This document describes a proposed algebraic model for simulating two-phase slug flow with heat transfer. The model uses a one-dimensional, stationary approach to calculate key hydrodynamic and heat transfer parameters for a unit cell of slug flow, which consists of a liquid slug and elongated gas bubble. Mass, momentum and energy balances are applied to each component of the unit cell to develop an implicit algebraic equation system. An iterative process is used to solve for the unit cell, which is then propagated along the pipe while accounting for pressure and temperature gradients. The temperature profile output can be used to calculate a two-phase heat transfer coefficient, which is compared to existing correlations in the literature. Results show good agreement with reported data.
Transfer velocities for a suite of trace gases of emerging biogeochemical im...Martin Johnson
Authors M. T. Johnson, P. S. Liss, T.G. Bell and C.Hughes and J. Woeltjen
Paper given at the 6th International Symposium on Gas Transfer at Water Surfaces, Kyoto, Japan, May 2010.
A Black-Oil Model for Primary and Secondary Oil-Recovery in Stratified Petrol...Anastasia Dollari
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processes ranging from the natural pressure-driven fluid expansion, solution gas-drive to more elaborate pressure maintenance strategies relying on water-flooding and water-alternating-gas in a typical stratified and anisotropic petroleum reservoir. Oral presentation in COMSOL Conference, Lausanne 2018.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Presentation of 2 datasets of H2 and its isotopic composition:
- Time series from six background stations
- Dataset collected in the UTLS with a commercial aircraft
Transfer velocities for a suite of trace gases of emerging biogeochemical im...Martin Johnson
Authors M. T. Johnson, P. S. Liss, T.G. Bell and C.Hughes and J. Woeltjen
Paper given at the 6th International Symposium on Gas Transfer at Water Surfaces, Kyoto, Japan, May 2010.
A Black-Oil Model for Primary and Secondary Oil-Recovery in Stratified Petrol...Anastasia Dollari
In this contribution, we present the application of the black-oil model in common oil recovery
processes ranging from the natural pressure-driven fluid expansion, solution gas-drive to more elaborate pressure maintenance strategies relying on water-flooding and water-alternating-gas in a typical stratified and anisotropic petroleum reservoir. Oral presentation in COMSOL Conference, Lausanne 2018.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Presentation of 2 datasets of H2 and its isotopic composition:
- Time series from six background stations
- Dataset collected in the UTLS with a commercial aircraft
Phase equilibria: phase, components and degrees of freedom. The phase rule and its
thermodynamic derivation. The phase diagrams of water and sulphur systems, partially
miscible liquid pairs: the phenol and water and nicotine-water systems. Completely
miscible liquid pairs and their separation by fractional distillation. Freeze drying
(lyophilization).
Tomographic inverse estimation of aquifer properties based on pressure varia...Velimir (monty) Vesselinov
Vesselinov, V.V., Harp, D., Koch, R., Birdsell, K., Katzman, K., Tomographic inverse estimation of aquifer properties based on pressure variations caused by transient water-supply pumping, <em>AGU Meeting</em>, San Francisco, CA, December 15-19, 2008.
Phase equilibria: phase, components and degrees of freedom. The phase rule and its
thermodynamic derivation. The phase diagrams of water and sulphur systems, partially
miscible liquid pairs: the phenol and water and nicotine-water systems. Completely
miscible liquid pairs and their separation by fractional distillation. Freeze drying
(lyophilization).
Tomographic inverse estimation of aquifer properties based on pressure varia...Velimir (monty) Vesselinov
Vesselinov, V.V., Harp, D., Koch, R., Birdsell, K., Katzman, K., Tomographic inverse estimation of aquifer properties based on pressure variations caused by transient water-supply pumping, <em>AGU Meeting</em>, San Francisco, CA, December 15-19, 2008.
Heat Transfer Analysis of Refrigerant Flow in an Evaporator TubeIJMER
the paper aim is to presenting the heat transfer analysis of refrigerant flow in an evaporator
tube is done. The main objective of this paper is to find the length of the evaporator tube for a pre-defined
refrigerant inlet state such that the refrigerant at the tube outlet is superheated. The problem involves
refrigerant flowing inside a straight, horizontal copper tube over which water is in cross flow. Inlet
condition of the both fluids and evaporator tube detail except its length are specified. here pressure and
enthalpy at discrete points along the tube are calculated by using two-phase frictional pressure drop model.
