\label{phase:eq:Gtau}
\end{align}
or in a simplified form as
\begin{align}
\dfrac{dU_{y} }{dx} =
\dfrac{\rho_L\,g\,(xh) }{\mu_L} + \dfrac{\tau_i}{\mu_L}
\label{phase:eq:GtauS}
\end{align}
Equation \eqref{phase:eq:GtauS} can be integrate to yield
\begin{align}
U_{y} = \dfrac{\rho_L\,g} {\mu_L} \left(\dfrac{x^2}{2} h\,x \right)
+ \dfrac{\tau_i\,x}{\mu_L} + C_2
\label{phase:eq:Uyproife}
\end{align}
The liquid velocity at the wall, [$U(x=0) =0$], is zero
and the integration coefficient can be found to be
\begin{align}
C_2 = 0
\label{phase:eq:zeroWall}
\end{align}
The liquid velocity profile is then
Liquid Velocity
\begin{align}
\label{phase:eq:U1proife}
U_{y} = d\dfrac{\rho_L\,g} {\mu_L} \left(\dfrac{x^2}{2} h\,x \right)
+ \dfrac{\tau_i\,x}{\mu_L}
\end{align}
The velocity at the liquid–gas interface is
\begin{align}
U_{y}(x=h) = \dfrac{\tau_i\,h}{\mu_L}  \dfrac{\rho_L\,g\,h^2} {2\,\mu_L}
\label{phase:eq:Uproife2}
\end{align}
The velocity can vanish (zero) inside the film in another point which can be obtained from
\begin{align}
0 = \dfrac{\rho_L\,g} {\mu_L} \left(\dfrac{x^2}{2} h\,x \right)
+ \dfrac{\tau_i\,x}{\mu_L}
\label{phase:eq:UproifeZero}
\end{align}
The solution for equation \eqref{phase:eq:UproifeZero} is
\begin{align}
\label{phase:eq:UproifeZeroPoint1}
\left. x\,\right_{@ U_L=0} = 2\, h  \dfrac{2\,\tau_i}{\mu_L\,g\,\rho_L}
\end{align}
The maximum $x$ value is limited by the liquid film thickness, $h$.
The minimum shear stress that start to create reversible velocity is obtained when $x=h$ which is
\begin{align}
\label{phase:eq:UproifeZeroPoint}
0 = \dfrac{\rho_L\,g} {\mu_L} \left(\dfrac{h^2}{2} h\,h \right)
+ \dfrac{\tau_i\,h}{\mu_L}
\ \nonumber \hookrightarrow
{\tau_i}_{0} = \dfrac {h\,g\,\rho_L} {2}
\end{align}
If the shear stress is below this critical shear stress ${\tau_i}_{0}$ then no part of the liquid will have a reversed velocity.
The notation of ${\tau_i}_{0}$ denotes the special value at which a starting shear stress value is obtained to have reversed flow.
The point where the liquid flow rate is zero is important and it is referred to as initial flashing point.
The flow rate can be calculated by integrating the velocity across the entire liquid thickness of the film.
\begin{align}
\dfrac{Q}{w} = \int_0^h U_{y}dx =
\int_0^h \left[ \dfrac{\rho_L\,g} {\mu_L} \left(\dfrac{x^2}{2} h\,x \right)
+ \dfrac{\tau_i\,x}{\mu_L} \right] dx
\label{phase:eq:flowRate}
\end{align}
Where $w$ is the thickness of the conduit (see Figure 13.15).
Integration equation \eqref{phase:eq:flowRate} results in
\begin{align}
\dfrac{Q}{w} =
\dfrac{h^2 \left( 3\,\tau_i2\,g\,h\,{\rho_L} \right) }
{6\,\mu_L}
\label{phase:eq:QB}
\end{align}
It is interesting to find the point where the liquid mass flow rate is zero.
This point can be obtained when equation \eqref{phase:eq:QB} is equated to zero.
There are three solutions for equation \eqref{phase:eq:QB}.
The first two solutions are identical in which the film height is $h=0$ and the liquid flow rate is zero.
But, also, the flow rate is zero when $3\,\tau_i=2\,g\,h\,{\rho_L}$.
