Process Heat Transfer 2022-4.pptx

PROCESS HEAT TRANSFER
(Basics)
1. Nuclear Heat Transport by El-Wakil
(Main)
1. Nuclear Systems 1 & 2 by Kazimi
2. Boiling, Condensation, and Gas-Liquid Flow by Whalley
(Reference)
1. Convective Boiling & Condensation by Collier & Thom
2. 2-Phase Flow, Boiling & Condensation in Conventional & Miniature
Systems by Ghiaasiaan
3. Introduction to Nuclear Power by Hewitt and Collier

HEAT TRANSFER MATERIAL ALREADY COVERED
1. Introduction
1. Relationship Between Thermodynamics and Heat Transfer
2. Modes of Heat Transfer
2. Conduction Heat Transfer
1. Fourier’s Law of Heat Conduction
2. The Conduction Equation (The Differential Form)
3. Special Forms of Conduction Equation
4. BC & IC required for the conduction eq. solution
5. Applications of Conduction equation
6. Case studies with constant and variable thermal conductivity
7. Conduction heat transfer with heat generation and its
application to nuclear reactors

3. Convection Hear Transfer
1. Newton’s Law of cooling
2. Free or Forced Convection Differential form of Convection
equation & its approx. solution
3. Dimensionless form of the convection equation
4. Nusselt , Prandtl, Reynolds and Grashoff Numbers
5. Empirical correlations for external & internal forced convection
6. Empirical correlations for free convection
7. Concept of equivalent diameter
8. Effect of friction on heat transfer
9. Fluid flow power requirement for a given heat load and its
application to nuclear reactors
HEAT TRANSFER MATERIAL ALREADY COVERED

Course Outlines to be covered
1. TWO PHASE FLOW DYNAMICS
1. Introduction to multiphase flow dynamics and its implications on
nuclear reactors
2. Flow regimes & their identification in vertical and horizontal flow
3. Transition from one regime to another.
4. Two phase flow models
3. Concept of static and flow quality, void fraction and slip ratio
4. Boiling & non boiling heights for P prediction in 2 phase flow
5. Empirical & theoretical models available for 2 phase p
6. Critical flow and its importance in nuclear reactor safety
7. Thermal and Flow instabilities in Nuclear reactors

Course Outlines to be covered (Contd.)
2. BOILING HEAT TRANSFER
1. Introduction to boiling heat transfer and its importance in Nuclear
Reactors
2. Static and flow boiling curve, its implications on nuclear reactors
3. Heat transfer regions and development of void fraction/quality
4. Boiling Heat transfer coefficients correlation
5. Critical conditions or Boiling Crisis
6. Critical Heat Flow (CHF) & its importance in Nuclear reactor safety
7. Difference between DNB and Dryout
8. Heat transfer correlations for pool boiling heat transfer
9. Heat transfer correlations for flow boiling heat transfer
10.Design Considerations due to critical conditions

Course Outlines to be covered (Contd.)
3. SINGLE HEATED CHANNEL: STEADY-STATE ANALYSIS
1. Steady-State Single-Phase Flow in a Heated Channel
1. Solution of the Energy Equation for a Single-Phase Coolant
and Fuel Rod (PWR Case)
1. Coolant Temperature, Cladding Temperature
2. Fuel Centerline Temperature
2. Solution of the Momentum Equation to Obtain Single-Phase
Pressure Drop
3. Steady-State Two-Phase Flow in a Heated Channel Under
Fully Equilibrium (Thermal and Mechanical Conditions
4. Solution of the Energy Equation for Two-Phase Flow (BWR
Case with Single-Phase Entry Region)
5. Solution of the Momentum Equation for Fully Equilibrium
Two- Phase Flow Conditions to Obtain Channel Pressure Drop
(BWR Case with Single-Phase Entry Region)
1. Δ pacc, Δ pgrav, Δ pfric, and Δpform

THERMAL DESIGN PRINCIPLES
The thermal performance is dictated by the bounds of the maximum allowable
primary coolant outlet temperature and the minimum achievable condenser
coolant inlet temperature. Because this atmospheric heat sink temperature is
relatively fixed, improved thermodynamic performance requires increased
reactor coolant outlet temperatures.

THERMAL DESIGN PRINCIPLES
(Contd.)

THERMAL DESIGN PRINCIPLES(Contd.)

THERMAL DESIGN LIMITS (Contd.)

THERMAL DESIGN LIMITS(Contd.)
Normal operation refers to reactor operation at full or partial
power. This means that the reactor is operating within specified
operational limits and conditions. During normal operation, water
flows through the inlet pipes (the “cold legs”) to the reactor vessel,
down through the annular space around the core, up through the
core, and out through the vessel outlet pipes (hot legs) to the
steam generator.

Blowdown (0 – 30s) happens when a large break occurs in one of
the cold legs, the contents of the reactor vessel and the primary
loops are blown down through the break. In the reactor vessel, the
water flashes into steam and at the same time the pressure in the
reactor vessel drops to the saturation pressure. Reduction of core
flow and an increase in the local quality are conservatively
assumed to result in departure from nucleate boiling (DNB) or dry-

During refill (30 – 40s), despite the fact that ECCS flow is entering
the intact inlet pipe, the steam being generated in the vessel flows
in the opposite direction to the ECCS liquid flow, forcing the water
to bypass the upper part of the inlet annulus. After further
depressurization, the steam flow up the annulus drops to a value
which is low enough to allow the ingress of ECCS water, and the
lower plenum starts filling up.

Reflood phase (40 – 250s) begins when the lower plenum is filled
and the fuel elements start to rewet from the bottom upward. A
constant liquid head is maintained in the inlet annulus and excess
ECCS water overflows through the break.

The critical heat flux (CHF) phenomenon results from a relatively sudden
reduction of the heat transfer capability of the two-phase coolant. The resulting
thermal design limit is expressed in terms of the departure from nucleate
boiling condition for PWRs and the critical power condition for BWRs. For
fuel rods, where the volumetric energy-generation rate q"'(r, z, t) is the
independent parameter, reduction in surface heat transfer capability for
nominally fixed bulk coolant temperature (Tb) and heat flux causes the clad
temperature to rise
Physically, this reduction occurs because of a change in the liquid-vapor flow
patterns at the heated surface. At low void fractions typical of PWR operating
conditions, the heated surface, which is normally cooled by nucleate boiling,
becomes vapor-blanketed, resulting in a clad surface temperature excursion by
departure from nucleate boiling (DNB). At high void fractions typical of BWR
operating conditions, the heated surface, which is normally cooled by a liquid
film, overheats owing to film dryout (DRYOUT).

The ratio between the predicted correlation heat flux and the
actual operating heat flux is called the departure from nucleate
boiling ratio (DNBR).

THERMAL PERFORMANCE
The design performance of a power reactor can be characterized by two
figures of merit: the power density (Q"') and the specific power.
1. Power density is the measure of the energy generated relative to
the core volume. Because the size of the reactor vessel and hence the
capital cost are nominally related to the core size, the power density is
an indicator of the capital cost of a concept.

THERMAL PERFORMANCE(Contd.)
Power density
of a triangular
array is 15. 5%
greater than
that of a square
array for a
given pitch
For light-water reactors, the simpler square array is more
desirable, as the necessary neutron moderation can be provided
by the looser-packed square array.

THERMAL PERFORMANCE(Contd.)
2. Specific power is the measure of the energy generated per unit mass of fuel
material. It is usually expressed as watts per gram of heavy atoms. This
parameter has direct implications on the fuel cycle cost and core inventory
requirements. For the fuel pellet shown, the specific power, (watts per grams of
heavy atoms), is:

Assignments I and II (Deadline: 10 Feb 2023)
1. Solve the following question.
2. Solve problems 2 – 1 to 2 – 5 of Nuclear Systems I (Todreas and Kazimi).

THERMODYNAMICS OF NUCLEAR ENERGY
CONVERSION SYSTEMS
Separate set of power point slides will be presented and
discussed

Chapter No 4 of Nuclear Systems (vol 1)
by
Kazimi and Todreas
contains overview of transport equations for single phase
flows. Please go through them as your reading
assignment.
Reading Assignment

by
Kazimi and Todreas
Fuel Coolant Interaction
ASSIGNMENT – III (Deadline: 3 March 2023)
We may define either one control mass, consisting of the combined mass of the
fuel and the coolant, or two control masses, one consisting of the mass of the
fuel and the other consisting of the mass of the coolant. We choose the second
representation because it is easier to define the T-s diagrams for a one-
component (two-phase) fluid (i.e. , the coolant or fuel alone) then it is to define
the T-S diagram for the two-component coolant-fuel mixture.
Question: Derive the expression for:
1. The equilibrium temperature, Te for both of the above mentioned cases.
2. Expression for isentropic and adiabatic expansion of coolant-fuel mixture
as two independent systems.
3. Expression for above expansion for coolant-fuel mixture as one system in

by
Kazimi and Todreas
Thermodynamic Analysis of a Simplified PWR
ASSIGNMENT – IV (Deadline: 3 March 2023)
Derive Equations 6-58, 6-63c, 6-64, 6-66, 6-72b, 6-78 and
6-84.