Predicted values were compared using another different pressure drop model. A computer-code using
Turbo C has been developed for performing the entire calculation
Natural Convection and Entropy Generation in Γ-Shaped Enclosure Using Lattice...A Behzadmehr
This work presents a numerical analysis of entropy generation in Γ-Shaped enclosure that was submitted to the natural convection process using a simple thermal lattice Boltzmann method (TLBM) with the Boussinesq approximation. A 2D thermal lattice Boltzmann method with 9 velocities, D2Q9, is used to solve the thermal flow problem. The simulations are performed at a constant Prandtl number (Pr = 0.71) and Rayleigh numbers ranging from 103 to 106 at the macroscopic scale (Kn = 10-4). In every case, an appropriate value of the characteristic velocity is chosen using a simple model based on the kinetic theory. By considering the obtained dimensionless velocity and temperature values, the distributions of entropy generation due to heat transfer and fluid friction are determined. It is found that for an enclosure with high value of Rayleigh number (i.e., Ra=105), the total entropy generation due to fluid friction and total Nu number increases with decreasing the aspect ratio.
Basic Study on Solid-Liquid Phase Change Problem of Ice around Heat Transfer ...IJERDJOURNAL
Abstract:- Phase change heat transfer around heat transfer tubes is one of the basic problem of an ice heat storage exchanger. It can lead to decrease of thermal storage efficiency and damage of heat transfer tubes if continued freezing further after the ice has bridged because of the generated ice thermal resistance and volume expansion. In this study, we focused on freezing phenomena of phase change material (PCM) between two heat transfer tubes, which can simulate an inside structure of ice heat storage exchangers. Bridging time between two heat transfer tubes was studied numerically. We used water as the PCM, which is filled in the water container. Two horizontal elliptical tubes were used as heat transfer tubes in order to observe the influence of natural convection. Single-domain calculation model was used to calculate arbitrary shape of the two tubes during the ice freezing process. We changed arranged angle and relative position of the tubes to investigate impact of the tube arrangement on freezing phenomenon. In order to confirm the accuracy of our analysis, analytical results were compared with experimental results at the same conditions. Results show that the bridging time was not simply in proportional to the initial temperature of water when considered the natural convection influenced by such as density inversion of water. Moreover, we found that when we set the temperature of tube wall and initial temperature of water as the parameters, bridging time has a similar trend with distance between the axes. Therefore, it is possible to predict the bridging time for elliptical heat transfer tube.
Study on Air-Water & Water-Water Heat Exchange in a Finned Tube ExchangerAnamika Sarkar
Heat exchange characteristics between hot and cold water as well as cold air and hot water have been investigated in a finned tube heat exchanger. The exchanger was operated in both parallel and counter-flow modes. For water-water heat exchange, the tube side (hot water) fluid flow rate was varied to observe the effects on heating the cold water. Conversely, for air-water system, the shell side (cold air) flow rate was varied to evaluate the cooling effectiveness of air in the finned tube exchanger. COMSOL Multiphysics 5.6 was used to simulate the system in 3-D mode, and the temperature profiles along with heat flux and velocity streamlines were evaluated. The NTU and effectiveness for varying fluid flow rates for both water-water and air-water systems were calculated. NTU values were considerably higher for air-water heat exchange and showed a decreasing trend with increasing fluid flow rate. To ascertain the reliability of the simulation models, the experimental and simulated results were compared. To evaluate the performance of the fins, the fin efficiencies and effectivenesses were calculated and the values were notably higher in case of air-water system. This is consistent with established literature. Also, increasing shell-side fluid flow rate led to a reduction in fin efficiency and effectiveness, which is again consistent with literature.