This request is identical to the demand in which
Shear Strees
\begin{align}
\label{phase:eq:tau_iCriticla}
{\tau_i}_{\text{ critical} } = \dfrac{2\,g\,h\,{\rho_L}}{3}
\end{align}
This critical shear stress, for a given film thickness, reduces the flow rate to zero or effectively ``drying'' the liquid
(which is different then equation \eqref{phase:eq:UproifeZeroPoint}).
For this shear stress, the critical upward interface velocity is
Critical Velocity
\begin{align}
\label{phase:eq:UatInterfaceCritical}
\left.{U}_{critical} \right_{interface} =
\overbrace{\dfrac{1}{6}}^{\left( \dfrac{2}{3}  \dfrac{1} {2} \right)}
\left( \dfrac{\rho_L\,g\,h^2}{\mu_L} \right)
\end{align}
The wall shear stress is the last thing that will be done on the liquid side. The wall shear stress is
\begin{align}
\left.\tau_{L}\right_{@ wall} = \mu_L
\left. \dfrac{dU}{dx} \right_{x=0} = \mu_L \left(
\dfrac{\rho_L\,g}{\mu_L} \left( \cancelto{0}{2\,x}  h \right)
+ \overbrace{\dfrac{2\,g\,h\,{\rho_L}}{3}}^{\tau_i} \dfrac{1}{\mu_L}
\right)_{x=0}
\label{phase:eq:lwallStress}
\end{align}
Simplifying equation
be equal $g\,h\,\rho_L$ to support the weight of the liquid.}
becomes (notice the change of the sign accounting for the direction)
\begin{align}
\left.\tau_{L}\right_{@ wall} = \dfrac{g\,h\,{\rho_L}}{3}
\label{phase:eq:lwallStressF}
\end{align}
Again, the gas is assumed to be in a laminar flow as well.
The shear stress on gas side is balanced by the pressure gradient in the $y$ direction.
The momentum balance on element in the gas side is
\begin{align}
\dfrac{d{\tau_{xy}}_G}{dx} = \dfrac{dP}{dy}
\label{phase:eq:GasTau}
\end{align}
The pressure gradient is a function of the gas compressibility.
For simplicity, it is assumed that pressure gradient is linear.
This assumption means or implies that the gas is incompressible flow.
If the gas was compressible with an ideal gas equation of state then the pressure gradient is logarithmic.
Here, for simplicity reasons, the linear equation is used.
In reality the logarithmic equation should be used ( a discussion can be
found in ``Fundamentals of Compressible Flow'' a Potto project book).
Thus, equation \eqref{phase:eq:GasTau} can be rewritten as
\begin{align}
\dfrac{d{\tau_{xy}}_G}{dx} = \dfrac{\Delta P}{\Delta y} = \dfrac{\Delta P}{L}
\label{phase:eq:GasTauA}
\end{align}
Where $\Delta y = L$ is the entire length of the flow and $\Delta P$ is the pressure difference of the entire length.
Utilizing the Newtonian relationship, the differential equation is
\begin{align}
\dfrac{d^2U_{G}}{dx^2} = \dfrac{\Delta P}{\mu_G\,L}
\label{phase:eq:d2Udx}
\end{align}
Equation \eqref{phase:eq:d2Udx} can be integrated twice to yield
\begin{align}
U_{G} = \dfrac{\Delta P}{\mu_G\,L} \, x^2 + C_1\, x + C_2
\label{phase:eq:Gvelocity}
\end{align}
This velocity profile must satisfy zero velocity at the right wall.
The velocity at the interface is the same as the liquid phase velocity or the shear stress are equal.