BASIC CONCEPTS & TERMINOLOGIES
-1
Why heat transfer with phase change is important in nuclear reactors ?
Liquids are subjected to high heat fluxes. Resulting in
1. Liquids attaining as high heat transfer rates a benefit
2. Void formed, not good in nuclear reactors, affects moderation / pumping
3. Heat fluxes, limited due to burnout (leading to structural instability)
4. Saturation or superheated temperatures
5. Causing liquids to boil due to phase change
6. Causing dissolved gases to come out of the liquid forming bubbles
7. Causing other phenomena which generate bubbles
8. Bubble formation cause agitation and turbulence, resulting in
Hence, an in-depth knowledge of phase-change behavior of working
fluid can result in better performance of nuclear reactors
1. Two-phase coolant thermal conditions in accidental loss of coolant
2. Provision of sufficient safety margin between anticipated transient heat
fluxes and critical boiling heat fluxes

-2
TERMS USED TO DESCRIBE PHASE CHANGE PROCESSES
VAPORIZATION
Conversion of liquid into vapors
EVAPORATION
Conversion of liquid into vapors below boiling point
BOILING
Formation of vapors within a liquid phase at and above boiling point
TWO-PHASE FLOW
Where two or more phases move simultaneously in a channel
CONDENSATION
Reverse of evaporation

-3
CLASSIFICATION OF BOILING
BASED ON THE LOCATION OF BOILING PHENOMENON
BASED ON THE MECHANISM OF BOILING
BASED ON THE TEMPERATURE CONDITIONS OF FLUID

-4
BASED ON THE LOCATION OF BOILING PHENOMENON
Boiling due to heat added to the
liquid by a surface in contact with
or submerged within the liquid
Boiling due to heat generation
within the liquid by chemical /
nuclear reaction or
electrical/mechanical energy
dissipation
POOL BOILING
NON-FLOW
VOLUME OR BULK BOILING
CAN BE NON-FLOW
OR FLOW BOILING

-5
BASED ON THE MECHANISM OF BOILING
Bubbles are formed around a
small nucleus of vapors or gas
There is a formation of a
continuous film of vapor that
blankets the heating surface
NUCLEATE BOILING FILM BOILING
Partial film boiling, Partial nucleate boiling, Transition film boiling,
Unstable nucleate film boiling etc.
Pool nucleate boiling or
volume nucleate boiling
There is only pool film boiling rather
than volume film boiling
Under certain conditions nucleate and film boiling coexist

-6
BASED ON THE TEMPERATURE OF LIQUID
bulk of the liquid is at
saturation temperature
bulk of the liquid is subcooled
SATURATED BOILING SUBCOOLED BOILING
In saturated boiling, the
bubbles rise to the liquid
surface where they are
detached
In sub-cooled boiling, the bubbles
begin to rise but may collapse before
they reach the liquid surface
.
to generate bubbles the heating surface must be at
temperature > saturation, consequently some of the liquid
immediately adjacent to that surface, is superheated
An important
point

-7
HENCE
BOTH SATURATED AND SUBCOOLED BOILING CAN BE
NUCLEATE OR FILM BOILING
WHEREAS VOLUME OR BULK BOILING CAN ONLY BE
SATURATED BOILING
TWO-PHASE FLOW CLASSIFICATION
DIABATIC
WITH BOILING
ADIABATIC
WITHOUT BOILING
BASED ON HEAT TRANSFER

-8
Single component flow
Water and steam
Two component flow
Water and air
BASED ON COMPONENTS
IN BOTH THESE TYPES OF TWO-PHASE FLOW
1. Vapor and liquid flow at different velocities
2. Vapors normally flow faster than liquid
3. The ratio of their velocities is called slip ratio

GAS LIQUID GENERAL FLOW REGIMES
DISPERSED
FLOW
STRATIFIED
FLOW

2 PHASE FLOW PATTERNS EXPERIMENTAL
SETUPS

Adiabatic Vertical 2 Phase Flow
Patterns
ADIABATIC FLOW
Flow is
• co-current and
steady-state in
• a long tube with
low or moderate
but constant liquid
volumetric flow
rate QL
• The gas volumetric
flow rate QG is
started from a very
low value and is
gradually increased

Adiabatic Horizontal 2 Phase Flow
Patterns

Adiabatic Vertical 2 Phase Flow Patterns. -1
Bubbly flow pattern characteristics
Liquid
Gas
• Distorted-spherical and discrete bubbles
• Move in a continuous liquid phase
• Bubbles have little interaction at very low QL
• Increase in number density as QG is increased
• At higher QG rates, bubbles interact, leading
to their coalescence and breakup

Dispersed Bubbly flow pattern characteristics
Adiabatic Vertical 2 Phase Flow Patterns. - 2
Flow regimes associated with very high QLiquid
• Due to very large liquid and mixture velocities the slip
velocity between the two phases is often small in comparison
with the average velocity of either phase
• The effect of gravity is relatively small as long as the void
fraction is small enough to allow the existence of a
continuous liquid phase
• Highly turbulent liquid flow does not allow existence of large
gas chunks & shatters gas into small bubbles
• The bubbles are quite small and nearly spherical
Dispersed
Bubbly Flow

Adiabatic Vertical 2 Phase Flow Patterns. – 3
Liquid
Gas
• Large QG discrete bubbles coalesce to form very large
bubbles
• The slug / plug flow regime then develops; dominated
by bullet-shaped bubbles (Taylor bubbles)
• Approximately hemispherical caps and are separated
from one another by liquid slugs.
• The liquid slug often contains small bubbles
• A Taylor bubble approximately occupies the entire cross
section and is separated from the wall by a thin liquid
film
• Taylor bubbles coalesce and grow in length until a
relative equilibrium liquid slug length (Ls/D ∼ 16) in
common vertical channels is reached
Slug / Plug flow pattern characteristics

• At further higher QG large Taylor bubbles are disrupted
to form churn (froth) flow
• A chaotic motion of the irregular-shaped gas pockets
takes place, where the interface shape cannot be defined
• Both phases
• May appear to be contiguous
• Have incessant churning and oscillatory backflow
• Liquid near the tube wall continually pulses up and down
• Churn flow also occurs at the entrance of a vertical
channel, before slug flow develops.
Churn flow pattern characteristics

• Annular-dispersed (annular-mist) flow replaces churn
flow at even higher QG
• A thin liquid film, often wavy, sticks to the wall while
gas occupies the core often with entrained droplets
• In common pipe scales, the droplets are typically 10–
100 μm in diameter
• The annular-dispersed flow regime is usually
characterized by continuous impingement of droplets
onto the liquid film and simultaneously an incessant
process of entrainment of liquid droplets from the liquid
film surface
Annular flow pattern characteristics

FLOW REGIME MAPS. -1
• Flow pattern/regimes maps are an attempt, on a 2-D graph, to
separate the space into areas corresponding to the various flow
patterns
• Simple flow pattern maps use the same axes for all flow patterns
and transitions
• Complex maps use different axes for different transitions
• There are flow maps for vertical as well as for horizontal flow
• Several maps are available in the literature
• There are quite a few limitations in the use of these maps
• The flow regimes considered are the major and easily
distinguishable flow patterns.
• Transition from one major flow regime to another is never sudden
• Each pair of major flow regimes are separated from one another by
a relatively wide transition zone

VERTICAL FLOW REGIME MAPS. -3
The Hewitt and Roberts (1969) map for vertical up flow in a tube
 
 
2
2
2
2
1
,
superficial velocity
volumetric flow rate
tube cross-sectional area
G G
G
L L
L
Gx
j
G x
j
where
j





 

 



     
2
2 2
1
( )
G L
G L
G L
G x
G Gx Q Q
Q
j j j
A A A
  

     

FLOW REGIME MAPS. -4 (vertical flow)
Gover and Aziz 1972,
vertical two phase flow map,
2.6 cm ID

HORIZANTLE TWO PHASE FLOW PATTERNS. - 1
Bubbly Flow
Increasing Gas flow Rate
Plug Flow Stratified Flow
Wavy Flow Slug Flow Annular Flow

HORIZONTAL 2-PHASE FLOW PATTERNS. - 1
1. Bubbly flow, in which the gas bubbles tend to flow along the top of
the tube
2. Plug flow, in which the individual small gas bubbles have
coalesced to produce long plugs
3. Stratified flow, in which the liquid—gas interface is smooth. Note
that this flow pattern does not usually occur, the interface is almost
always wavy as in wavy flow
4. Wavy flow, in which the wave amplitude increases as the gas
velocity increases
5. Slug flow, in which the wave amplitude is so large that the wave
touches the top of the tube
6. Annular flow, which is similar to vertical annular flow except that
the liquid film is much thicker at the bottom of the tube than at the
top

FLOW REGIME MAPS. -2
1
2
g l
air water
 

 
 
  
 
1
2 3
water l water
water l
  

  
 
 
 
  
 
 
 
Baker’s map for horizontal 2 phase flow in a tube

FLOW REGIME MAPS. -5 (horizontal flow)
Govier and Aziz 1972 Horizontal two phase flow map, 2.6 cm ID

FLOW REGIME IDENTIFICATION SUMMARY
1. Void distribution depends on the pressure, channel geometry and its
orientation, gas and liquid flow rates.
2. Flow regimes in vertical flows are bubbly, slug, churn & annular
3. Bubbly regime have the presence of dispersed vapor bubbles in a
continuous liquid phase
4. Slug flow is distinguished by the presence of gas plugs separated by
liquid slugs.
5. Churn flow is more chaotic but of same basic character as the slug flow
6. Annular flow is distinguished by the presence of a continuous core of gas
surrounded by an annulus of the liquid phase
7. If the gas flow in the core is sufficiently high, it may carry liquid droplets
referred as an annular-dispersed flow
8. Hewitt and Roberts [19] also suggested that the droplets can gather in
clouds forming an annular-wispy regime
9. The liquid droplets are torn from the wavy liquid film, get entrained in
the gas core, and can be de-entrained to join the film downstream of the
point of their origin.

FLOW REGIME TRANSITION – DIABATIC
FLOW

FLOW REGIME TRANSITION – DIABATIC
FLOW
Vvs is the
superficial
velocity of that
portion of the
gas that flows
above the
interface as
additional gas
flows as bubbles
with the vapor in
the boiling case;
Hfg is latent
heat of
vaporization.

ASSIGNMENT
INDIVIDUAL ASSIGNMENT AS ASSIGNED IN CLASS
SUBMISSION DEADLINE
2 MAY 2022
15 MINUTE PRESENTATION ALONG WITH HARD
COPY SUBMISSION BY EACH STUDENT

TRANSPORT EQUATIONS FOR TWO-PHASE
FLOWS
1. Macroscopic Versus Microscopic Information
2. Multicomponent Versus Multiphase Systems
3. Mixture Versus Multifluid Models
i. The homogeneous equilibrium model (HEM) is the simplest of the
mixture models. It assumes that there is no relative velocity between
the two phases (i .e. homogeneous flow) and that the vapor and
liquid are in thermodynamic equilibrium. In this case the mass,
momentum, and energy balance equations of the mixture are
sufficient to describe the flow.
ii. Mixture models other than HEM add some complexity to the two-
phase flow description. By allowing the vapor and liquid phases to
have different velocities but constraining them to thermal
equilibrium, these methods allow for more accurate velocity
predictions. Alternatively, by allowing one phase to depart from
thermal equilibrium, enthalpy prediction can be improved. In this
case four or even five transport equations may be needed as well as
a number of externally supplied relations to specify the interaction
between the two phases.

FLOWS (Contd.)
In the two-fluid model, three conservation equations are written for both
the vapor and the liquid phases. Hence the model is often called the six-
equation model . This model allows a more general description of the
two-phase flow. However, it also requires a larger number of constitutive
equations. The most important relations are those that represent the
transfer of mass (Γ), transfer of energy (Qs) and transfer of momentum
(Fs) across liquid-vapor interfaces. The advantage of using this model is
that the two phases are not restricted to prescribed temperature or
velocity conditions. Extensions of the two-fluid model to multifluid
models- in which vapor bubbles, a continuous liquid, a continuous vapor,
and liquid droplets are described by separate sets of conservation
equations-are also possible but have not been as widely applied as the
simpler two-fluid model .

FLOWS (Contd.)

PREAMBLE 2 PHASE MODELS
FOR A HOMOGENOUS FLOWING TWO PHASE MIXTURE
Vapor mass flow fraction of total flow is called flow quality (x)
Flow quality in a one-dimensional flow is defined as
v
z
v l
m
x
m m


• Concept of flow quality is used in the analysis of predominantly 1-D 2-
phase flow
• Many correlations that are extensively used in two-phase flow and heat
transfer are given in terms of the flow quality
• Flow quality becomes particularly useful when thermodynamic
equilibrium between the two phases is assumed
• In that case flow quality can be obtained from energy balances of the flow.
However, additional info/correlations are needed to calculate α
• With 2-D flow the quality at a given plane has two components and is not
readily defined by a simple energy balance
• For 2 or 3-D flow, the flow quality becomes less useful.
• Little literature exists, on the effect of the vectorial nature of quality on
flow conditions

Two Phase Flow Modelling
• Rigorous modelling of gas–liquid 2P flow based on the solution of local and
instantaneous conservation principles is generally very complicated
• Simplified models  idealization, time & volume averaging are usually used
• Simplified multiphase flow conservation eqs can be obtained in many ways
a. Each point in mixture is simultaneously occupied by both phases
b. Developing control-volume-based balance equations
c. Performing some form of averaging (time, volume, flow area OR composite)
on local & instantaneous conservation equations
d. Postulating a set of conservation equations based on physical insight.
• 2 degrees of freedom can be either allowed or disallowed by various models
a. Thermal non-equilibrium, which allows one or both phases to have T  Tsat
b. Unequal velocities, where the two phases can have different velocities S >1
Most widely used is the averaging method. Lead to flow parameters that are
1. Measurable with available instrumentation
2. Continuous
3. In case of double averaging have continuous first derivatives

Two Phase Flow Models -1
Various two–phase flow models
1. Homogeneous mixture model:
1. The simplest two-phase flow model, and it essentially treats the two-phase
mixture as a single hypothetical fluid
2. The two phases are assumed to be well mixed and have the same velocity
at any location, S = 1
3. Thus, only one momentum equation is needed
4. If in a single-component flow, thermodynamic equilibrium is also assumed
between the two phases everywhere, the homogeneous–equilibrium
mixture model results
5. 2 phases may not be at thermodynamic equilibrium, e,g. flashing liquids &
condensation of vapour bubbles surrounded by sub-cooled liquid
6. Solution of conservation equations is more complicated than single-phase
flow
7. Fluid mixture is compressible; thermo-physical properties can vary with
time & position
8. Particularly useful for high pressure and high flow rate conditions

1D Steady Homogeneous Equilibrium
Flow

1D Steady Homogeneous Equilibrium
Flow
Prove it.

Correlation for Heat Transfer in Vertical
Pipes

Generalized Correlation for Heat
Transfer
In order to handle the effects of various flow patterns and
inclination angles on the two-phase heat transfer data with only
one correlation, Ghajar and Kim, introduced the flow pattern factor
(FP) and the inclination factor (I). The void fraction (α), which is the
volume fraction of the gas-phase in the tube cross-sectional area,
does not reflect the actual wetted-perimeter (SL) in the tube with
respect to the corresponding flow pattern. For instance, the void
fraction and the non dimensionalized wetted perimeter of annular
flow both approach unity, but in the case of plug flow the void
fraction is near zero and the wetted-perimeter is near unity.
However, the estimation of the actual wetted-perimeter is very
difficult due to the continuous interaction of the two phases in the
tube. Therefore, instead of estimating the actual wetted-perimeter,
modeling the effective wetted-perimeter is a more practical
approach. In their model, Ghajar and his co-workers have ignored
the influence of the surface tension and the contact angle of each
phase on the effective wetted-perimeter.

Transfer
The wetted-perimeter at the equilibrium state, which can be
calculated from the void fraction, is

Transfer
The shape of the gas-liquid interface at the equilibrium state based on
the void fraction (α) is far different from the one for the realistic case.
The two-phase heat transfer correlation weighted by the void fraction
(1−α), is not capable of distinguishing the differences between different
flow patterns. Therefore, in order to capture the realistic shape of the
gas-liquid interface, the flow pattern factor (FP), an effective wetted-
perimeter relation, which is a modified version of the equilibrium wetted-
perimeter, is proposed
The term (FS) appearing in is referred to as shape factor which in essence is a
modified and normalized Froude number. FS is applicable for slip ratio greater
than or equal to one. The shape factor (FS) is defined as

Transfer
In order to account for the effect of inclination, researchers proposed the
inclination factor
The correlation does not account for the surface tension
force !!!

Transfer

Transport Equations for Two Phase Flows

Transport Equations for Two Phase Flows
Homogeneous Equilibrium Model

Mixture Equations for 1D Flows

Mixture Equations for 1D Flows
What would be the final form of these equations if

Two Fluid Type Transport Equations

Two Fluid Type Transport Equations
Assignment
Derive momentum balance for entire volume V by adding up
two control volumes i.e. Vv and Vl.

1D Space Averaged Transport Equations

QUALITY & VOID FRACTION
For A Stationary Homogeneous Two-Phase Mixture
x
 
Mass of vapors in mixture
Mass static quality
Total mass of mixture
mixture
of
volume
Total
mixture
in
vapors
of
Volume
Fraction
Void 

Relationship between ‘x’ and ‘’ can be established by assuming a unit
mass. From thermodynamics the volume of this unit mass is
fg
f xv
v
mass
volume

  







 









 




f
g
g
f
fg
f
g
x
x
v
v
x
x
xv
v
xv



1
1
1
1
1
1
For x = 2% steam water system  = 97.1% at 1 atmosphere
(prove it!)
 
v
v
st
v l v l
V
m
x
m m V

 
 
 
<xst > is a volume-average property by definition; therefore <xst>  xst.
v
v
v l
V
V V
 
 

Volume fraction of the gaseous phase is normally called the void fraction

Fundamental Void Fraction-Quality-Slip
Relation
SLIP RATIO is;
f
g
g
f
f
g
v
v
A
A
x
x
V
V
S



1 







1
g
f
g
f
g
A
A
A
A
A
SLIP RATIO, QUALITY &
VOID FRACTION can be
related as;
1
(1)
1
g g
f f
V v
x
S
V x v



 

1 1
(2)
1
1 1
1 f
g
x
v
x
S x
x v


 

   

    
  
    
  
1 1
1 1 (3)
x

 

 
  
 
If S = 1, there can be two cases
1. A non flow system
2. A homogeneously flowing
system
g
f
1
1 x
1
x




 

 
 
1
2 3
water l water
water l
  

  
 
 
 
  
 
 
 

Relation

Relation
 
 
1
1
1
v
l
x
S
x


 



1 1
2
st
st
x x
S
x x
 

 
 
 
 
1 1
3
S
 
 
 
 
 
 
1 1
4
v
l
x
x
 
 
 

 
 
1 1
5
st v
st l
x
x
 
 
 


QUALITY & VOID FRACTION
Important Inferences:
1. For constant x,  decreases with pressure. x =  at critical pressure
2. For a constant pressure d/dx decreases with quality
3. For low ‘x’ as in BWRs, d/dx increases with decreasing pressure,
becomes exceptionally large affecting reactor stability
Assignment
• Show by plotting a graph between x and  that the void fraction
decreases with a higher slip ratio for fixed vapor-liquid density
ratio.
• Show by plotting a graph between x and  the effect of increasing
vapor-liquid density ratio for a system assumed to be homogeneous
and at equillibrium.