Mathematically these boundary conditions are
\begin{align}
U_{G}(x=D) = 0
\label{phase:eq:UGDzero}
\end{align}
and
\begin{align}
\label{phase:eq:UGhUL}
\begin{array}{rcrr}
U_{G}(x=h) &= U_L(x=h) & \;(a) & \qquad or \
\tau_G (x=h) &= \tau_L (x=h) & \;(b) &
\end{array}
\end{align}
Applying B.C. \eqref{phase:eq:UGDzero} into equation
qref{phase:eq:Gvelocity} results in
\begin{align}
\label{phase:eq:C2ini}
U_G = 0 = \dfrac{\Delta P}{\mu_G\,L} \, D^2 + C_1\, D + C_2
\Longrightarrow C_2 =  \dfrac{\Delta P}{\mu_G\,L} \, D^2 + C_1\, D
\end{align}
Which leads to
\begin{align}
U_{G} = \dfrac{\Delta P}{\mu_G\,L} \, \left(x^2 D^2\right) +
C_1\, \left( x  D \right)
\label{phase:eq:GvelocityIMProved}
\end{align}
At the other boundary condition, equation \eqref{phase:eq:UGhUL}(a), becomes
\begin{align}
\dfrac{\rho_L\,g\,h^2}{6\,\mu_L} =
\dfrac{\Delta P}{\mu_G\,L} \, \left(h^2 D^2\right) +
C_1\, \left( h  D \right)
\label{phase:eq:}
\end{align}
The last integration constant, $C_1$ can be evaluated as
\begin{align}
C_1 =
\dfrac{\rho_L\,g\,h^2}{6\,\mu_L \, \left( h  D \right)} 
\dfrac{\Delta P \, \left(h +D\right) }{\mu_G\,L}
\label{phase:eq:C1L}
\end{align}
With the integration constants evaluated, the gas velocity profile is
\begin{align}
U_{G} = \dfrac{\Delta P}{\mu_G\,L} \, \left(x^2 D^2\right) +
\dfrac{\rho_L\,g\,h^2 \left( x  D\right) }{6\,\mu_L \left( h  D \right)} 
\dfrac{\Delta P \, \left(h +D\right) \left( x  D\right) }{\mu_G\,L}
\label{phase:eq:GvelocityIMProvedF}
\end{align}
The velocity in equation \eqref{phase:eq:GvelocityIMProvedF} is equal to
the velocity equation \eqref{phase:eq:Uproife2} when ($x=h$).
However, in that case, it is easy to show that the gas shear stress is not equal to
the liquid shear stress at the interface (when the velocities are assumed to be the equal).
The difference in shear stresses at the interface due to this assumption,
of the equal velocities, cause this assumption to be not physical.
The second choice is to use the equal shear stresses at the interface,
condition \eqref{phase:eq:UGhUL}(b).
This condition requires that
\begin{align}
\mu_G \dfrac{dU_G}{dx} =
\mu_L \dfrac{dU_L}{dx}
\label{phase:eq:tauLtauG}
\end{align}
The expressions for the derivatives are
\begin{align}
\overbrace{\dfrac{2\,h\,\Delta P}{L} + \mu_G \, C_1}^{gas\;side} =
\overbrace{\dfrac{ 2\, g\,h\,\rho_L} {3}}^{liquid\;side}
\label{phase:eq:dTauLG}
\end{align}
As result, the integration constant is
\begin{align}
C_1 = \dfrac{ 2\, g\,h\,\rho_L} {3\,\mu_G} 
\dfrac{2\,h\,\Delta P}{\mu_G\,L}
\label{phase:eq:C1tau}
\end{align}
The gas velocity profile is then
\begin{align}
U_{G} = \dfrac{\Delta P}{\mu_G\,L} \, \left(x^2 D^2\right) +
\left( \dfrac{ 2\, g\,h \,\rho_L} {3 \, \mu_G} 
\dfrac{2\,h\,\Delta P}{\mu_G\,L} \right) \left( x  D \right)
\label{phase:eq:UGtau}
\end{align}
The gas velocity at the interface is then
\begin{align}
\left. U_{G} \right_{@ x=h} =
\dfrac{\Delta P}{\mu_G\,L} \, \left(h^2 D^2\right) +
\left( \dfrac{ 2\,g\,h \,\rho_L} {3\,\mu_G} 
\dfrac{2\,h\,\Delta P}{\mu_G\,L} \right) \left( h  D \right)
\label{phase:eq:UatInterfacetau}
\end{align}
This gas interface velocity is different than the velocity of the liquid side.
The velocity at interface can have a ``slip'' in very low density and for short distances.
The shear stress at the interface must be equal, if no special effects occurs.
Since there no possibility to have both the shear stress and velocity on
both sides of the interface, different thing(s) must happen.
It was assumed that the interface is straight but is impossible.