Mixture Density
BASIC TWO-PHASE PROPERTIES
g f
m m
V


 1
v l
    
  
Similarly, the average phasic
density becomes
 
1
,
1
st st
v l
x x
 
 
 

 

For A Homogeneous Flowing Mixture at Thermodynamic Equilibrium
Flow (mixing-cup) enthalpy  
1
m v l
h xh x h

  
Flow thermodynamic (or equilibrium)
quality is given by relating the flow
enthalpy to the saturation liquid and
vapor enthalpies
m f
e
g f
h h
x
h h




Flow quality can be used to
define the equilibrium quality
The two-phase mixture
density in a volume is:
 
1
g l f
e
g f
xh x h h
x
h h
  


Saturation
conditions replace
local for liquid
and vapor phase

Drift Flux Model
1. A thermal equilibrium mixture model with an algebraic relation between
the velocities (or a slip ratio) of the two phases
2. Drift flux model (DFM, Zuber–Findlay model) is one of the most widely
used diffusion model & more often used for void fraction calculations
3. In these models the liquid and gas phases constitute the two domains
4. This model is different from HEM model only in allowing the two phases
to have different velocities that are related via a predetermined relation
5. Hence, only a single momentum equation is required
6. This is made possible by obtaining the relative (slip) velocity between the
two phases, or the relative velocity of one phase with respect to the
mixture, from a model or correlation
7. Slip-velocity relation is usually algebraic (rather than by a differential
eq.).
8. This model is useful for low pressure and/or low flow rate flows under
steady state or near- steady-state conditions

Drift Flux Model
Considers the average velocity of the vapor in the channel.
To define the drift flux, first the two-phase volumetric velocity (j) and
the local drift velocity of the vapor (υvj) is defined.
υ𝐯 = 𝒋 + υ𝐯𝒋
Where; 𝒋𝐯 = 𝜶υ𝐯 = 𝜶𝒋 + 𝜶 υ𝐯 − 𝒋 , averaging over area
𝒋𝐯 = 𝜶𝒋 + 𝜶 υ𝐯 − 𝒋 >0, the drift flux
Physically represents the rate at which
vapor passes through a unit area (normal to
the channel axis) that is already traveling
with the flow at a velocity j
Superficial velocity.
Velocities each fluid
phase would have if all
the cross-sectional area
of the pipe were
available for the fluid
to flow alone
Local vapour velocity
= volumetric flow rate of vapor/ flow cross sectional area of
vapor

Drift Flux Model
Can there be situations where Vvj ~ 0 ?
Can there be situations where C0 = 1 ?

Slip ratio correlations
Several correlations for calculation the slip, S, and the void fraction
are presented in literature.
A few of the most recommended in order of decreasing accuracy.
 
 
   
1 2
2
0 22
0 19
1
0 08
0 51
2
2
1
1 1
1 1
1 578
0 0273
.
.
.
.
,
.
.
surface tension
L
G
l
l v
H L
L G
H H L
L L G
x
y
S E yE if y
yE x
where
x
y with
x x
GD
E
G D GD
E


 

  

 

  




   
 
 
  
 
 
  
 
   
 
  
  
  
    

CISE correlation,
Premoli et al., 1970

 
0.1
0.67
1 1
1 1
g f
v v
g
f
v
x v 
 
   
 
  
   
   
 
   
Von Glahn (1962) empirical
correlation based on data of
lot of workers
 
1
1
1 1
1 1
1
1
G
G G L
L L
x
K
x
x x
K K
x
x x
K
x

  

 

 

 

 
 
 
     
 
   
   
 

 
    
 
 
 
 
 
Smith’s void correlation
Where K = 0.4 in order to achieve good agreement with experimental data
1 1 L
G
S x


 
  
 
 
Chisholm’s slip correlation
3
L
G
S



Zivi’s slip correlation
1
S 
Homogeneous flow model

Several void–quality correlations, including Zivi’s, can be represented in the
following generic form (Butterworth, 1975)
1
1
1
p r
q
G L
L G
x
A
x
 

 

 
 
  
 
 
     
 
 
 
   
 
 
Latest and a very comprehensive review of the quality-void fraction-slip ratio
correlation (around 70) is given by Woldesemayat and Ghajar (2007)
Correlation A p q r
Homogeneous flow model 1 1 1 0
Zivi (1964) 1 1 0.67 0
Turner and Wallis (1965) 1 0.72 0.40 0.08
Lockhart and Matinelli (1949) 0.28 0.64 0.36 0.07
Thom (1964) 1 1 0.89 0.18
Baroczy (1963 1 0.74 0.65 0.13
Constants in various slip ratio correlations (Butterworth, 1975)

HEM Pressure Drop Correlations
For incompressible liquid and gas phase assumption

HEM Pressure Drop Correlations
At same
mass flux
Gm
Same Re
dependence

Pressure Drop in 2 Phase Flow - 1
Total ac fr st
dP dP dP dP
dz dz dz dz
       
  
       
       
 
2
1
sin
w f
Total
d AG v p
dp G
g
dz A dz t A

 
 

 
    

 
 
Acceleration pressure drop due to
change in momentum as phase
change occurs
2
,
8
w TP
G
where f



Wetted perimeter

The Friction Pressure Drop in 2 Phase
Flow - 1
• 2 phase flow P > single phase pressure drop for same L & G
• Difference depends on the flow patterns discussed earlier due to increased
speeds of various phases
• Difference in the two pressure drops can be found by
         
,2 exp, ,2 ,1 ,
( )
fr P Total P ac others fr P ONLY
P P P P P A
        
• Need a concept to find the (P)fr,2P from the (P)fr,1P,ONLY
• Concept is the development of the two-phase multipliers  2
• By definition
•  2 > 1 always
• There is one hidden problem in equation A
• What fluid you will use to determine the (P)fr,1P,ONLY ‘liquid’ or ‘liquid
only’ or ‘gas’ or ‘gas only’
• In fact, researchers have reported  2 for all cases
• Hence,  2 is reported as  L
2 OR  Lo
2 OR  G
2 OR  Go
2
• Generally, the liquid only or gas/vapor only is adapted
   
2
,2 ,1 ,
fr P fr P ONLY
P P
   

The Friction Pressure Drop in 2 Phase Flow - 2
• Concept of 2 needs information about the thermodynamic condition
of the fluid (for Pfr,1P,ONLY) as well as its flow properties
• Thermodynamic state of the single-phase liquid or vapor is always
taken as the saturated state
• Flow properties are calculated in terms of the superficial velocity of
the liquid only or vapor/gas only conditions for the same mass flow
rate through the same cross-sectional area
• Concept is very good but there are some further issues that need to be
addressed
• Hence in general the 2 depends on
• Phasic flow rates which in turn depends on the flow regimes.
Remember, the liquid vapor interaction in various flow
regimes. Each flow regime interaction has its own physics
• Thermodynamic conditions of the multi-phase flow
• Phasic thermodynamic properties and how they will be
calculated

The Friction Pressure Drop in 2 Phase Flow - 3
• Two phase multipliers cannot be reported as a function of Re
number or other parameters as in the case of single phase flow
• Experimental data has been reported for various thermodynamic and
geometric conditions as a function of various parameters such as
quality, void fraction, pressure, mass flux etc.
• This data can only be used by others only if dynamic similarity
exists between the experimental setups used and the equipment that
need to be designed by others
• Only way one can feel that the similarities exist is the matching of
the flow regimes, which in turn are not yet fully understood
WHAT IS THE WAY OUT
• Researchers have developed theoretical models which are regime
dependent for 2 and then by comparing the experimental data with
these models researchers have suggested various correction factors
to better predict the 2

2-P Frictional Pressure Drop in Homogeneous
Flow-1
Assumptions;
• The two phases are assumed to remain well mixed
• The two phases move with identical velocities everywhere S = 1
• A homogeneous mixture thus acts essentially as a singe-phase fluid that is
compressible and has variable properties
• Two-phase pressure drop is developed by analogy with single-phase flow
For a turbulent single-phase flow
2 2
1
1 1
,1 1
1 1
2 2
P
P P
fr P H H P
V
dp G
f f
dz D D


 
  
 
 
Assuming that the friction factor may be expressed in terms of the
Reynolds number by the Blasius equation
 
0.25
0.25
1
1
0.079 Re 0.079 H
P
P
GD
f


  
   
 

Flow-2
By analogy for a turbulent two-phase flow, one can write
2 2
2
2 2
,2 2
1 1
2 2
P
P P
fr P H H P
V
dp G
f f
dz D D


 
  
 
 
 
1
4
1
4
2 2
2
0.079 Re 0.079 H
P P
P
GD
f


  
   
 
 
 
1
2 2
1
1
P g f P
g f
x
x
x x
   
 

 

 
     
 
 
 
 
A mean two-phase viscosity 2P of the homogenized fluid is needed,
which must satisfy the following limiting conditions
2 2
0, ; 1,
P f P g
x x
   
   

Flow-3
Various forms of such relationships, quoted in literature
 
1
2
(1942)
1
P
g f
McAdamset al
x
x

 

 

 
 
 
 
 
2
(1960)
1
P g f
Cicchittiet al
x x
  
    
2 2
(1964)
1
P P g g f f
Dukler et al
xv x v
   
 
  
 
Substitute the relation of 2P density and f2P into the 2P Darcy equation
2
,2
LO
fr P Lo
dp dp
dz dz

   
  
   
   
1
4
1 1 1 L G
L
G G
x x
 

 

   
   

  
   
   
   
   
2
,2
GO
fr P Go
dp dp
dz dz

   
  
   
   
   
1
4
1 1
G G
L L
x x x x
 
 

   
   
   
   
2
,2
G
fr P G
dp dp
dz dz

   
  
   
   
   
1
4
7
4
1 1 1
G G
L L
x x x x
 
 


   
   
   
   
2
,2
L
fr P L
dp dp
dz dz

   
  
   
   
 
1
4
7
4
1 1 1 1 L G
L
G G
x x x
 

 


   
   

   
   
   
   
   

Flow-3
Values of the two-phase frictional multiplier for liquid only for the
homogeneous model steam-water system
1
4
2
1 1 1 L G
L
LO
G G
x x
 


 

   
   

   
   
   
   
   

The two-phase friction factor
Application of the homogeneous theory to experimental observations
Friction factor for use in the HEM can be calculated, by either
From single-phase flow correlations
OR, estimate directly from measured two-phase pressure drops
values of fTP in the range 0.0029-0.0033 have been suggested for
low-pressure flashing steam-water flow
(Benjamin and Miller 1942; Bottomley 1936-37; Allen 1951)
Values of about 0.005 for analysis of
circulation in high-pressure boilers
(Lewis & Robertson 1940; Markson et al. 1942)
and for petroleum pipe stills
(Dittus & Hildebrand 1942)

The two-phase friction factor
1. The experimental two-phase friction factor fTP was plotted against the
Reynolds number for all-liquid flow.
2. Large discrepancies from the single-phase friction factor were
observed at Reynolds numbers less than 2 x 105
3. Considerably better agreement with the normal single-phase flow
relationship if the experimental friction factor is plotted against the all-
liquid Reynolds number multiplied by the ratio of the inlet to outlet
mean specific volumes.
This is equivalent to defining a new average viscosity
1
1 Re 1
fg fg
f
f f f
v v
GD GD
x x
v v
 
 

     
 
     
     
 
     
 
     

• HEM performs reasonably/well-mixed configuration (dispersed bubbly).
• Deviates from experimental data for annular, slug, and stratified flows
• Empirical correlations remain the most widely applied method
• Most empirical correlations use the concept of ∅2that are applicable to all
flow regimes (flow regime transition effects are implicitly included.
• Original concept was proposed by Lockhart and Martinelli (1949) based
on a simple separated-flow model
• Separated flow model in general, ∅2 = 𝑓 𝐺, 𝑥, 𝑓𝑙𝑢𝑖𝑑 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠
• A number of empirical methods are available
• For separated flow model the total pressure drop is written as
2-P Frictional Pressure Drop in Separated Flow-1
r4
r2
r3

• In SFM a relationship is developed between (2) and the void
fraction () in terms of the independent flow variables.
• First achieved by Martinelli and co-workers.
• Martinelli model was successively developed in the period 1944-49
from a series of studies of isothermal two-phase two-component
flow in horizontal tubes.
• Initially limited to the frictional component resulting in a paper by
Lockhart and Martinelli (1949), proposing a generalized method for
calculating the frictional pressure gradient for isothermal two-
component flow.
• Later extended their work for the estimation of the accelerative
component and resulted in the well-known Martinelli-Nelson ( 1948)
method for the prediction of pressure drop during forced circulation
boiling and condensation.
2-P Frictional Pressure Drop in Separated Flow-1

• A definite portion of the flow area is assigned to each phase, assuming
conventional friction pressure-drop equations can be applied to the flow
path of each phase.
• Interaction between the phases is thus ignored (which led to an
inconsistent result for the relationship between dp/dz and 
• Four flow regimes were defined on the basis of the behaviour of the flow
(viscous or turbulent) passing alone through the channel.
• Liquid and gas phase pressure drops were considered equal irrespective of
the details of the particular flow pattern.
• As initially only frictional component of the pressure drop was considered
hence the acceleration and static head components being assumed
negligibly small
• Therefore, the frictional pressure drop in the gas phase must equal the
frictional drop in the liquid phase irrespective of the flow pattern details
Lockhart-Martinelli ( 1 949) correlation

The Lockhart-Martinelli Method -1
Martinelli and his co-workers argued that the two-phase friction multipliers
ϕ2
L and ϕ2
G can be correlated uniquely as a function of a parameter X, where
,
2
,
fr L
fr G
dp
dz
X
dp
dz
 
 
 

 
 
 
This was verified using their experimental data. The
resulting graphical correlation is shown where friction factor
multipliers are plotted against X for all four flow regimes
Remember  not 2 is
plotted against X not X2

 
2 2
2 2 1
f f
f
f x G v
D

 

 
 
 
2 2
2 2 g g
g
f x G v
D

 
 
 
,
2
,
f fr
g fr
dp
dz
X
dp
dz
 
 
 

 
 
 
2 2
, ,
g f
fr g fr f fr
dP dP dP
dz dz dz
 
     
 
     
     
2
,
2
2
,
f fr g
f
g fr
dp
dz
X
dp
dz


 
 
 
 
 
 
 
By definition we know that pressure
gradient can be written as

 
2 2
2
2 2
2 1
2
f f
g g
f x G v
D
X
f x G v
D
 

 
 
 

 
 
 
n
f f
f
f
u D
f K



 
  
 
 
n
g g
g
f
u D
f K



 
  
 
 
Simplification
leads to
For non circular
geometry D = De
Using Blasius
correlation where
n= 0.25
2
2 1
n n
f g
g f
x
X
x
 
 

   

 
    
 
   
 
   
0.25 1.75
2 1
f g
g f
x
X
x
 
 
   

 
    
 
   
 
   

1. Turbulent (L)-Turbulent (G) ϕtt C = 20
2. Viscous (L)-Viscous (G) ϕvv C = 5
3. Turbulent (L)-Viscous (G) ϕtv C = 10
4. Viscous (L)-Turbulent (G) ϕvt C = 12
Curves are well represented by these when C has the following values in
the following equations
How to use the Lockhart-Martinelli correlation to calculate the two-phase
friction pressure gradient
• Calculate the friction pressure gradients for each phase alone flowing in
the channel at its actual flow rate and then use the Figure OR use the
above correlations in terms of X to get the multipliers.
• Correlation was developed for horizontal two-phase flow of two-
component systems at low pressures (close to atmospheric) and its
application to situations outside this range of conditions is not
recommended
2
2
2 2
1
1
1
L
G
C
X X
CX X


  
  

It follows that if the parameter ϕf is a function of the parameter X then
the void fraction α must also be a function of X. The correlation
between α and X can be derived
2 2
, ,
fo f
fr fo fr f fr
dP dP dP
dz dz dz
 
     
 
     
     
 2
2 2
2 2
,
2 2 1
fo f f f
fo f
fr
f fr
f G v f G x v
dP
dz D D
 
 
  
 
   
 
     
     
 
0.25
1
0.079
f
f
G x D
f


 

  
 
 
0.25
0.079
fo
f
GD
f


 
  
 
 
0.25
1
1
f
fo
f
f x
 
  

 
 
2
2 2
1
f
fo f
fo
f
x
f
 
 
 1.75
2 2
1
fo f x
 
 
Lockhart-Martinelli also derived a relation between ∅𝑓
2
𝑎𝑛𝑑 ∅𝑓𝑜
2

The Martinelli-Nelson ( 1 948) correlation -1
The Lockhart-Martinelli correlation was related
1. To the adiabatic flow of low pressure air-liquid mixtures,
2. However, the information was purposely presented in a
generalized manner to enable the application of the model to
single component systems and to steam-water mixtures.
1. For the prediction of pressure drops during forced circulation boiling
Martinelli and Nelson (1948) assumed the flow regime would always
be ‘turbulent-turbulent’ i.e. C = 20
Features of Martinelli-Nelson correlation
2. Correlation of frictional pressure gradient is worked in terms of the
parameter which is more convenient for boiling and condensation
problems than
2
fo

2
f


3. Thermodynamic equilibrium was assumed to exist at all points in
the flow and the curve correlating was arbitrarily applied to
atmospheric pressure steam-water flow.
2
,
f tt

4. A relationship between ϕf and Xtt was established for the critical
pressure level by noting that as the pressure is increased towards
the critical point, the densities and viscosities of the phases
become similar. The relationship may be represented by
with the value of C=1.36
2
2
1
1
f
C
X X
   
5. Knowing the CURVES for critical and atmospheric pressure,
curves at intermediate pressures were established by trial and error
using the experimental data of Davidson et al. (1943) as a guide.
6. With this knowledge of f and Xtt for a number of pressures a
plot of 2
fo and mass quality x was made using   
2
2 2
1
fo f f fo
f f x
 
 

HEM
correlation

In order to calculate the total two phase pressure drop let us revisit the
separated flow model equation
2
2
0
2 1
x
fo f
fo acc grav
f G v L
p dx p p
D x

 
     
 
 

Need to solve this
term, an average
value of friction
factor multiplier
Martinelli & Nelson
evaluated this integral

Martinelli and Nelson used
the Lockhart - Martinelli
curve for steam-water flow
at Patm showed that, at the.
critical pressure α = β = x.
Knowing the α-Xtt curves
for both atmospheric and
critical pressures, curves at
intermediate pressures were
interpolated. These curves
were then transposed to give
values of α as a function of
mass quality x with pressure
as parameter