Then if the interface becomes wavy, the two conditions can co–exist.
The wall shear stress is
\begin{align}
\left.\tau_{G}\right_{@ wall} = \mu_G
\left. \dfrac{dU_G}{dx} \right_{x=D} = \mu_G \left(
\dfrac{\Delta P \, 2\, x}{\mu_G\, L}
+
\left( \dfrac{ 2\,g\,h \,\rho_L} {3\,\mu_G} 
\dfrac{2\,h\,\Delta P}{\mu_G\, L} \right)
\right)_{x=D}
\label{phase:eq:gwallStress}
\end{align}
or in a simplified form as
\begin{align}
\left.\tau_{G}\right_{@ wall} =
\dfrac{2\,\Delta P \, \left( D h \right) }{L}
+ \dfrac{ 2\,g\,h \,\rho_L} {3}
\label{phase:eq:gwallStressF}
\end{align}
The Required Pressure Difference
Fig. 13.16 General forces diagram to calculated the in a two dimension geometry.
The pressure difference to create the flooding (drying)
has to take into account the fact that the surface is wavy.
However, as first estimate the waviness of the surface can be
neglected.
The estimation of the pressure difference under the assumption of equal shear
stress can be applied.
In the same fashion the pressure difference under the assumption the
equal velocity can be calculated.
The actual pressure difference can be between these two assumptions
but not must be between them.
This model and its assumptions are too simplistic and the actual
pressure difference is larger.
However, this explanation is to show magnitudes and trends and hence
it provided here.
To calculate the required pressure that cause the liquid to dry, the total
balance is needed.
The control volume include the gas and liquid volumes.
Figure 13.16 describes the general forces that acts
on the control volume.
There are two forces that act against the gravity and two forces with
the gravity.
The gravity force on the gas can be neglected in most cases.
The gravity force on the liquid is the liquid volume times the liquid
volume as
\begin{align}
F_{gL} = \rho\,g\,\overbrace{h\,L}^{\dfrac{Volme}{w} }
\label{phase:eq:FgL}
\end{align}
The total momentum balance is (see Figure 13.16)
\begin{align}
F_{gL} + \overbrace{L}^{\dfrac{A}{w} }\, {\tau_w}_{G} =
\overbrace{L}^{\dfrac{A}{w} }\,{\tau_w}_ {L}
+ \overbrace{D\,\Delta P}^{\text{force due to pressure}}
\label{phase:eq:Lfriction}
\end{align}
Substituting the different terms into \eqref{phase:eq:Lfriction} result in
\begin{align}
\rho\,g\,L\,h + L\, \left(
\dfrac{2\,\Delta P \, \left( D h \right) }{L}
+ \dfrac{ 2\,g\,h \,\rho_L} {3} \right)
= L \,\dfrac{g\,h\,\rho_L}{3}
+ {D\,\Delta P}
\label{phase:eq:Gfriction}
\end{align}
Simplifying equation \eqref{phase:eq:Gfriction} results in
\begin{align}
\dfrac{4\,\rho\,g\,L\,h}{3} =
\left( 2\,h D \right) \Delta P
\label{phase:eq:DeltaP}
\end{align}
or
\begin{align}
\Delta P =
\dfrac{4\,\rho\,g\,L\,h}{3 \, \left( 2\,h D \right) }
\label{phase:eq:DeltaPF}
\end{align}
This analysis shows far more reaching conclusion that initial anticipation expected.
The interface between the two liquid flowing together is wavy.
Unless the derivations or assumptions are wrong, this analysis equation
qref{phase:eq:DeltaPF} indicates that when $D>2\,h$ is a special case
(extend open channel flow).
Multi–Phase Conclusion
For the first time multi–phase is included in a standard introductory textbook on fluid mechanics.
There are several points that should be noticed in this chapter.
There are many flow regimes in multi–phase flow that ``regular'' fluid
cannot be used to solve it such as flooding.
In that case, the appropriate model for the flow regime should be employed.
The homogeneous models or combined models like Lockhart–Martinelli can be employed in some cases.
In other case where more accurate measurement are needed a specific model is required.
Perhaps as a side conclusion but important, the assumption of straight line is
not appropriate when two liquid with different viscosity are flowing.