The Thom correlation -1
An alternative set of consistent values for the terms
Why there was a need Values were interpolated between Patm and Pcr
Thom ( 1964)
revised values
were derived
using an
extensive set of
experimental
data for steam-
water pressure
drops obtained
at Cambridge,
England, on
heated and
unheated
horizontal and
vertical tubes.
2
3
0
1
, and
x
fo
r dx x
x
 
 
  
 


Application of the SFM to experimental
observations -1
1. Lockhart-Martinelli-Nelson model used extensively for the correlation of
experimental pressure gradients and void fraction for both single- and
two-component gas-liquid flow.
2. Generally, separated flow model is capable of more accurate predictions
than the homogeneous model
3. Two general observations concerning the application of the Lockhart -
Martinelli correlation.
1. Widely recognized that the curves of experimental ϕf or ϕg versus X are
not smooth as shown. Discontinuities of slope are present, associated
with changes of flow pattern. (Sze-Foo Chen and Ibele 1962; Gazley
and Bergelin 1949; Charvonia 1961 ; Kegel 1948; Dukler 1949).
2. G effects the curves of ϕf versus X. Widely reported for steam-water
flow at high pressures. Original Martinelli-Nelson correlation
corresponds to a G= 500 - 1000 kg/m2s. Homogeneous model yields
values close to those obtained experimentally for mass velocities
G>2000 kg/m2s. (Isbin 1959; Sher and Green 1959; Muscettola 1963)

Application of the SFM to experimental
observations -2
Quantitative data illustrating the
effect.
• Zuber et al. (1967), Freon at
elevated pressures
• Hughmark and Pressburg
(1961) for low-pressure air-
liquid flow

Correlations for use independent of the Model
Attempts to correct existing models for the influence of mass velocity on the
frictional multiplier, have been published by Baroczy (1965), by
Chisholm (1968), and by Friedel (1979) and many others.
2
,
fo tt

THE BAROCZY CORRELATION
This method employs the use of two separate set of curves
Second: A plot of correction factor Ω expressed as a function of the same
physical property index for G = 339, 678, 2712, and 4068 kg/m2 s with ‘x’ as
parameter. This plot serves to correct the value of obtained from the first
plot to the appropriate value of G
2
,
fo tt

The method proposed by Baroczy was tested against data from a wide range
of systems including both liquid metals and refrigerants with satisfactory
agreement between the measured and calculated values.
First: A plot of as a function of a physical property index with ‘x’ as a
parameter for a reference G = 1356 kg/m2 s (1×106 1b/hr-ft2)
2
,
fo tt


THE BAROCZY CORRELATION plot-1

THE BAROCZY CORRELATION DATA-1

THE BAROCZY CORRELATION plot-2
2
2
( 1356)
,
2 fo f
fo G
fo fr
f G v
p dP
L dz D
 
  
  
 
 

CHISHOLM'S METHOD -1
2
2
1
1
f
C
X X
   
Chisholm and Sutherland proposed the following procedure to account for
effects of G for steam-water flow in pressure tubes at pressures above 3
MPa (435 psia /30 bar)
For Gm ≤ G*; (G* is a reference mass flux)
 
0.5 0.5 0.5
2
fg g f
g f g
v v v
C C
v v v
 
   
     
   
   
     
     
   
     
   
where
 
2
0.5 2 2 power coeff of Re in the ' 'relation
n
n f
 
 
   
 
*
2
G
C
G


2
2
1
1
f
C
X X
 
 
  
 
 
For Gm > G*; (G* is a reference mass flux)
0.5 0.5
g f
f g
v v
C
v v
   
 
   
   
   
where
2 2
1 1
1 1
C C
T T T T

 
 
    
 
 
   
2 2 1 2
1
n n
n
f f
g g
v
x
T
x v



   
 
    
     

     
CHISHOLM'S METHOD -2
C is obtained as in the case of
G ≤ G*
For rough tubes;
G*=1500kg/m3s, λ = 0.75
and n = 0.2
For smooth tubes;
G*=2000kg/m3s, λ = 1.0
and n = 0

COMPARISON OF METHOD -4
For G > G*, ψ <1
2
2
( )
fo
fo HEM





THE JONES CORRELATION -1
Jones(1961), an empirical approximation of Martinelli-Nelson ∅𝑓𝑜
2
for water
 
   
2 0.824
2
6 6
6
6 6
6
, 1.2 1 1.0
: , , /
1.36 0.005 0.1 0.000714
10 10
0.7
10
10 10
1.26 0.004 0.119 0.00028
10
l
fo m
v
m
m m
m
m m
m
p G x
where p G psia lb hr ft
G G
p p
G
for
p p
G G
G
for



 
 
 
 
   
 
 
 
 
 
 
 

   
   
   
   
 

 
 
   
   
   
   


0.7


 


THE ARMAND-TRESCHEV CORRELATION -1
Armand-Treschev (1959)
Model does not account for the G effect but describe the effect of the  in a
more precise way.
Database of the model is steam-water flowing in rough pipes 25.5 to 56 mm
in diameter at pressures between 1.0 and 18 MPa
   
 
 
 
1.75
2
1.2
1
0.9 0.5,
1
fo
x
for and
  


  

 
 
 
0.833 0.05ln 10p


 
 
 
 
 
1.75
2
1.75
0.025 0.055
0.9, 1
1
fo
p
for x
 


  

   
 
 
 
1.75
2
2
0.48 1
0.9 0.5,
1
1.9 1.48 10 ( )
fo n
x
for and
where n p in MPa
  



  

  

COMPARISON OF CORRELATIONS
2
1: 1 fg
fo
f
v
HEM x
v

 
   
 
 
0.25
2
2: 1 1 1
fg g
fo
f f
v
HEM x x
v



   
   
   
   
   
   
   
   
   

THE FRIEDEL CORRELATION -1
It was obtained by optimizing an equation for to using a large data
base of two-phase pressure drop measurements
2
fo

Friedel (1979), One of the most accurate two-phase Δp correlations
2 2 3
1 0.045 0.035
3.24
fo
A A
A
Fr We
  
where
 
2 2
1 1 f go
g fo
f
A x x
f


 
    
 
 
 
0.224
0.78
2 1
A x x
 
0.91 0.19 0.7
3 1
f g g
g f f
A
  
  
     
 
     
     
     
2 2
2
G G D
Fr We
gD 
 
 
0.91 0.19 0.7
0.24
2 0.78 0.045 0.035
1 3.24 1 1
f g g
fo
g f f
A x x Fr We
  

  
 
     
   
     
     
     
Substituting A2 and A3 to get the following relation

THE FRIEDEL CORRELATION -2
where fgo and ffo are the friction factors defined for the total mass velocity
G as all vapour and all liquid, respectively by the equation
 The correlation is valid for vertical upwards flow and for horizontal flow
2
/ /
/
2 fo go f g
fo go
f G v
dp
dz D
 
 
 
 
 D is the equivalent diameter
 σ is the surface tension

 the homogeneous density given by eq.
 Standard deviations are about 40-50 per cent, which is large with
respect to single-phase flows but quite good for two-phase flows.
 
1
f g f
v x v v


 
  
 
 For vertical, downward flow, Friedel’s correlation gives
 
0.9 0.73 0.74
0.29
2 0.8 0.03 0.12
1 48.6 1 1
f g g
fo
g f f
A x x Fr We
  

  
 
     
   
     
     
     
 
2
Re
0.25 0.86859ln
1.964ln Re 3.8215
fo
fo
fo
f

 
 
 
 
  

 
 
 
 

COMPARISON
HEM and Friedel friction pressure gradient correlations compared to
Gaspari et al. CISE data [24] for water at p = 7.14 MPa flowing in a 5.08
mm diameter tube. (HEM evaluated using Equation 11.83 with n = 0.2 and
ffo = 0.184 Re-0.2.)

THE WHALLEY CRITERIA
Whalley (1980) has evaluated separated flow models against a large
proprietary data bank and gives the following recommendation:
f g
 
For most fluids and operating conditions, ( ) is less than 1000 and
the Friedel correlation will be the preferred method.
f g
 
(c) For ( ) > 1000 and G < 100 kg/m2 s: Utilize the correlations of
Lockhart and Martinelli (1949) and Martinelli and Nelson (1948)
f g
 
(b) For ( ) > 1000 and G > 100 kg/m2 s: Utilize the most recent
refinement of the Chisholm (1973) correlation;
f g
 
(a) For ( ) < 1000: Utilize the Friedel (1979) correlation;

THE ACCELERATION Δp FOR TWO PHASE
FLOW
Net force acting due to acceleration pressure drop is cross-sectional area times
acceleration pressure drop = rate of change of linear momentum (ONLY 1-D)
a c f fe g ge t i
F p A m V m V mV
    
 
 
1
1
e t fe e t ge t i
a e fe e ge i
c c c
x mV x mV mV
p G x V x V V
A A A

 
       
 
   
 
 
 
1 1 1
1 1
f f e t f e t f e f
fe
fe fe e c e
m v x m v x m v x Gv
V
A A A
 
  
   
 
e g
ge i i
e
x Gv
V and V Gv

 
Finally
 
2 2
2 2
1
1
e e
a f g i
e e
x x
p G v v v r G
 
 

    
 

 
 
Acceleration multiplier

THE MINOR LOSSES FOR TWO PHASE FLOW
Similar to frictional pressure drop, often a two–phase multiplier is used
2P P in a sudden expansion using the definitions of hem
2–phase pressure drop in a sudden contraction
Homogeneous flow model has been found to do well in predicting
experimental data (Guglielmini, 1986)
2 2 1
c C
C A A and A A

 
Schmidt & Friedel (1997) experimental data for 2P P across contractions for
air-water mixtures, Freon 12, aqueous solution of glycerol, & an aqueous
solution of calcium nitrate. For D ≈ 17.2 to 44.2mm range resulting  ≈ 0.057
to 0.445. No vena–contracta was observed in these experiments.
0 0 0 0
L L G G L L G G
P P P P P
   
        
0, 0,
ex L ex L ex
P P 
    
 
2 2
0,
1
1
L L
L ex
G
x x
 

   
 

  
 
 
 
 
0,con 0,con
con L L
P P 
    
2
2
2
0,con
1 1
1 1
2
L
L c
G
P
C


 
 
 
     
 
 
 
 

THE MINOR LOSSES FOR TWO PHASE FLOW
Empirical correlations for two phase pressure drop and it is assumed
that the pressure drop associated with single-phase flow is known
For flow through orifices,
Beattie (1973) proposed
For flow through spacer
grids in rod bundles,
Beattie (1973) proposed
Chisholm (1967, 1981)
for two-phase pressure
drop in a bend
Martinelli’s factor
defined for the bend
where KL0 is the bend’s single–phase loss coefficient for the conditions when
all the mixture is pure liquid, R = bend curvature radius and D = pipe diam

Po
4
3
1
2
CRITICAL FLOW -1
Velocity
Length
5
0
0
0
1
2
V*
Pback
Pexit
Po
1
2
Pe
*
0 0
1
2
3
4
5
Pb
Pressure
Length

CRITICAL FLOW
There are two kinds of waves propagating inside a two-phase fluid:
pressure wave and void wave; the latter is also sometimes called a
density wave. The pressure wave is related to the compressibility of fluid
and is the same as a sound wave. The void wave exists even if the fluid
does not have compressibility and it moves with the fluid when the
velocities of the two phases are the same. The void wave stands for
propagation of void fraction fluctuation, and, therefore, it is a major
source of instability when evaluating flow stability of two-phase flow. If
unstable flow occurs in the core or steam generator in a nuclear power
plant, it causes reduction of critical heat flux or wall temperature
fluctuation of heat exchangers. Because of this, careful consideration in

1. This phenomenon occurs in both single- and two- phase flow
2. Phenomenon has long been observed in boiler and turbine systems,
flow of refrigerants and rocket propellants, and many others
3. In nuclear reactors, the phenomenon is of utmost importance in
safety considerations of both boiling and pressurized systems
4. A break in a primary coolant pipe causes two-phase critical flow in
either system since even in a pressurized reactor, the reduction of
pressure of the hot coolant from about 10 MPa to near atmospheric
causes flashing and two-phase flow
5. This kind of break results in a rapid loss of coolant and is considered
to be the maximum credible accident in power reactors built to date
6. An evaluation of the rate of flow in critical 2P systems is important
for the design of emergency cooling and for the determination of the
extent and causes of damage in accidents
CRITICAL FLOW
IMPLICATIONS OF 2 PHASE CRITICAL FLOW

Steady, 1-D Isentropic Flow With Area Change
Governing equation for Steady 1-D Flow
With Area Change;
Assumptions:
1. No body forces, gdz = 0
2. No friction, Ff = 0
3. No heat Transfer, adiabatic flow, Q = 0
4. No drag force, D = 0
5. No work done, W = 0
Continuity Eq.
constant
m AV

 
Momentum Eq.
0
dp VdV

 
Energy Eq.
2
constant
2
V
h H
  
Entropy Equation constant
s 

Two Phase Critical Flow -1
2
fric, 2
m
TP
HEM e m
G
f
dp
dz D 
 
   
 
Δptotal (HEM) = Δpfric + Δpacc + Δpgravity
2 2
acc,
1 1 1
m
m m
HEM f g f
d
dp d
G G x
dz dz dz

  
 
 
    
 
 
 
 
 
 
 
1
2
1
P m
g f
x
x
 
 


 

  
 
 
 
static,
cos
m
HEM
dp
g
dz
 
 

Set denominator
equal to zero.

2
2
, 2
cos
2
1
m
TP
m g m
e m
Total HEM g
m
G
f dx
G v g
D dz
dp
dv
dz
G x
dp
 

 
 
 
 
 
 

 
 
For a 2P critical flow condition, the requirement is that dp/ dz  ∞
 
2 1
m cr
g
dp
G
x dv
 
 
 
 
 
1
2
2
,SFM
2
2 2
2 2
2
2 2
1
2 2 1
2 1
1
cos
1
g
m
Total
g f
lo m
lo m
e f
m f g m
v
dp x
G
dz p
xv x v
f G dx
G
D dz
x x d
G v v g
dz


  

 




 
   
 

 
 
   

  
 
   
 
  
 
 
 
 
 
 

 
 
 
 
 

 
 
 
 
2
2
m cr
g
dp
G
x dv

 
Δptotal (SFM), (Reference KAZMI, P=491)

• Different results for the above 2 models  different slip ratio
• In flashing fluids, a certain length of flow in a valve or pipe is needed
before thermal equilibrium is achieved
• Before that particular length there should be non equilibrium
conditions in terms of both the velocity and the temperature
differences between the two fluids
• Hence length of the flowing section plays a critical role in
determining the flow rate at the exit
• In the absence of sub-cooling and non-condensable gases the length to
achieve equilibrium appears to be on the order of 0.1 m
Source D(mm) L/D L (mm)
Fauske (water) 6.35  6  100
Sozzi & Sutherland (water) 12.7  10  127
Flinta (water) 35  3  100
Uchida & Nariai (water) 4  25  100
Fletcher (freon II) 3. 2  33  105
Van Den Akker et al . (freon 12) 4  22 90
Marviken data (water) 500 > 0.33 < 166
Relaxation length
observed in various
critical flow
experiments with
flashing liquids

In long channels,
1. Residence time is sufficiently long and thermodynamic equilibrium
between the phases is attained
2. Liquid partially flashes into vapor as the pressure drops along the
channel, and the specific volume of the mixture v attains a
maximum value at the exit.
3. As v is a function of x & , it must be a function of the slip ratio S
In the absence of sub-cooling and non-condensable gases the length to
achieve equilibrium appears to be of the order of 0.1 m
• If flow lengths < 0.1m,
• Discharge rate increases strongly with decreasing length as the
degree of non-equilibrium increases
• More of the fluid remains in a liquid state

Equilibrium Models For Two Phase Critical
Flow
Total enthalpy of 2P mixture under
thermal equilibrium conditions undergoing
isentropic expansion can be written as:
   
2 2
1 1
2 2
g f
o g f
V V
h xh x h x x
     
Total entropy of 2P mixture is:  
1 o f
o g f
g f
s s
s xs x s x
s s

    

Combining the above equations to get Gcr in terms of the enthalpy of the 2P
mixture undergoing isentropic expansion under thermal equilibrium conditions
   
2 1
cr o g f
G h xh x h
  
   
 
 
1
1 2
2
1 1
g f
x S
x x
where x
S

 

 
 
 
 
 
   
 
 
 
 
 
 
 
 
 
, , ,
g
cr o o cr
f
V
S and G G h p p S
V
 
Thermodynamic properties are found at experimentally determined Pcrit
     
 
1 1
m v v m l l
xG V and x G V
   
   
Since we have:

CRITICAL FLOW IN LONG CHANNELS -1
• If the critical pressure is known, ρg , ρf , hg , s and hf are known, x can be
determined provided the slip ratio (S) is known
• There are three models available to use the value of slip ratio S
Moody model is based on
maximizing the specific kinetic
energy of the mixture with
respect to the slip ratio
HEM model
1
S   
1 3
Moody model
f g
S  
  
1 2
Fauske model
f g
S  

 
2 2
1
0
2 2
g f
xV x V
S
 


 
 
  

• If the critical pressure is known, ρg , ρf , hg , s and hf are known, x can be
determined provided the slip ratio (S) is known
• There are three models available to use the value of slip ratio S
HEM model
1
S   
1 3
Moody model
f g
S  
  
1 2
Fauske model
f g
S  

Fauske model is based on maximizing
the flow momentum of the mixture
with respect to the slip ratio
 
1 0
g f
xV x V
S

 
  
 


1. Having L/D, ratios between
0 (an orifice) and 40
2. Believed to be independent
of diameter alone
3. Critical pressure ratio was
found to be approx, 0.55 for
long channels in which the
L/D ratio exceeds 12, region
III
4. This is the region in which
the Fauske slip-equilibrium
model is applicable
Data obtained on 0.25 in. ID channels with sharp-edged entrances
The critical pressure ratio varies with L/D, for shorter
channels, but independent of the initial pressure in all cases

Solutions for the set of equations defining the Fauske slip-equilibrium model
Critical flow is described
by local conditions at the
channel exit. Flow is
seen to increase with
increasing pressure &
with decreasing quality
at the exit.
The Fauske model assumes thermodynamic equilibrium a case which due
to the duration of flow, applies to long flow channels.
Experimental data by many investigators showed the
applicability of the Fauske model to L/D ratios above 12

A comparison of the predicted flow rate from various models
• HEM model prediction is good for pipe lengths greater than 300 mm
and at pressures higher than 2.0 Mpa
• Moody' s model over predicts the data by a factor of 2
• Fauske's model falls in between
• When the length of the tube is such that L/D > 40
• HEM model appears to do better than the other models
• Generally, the predictability of critical two-phase flow remains
uncertain
• Results of one model appear superior for one set of experiments
but not others

CRITICAL FLOW IN SHORT CHANNELS - 1
Liquid flashing into vapor occurs when the liquid moves into a region at
pressure lower than psat, if thermal equilibrium is maintained,
Flashing can be delayed due to
1. lack of nuclei about which vapor bubbles may form
2. surface tension which retards their formation
3. heat-transfer problems
Such a situation is called a case of metastability
Metastability occurs in rapid expansions, particularly in short flow channels,
nozzles, and orifices.
The case of short channels has not been completely investigated analytically
For 0 < L/D <12 the critical pressure ratios depend on L/D, unlike long channels
The experimental data covers both long and short tubes, 0 < L/D < 40
Thermal non-equilibrium cases

For orifices (L/D = 0) the experimental
data showed that because residence
time is short flashing occurred outside
the orifice and no critical pressure
existed.
The flow is determined from the incompressible flow orifice equation
For Region 1, 0 < L/D < 3,
the liquid immediately
speeds up and becomes a
metastable liquid core jet
where evaporation occurs
from its surface
 
0.61 2 o b
G p p

 
 
0.61 2 o cr
G p p

 
The flow is determined from the incompressible flow orifice
equation but pb changes to pc which can be obtained from

For 3 < L/D < 12, the metastable liquid
core breaks up, resulting in high-
pressure fluctuations. The flow is less
than would be predicted.
experimental critical flows for region II.

It will have some effect on rounded-entrance channels. The existence of
gases or vapor bubbles will affect the flow also, since they will act as
nucleation centers.
All the above data were obtained on sharp entrance channels
In rounded entrance channels, the metastable liquid remains more in
contact with the walls and flow restriction requires less vapor
For 0 < L/D < 3 channels, such as nozzles, the rounded entrances result in
higher critical pressure ratios than indicated earlier as well as greater flows.
The effect of rounded entrances is negligible for long channels (L/D > 12)
and the slip equilibrium model can be used
The effect of L/D ratio on flow diminishes between 3 and 12.
The condition of the wall surface does not affect critical flow in sharp
entrance channels, since the liquid core is not in contact with the walls and
evaporation occurs at the core surface or by core break up

In the absence of significant frictional losses, Fauske proposed
1
fg
cr
fg f
h
G
v NTc
 N, a non-equilibrium
parameter
2
2 2
10
2
fg
f fg f
h
N L
p K v Tc

 

where Δp = po - pb
K = discharge coefficient (0.61 for sharp edge)
L = length of tube, ranging from 0 to 0.1 m
For large values of L (L  0.1 m) , N =1.0 and above eq. reduces to
1
fg
cr
fg f
h
G
v Tc

When the properties are evaluated at po the value of
Gcr predicted by this eq. is called the equilibrium rate
model (ERM)

A comparison of ERM model with experimental data and other models

• The effect of sub-cooling on the discharge rate is simply obtained by
accounting for the increased single-phase pressure drop [po - p(To)]
resulting from the sub-cooling
• For flow geometries where equilibrium rate conditions prevail for
unsaturated inlet conditions (L  0.1 m) , the critical flow rate is
Good agreement between this prediction and various data including the
large-scale Marviken data
If sub-cooling is zero [p(To) = po] , the critical flow rate is approximated
by ERM model
  2
2
cr o o l ERM
G p p T G

 
 
  Eq. A

Comparison of Marviken test 4 (D =
509 mm and L/D = 3.1) and calculated
values based on Eq. A
Comparison of typical Marviken data
(D ranging from 200 to 509 mm and L
ranging from 290 to 1809 mm) and
calculated values using Eq. A

Two-Phase Pressure Drop Characteristics
• Total P drop is a sum of friction + gravity + acceleration pressure drops
• Total channel P-flow rate behavior, however does not show a simple
increasing behavior when considering the entire flow rate range
• For a wide range of pressure and heating rates, the p vs m characteristic
curve has a shape that can lead to multiple solutions and instabilities
• Two-phase flow systems are prone to a number of instability and
oscillation phenomena
• For a fixed set of boundary conditions, there are often multiple solutions
for the steady-state operation of a boiling/two-phase flow system, some of
which are unstable
• Small perturbations can cause a system that has multiple solutions for the
given boundary conditions to move from one set of operating conditions to
an entirely different set or to oscillate back and forth among two or more
unstable operating conditions
• Two-phase flow instability is of great concern for BWRs, steam generators
and boilers, heat exchangers, and cryogenic equipment, among others, and
has been extensively studied

Two-Phase Pressure Drop Characteristics

BUBBLE STATICS AND DYNAMICS -1
BUBBLE FORMATION REQUIRES LIQUID SUPERHEAT
(to what degree later)
Bubble Formation Aided By So Called Nucleation Aids
1. DISSOLVED GASES OR VAPORS PRESENT IN LIQUID
 Prominent in nuclear reactors
 Charged particles presence in bubbles aid bubble motion
 ‘h’ in BWR > ‘h’ in conventional boilers
1. CAVITIES OR CREVICES ON THE SOLID SURFACE
 Never completely filled due to surface tension
 Retain gases
 Centers of high temperature
the presence of ionization radiations

1. WETTING CHARACTERISTICS OF SOLID SURFACE
Depends on the interaction of solid, liquid and gas at interface
For a bubble resting on a solid surface there are three interfaces
1. Solid-liquid interface, function of liquid and solid surface properties
2. Liquid-vapor interface, function of liquid and vapor phase properties
3. Solid-vapor interface, function of vapor and solid surface properties
Making a force balance along the surface reveals
cos
gs fs fg
   
 
Term Wetting Always Refer To The Liquid Phase

If  = 90, borderline wetted and unwetted surface
If  > 90, unwetted surface
If  < 90, wetted surface

SURFACES DESIRABLE FOR BUBBLE FORMATION ???
First study Bubble Growth Depends on 
If  > 90, facilitates
bubble grows / larger bubbles
 > 90,
lesser superheat required
 > 90, larger bubbles,
film is readily formed,
Film, lesser heat transfer,
not desired
If  < 90,
More superheat required
 < 90,
Smaller Bubbles
 < 90
Bubble more easily detached
No Film, higher heat
transfer, desired

Degree of superheat required
for heterogeneous nucleation
Homogeneous & Heterogeneous
Nucleation
Degree of superheat required
for homogeneous nucleation

To form a bubble some degree of superheat is required
(ODD isn’t it)
Consider a stable floating bubble of radius ‘r’ in a saturated liquid
 
2
4
b l fg
D
p p D

 
 
4 fg
b l
p p
D

 
Bubble is at thermal equilibrium
Tb = Tl
But pb > pl to overcome
the surface tension force
Since pb corresponds to at
least Tsat,
Hence, Tl > Tsat
Liquid is superheated
A force balance reveals

What degree of superheat is required
*
4 2
fg fg
b l
p p
D r
 
  
pg = vapor pressure of
liquid inside bubble
pg = pb & Tg = Tb vapor pressure and
vapor temperature can be related
using Clausius Clapeyron eq.
 
fg
sat sat g f
h
dp
dT T v v
 

 

 
Assume vapors exist
at Tsat (pg/b ) inside
the bubble & vg >> vf
 
fg
b
b b g
h
dp
dT T v

2
fg
b
b
b b
h
dp
dT
p RT

Integrate between limits
of pb to pl & Tb to Tsat
using Ideal Gas
relation to get
g g g
p v RT


1 1
ln fg
b
l b sat
h
p
p R T T
   
  
   
   
*
4 2
fg fg
b l
p p
D r
 
  
*
2
ln 1 fg
b sat
b sat
fg l
RT T
T T
h p r

 
  
 
 
ln
b sat b
b sat
fg l
RT T p
T T
h p
 
   
 
*
2 fg sat
b sat
fg b
T
T T
h r


 
If rc  the order of molecular dimension  Tg – Tsat quite large,
Combine to get
 
*
2 fg sat
fg g b sat
T
r
h T T




b
*
2
if 1 and RT
fg
b l b b fg
l
p p p v v
p r

 

 For water, predicted Tg – Tsat  430oF,
 Measured Tg – Tsat  16oF
 Thanks to dissolved/trapped gases,
which reduce required vapor pressure for
bubble mechanical equilibrium, i.e.
4 2
fg fg
g vap l
c
p p p
D r
 
   
Bubble Detachment
Process

• Micro-cavities  10-3 mm at the surfaces act as gas storage volumes
• Such surfaces exist at solid surfaces or on suspended bodies
• Vapor exist in contact with sub-cooled liquid, if angular opening of
the crack is small (micro-cavity) surface tension effect
• Solid surfaces contains a large number of micro-cavities with a
distribution in sizes
• Boiling at the surface can begin if the Tcoolant near the surface is high
enough that the pre-existing vapor at the cavity site may attain
sufficient pressure to initiate the growth of a vapor bubble at that site.

The various stages in the pool boiling
curve

Jets and Columns
Nucleate boiling at high heat flux

BOILING REGIMES
BOILING HEAT TRANSFER - History
1. Nukiyama (1934)
Performed an experiment using an electrically heated platinum wire
immersed in water – BOILING CURVE
2. Gaertner (1965)
Vapour structures in nucleate boiling
HOW DO YOU THINK BOILING PHENOMENA GOES AS
TEMPEATURE OF THE HEATING SURFACE CHANGES

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