Second Law Analysis of Fluid Flow and Heat Transfer through Porous Channel wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
HEAT TRANSFER CORRELATION FOR NON-BOILING STRATIFIED FLOW PATTERN | J4RV3I11006Journal For Research
In chemical industries two phase flow is a process necessity. A better understanding of the rates of momentum and heat transfer in multi-phase flow conditions is important for the optimal design of the heat exchanger. To simplify the complexities in design, heat transfer coefficient correlations are useful. In this work a heat transfer correlation for non- boiling air-water flow with stratified flow pattern in horizontal circular pipe is proposed. To verify the correlation, heat transfer coefficients and flow parameters were measured at different combinations of air and water flow rates. The superficial Reynolds numbers ranged from about 2720 to 5740 for water and from about 563 to 1120 for air. These experimental data were successfully correlated by the proposed two-phase heat transfer correlation. It is observed that superficial.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Numerical Modeling and Simulation of a Double Tube Heat Exchanger Adopting a ...IJERA Editor
The double tube heat exchangers are commonly used in industry due to their simplicity in design and also their
operation at high temperatures and pressures. As the inlet parameters like temperatures and mass flow rates
change during operation, the outlet temperatures will also change. In the present paper, a simple approximate
linear model has been proposed to predict the outlet temperatures of a double tube heat exchanger, considering it
as a black box. The simulation of the heat exchanger has been carried out first using the commercial CFD
software FLUENT. Next the linear model of the double tube heat exchanger based on lumped parameters has
been developed using the basic governing equations, considering it as a black box. Results have been generated
for outlet temperatures for different inlet temperatures and mass flow rates of the cold and hot fluids. The results
obtained using the above two methods have then been discussed and compared with the numerical results
available in the literature to justify the basis for the assumption of a linear approximation. Comparisons of the
predicted results from the present model show a good agreement with the experimental results published in the
literature. The assumptions of linear variation of outlet temperatures with the inlet temperature of one fluid
(keeping other inlet parameters fixed) is very well justified and hence the model can be employed for the
analysis of double tube heat exchangers.
An Offshore Natural Gas Transmission Pipeline Model and Analysis for the Pred...IOSRJAC
The purpose of this paper is to model and analyze an existing natural gas transmission pipeline – the 24-inch, 5km gas export pipeline of the Amenam-Kpono field, Niger Delta, Nigeria – to determine properties such as pressure, temperature, density, flow velocity and, in particular, dew point, occurring at different segments of the pipeline, and to compare these with normal pipeline conditions in order to identify the segments most susceptible to condensation/hydrate formation so that cost-effective and efficient preventive/remedial actions can be taken. The analysis shows that high pressure and low temperature favor condensation/hydrate formation, and that because these conditions are more likely in the lower half of the pipeline system, remedial/preventive measures such as heating/insulation and inhibition injection should be channeled into that segment for cost optimization..
The dependencies of total pressure, velocity, vorticity, turbulent length, turbulent dissipation, turbulent viscosity, turbulent energy and turbulent time of moving fluid from a straight pipe length of a circular cross section are presented in graphical and mathematical forms. Changing analysis of considered parameters was performed at mass flow rates of 0.45, 1.0 and 1.5 kg/s. A transition boundary of laminar flow of fluid to turbulent flow is at the distance of 2/5 of length from the inlet of the pipe (at accepted total length of the pipe of 1000 mm).
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
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Simulacion de flujo bifasico intermitente con transferencia de calor
1. P ro g r am a d e Pó s G r a d u a ç ão em En g en h a r ia
M ecân ic a e de M at e ri ai s
III SEMINÁRIO ANUAL DO PROGRAMA DE PÓS-GRADUAÇÃO EM
ENGENHARIA MECÂNICA E DE MATERIAIS
30 de Agosto a 03 de Setembro de 2010 – Curitiba – Paraná - Brasil
ALGEBRAIC MODEL FOR THE SIMULATION OF TWO-PHASE SLUG
FLOW WITH HEAT TRANSFER
César D. Perea Medina, cesar.perea.medina@gmail.com1
Rigoberto E. M. Morales, rmorales@utfpr.edu.br1
Silvio L. M. Junqueira, silvio@utfpr.edu.br1
1
LACIT/PPGEM/UTFPR, Av. Sete de Setembro 3165, CEP. 80230-901, Curitiba-PR-Brasil
Abstract: Two-phase flows with heat transfer are found in many engineering applications. One of them is the
conduction of oil and gas in the deep ocean, where exists a temperature gradient due to the difference between the
temperature in the source and the temperature in the surrounding environment. In liquid-gas two-phase flows, one of
the most frequent patterns is the slug flow. This flow pattern is characterized by the alternate succession of two
structures: an aerated slug and an elongated gas bubble, which constitutes a unit cell. In spite of the unit cell
properties variation with time, it can be modeled as stationary if mean time values are used. In that context, the present
work proposes a mechanistic one-dimensional stationary model for the calculation of the main hydrodynamical and
heat transfer parameters of slug flow. Based on mass, momentum and energy balances on the unit cell, an implicit
algebraic equation system will be obtained. The solution for a unit cell is found through an iterative process and then
propagated along the pipe, assuming that the pressure and temperature gradients are linear. As a result, geometric
characteristics, phase velocities, pressure and temperature along the pipe can be known. From the temperature profile,
the two-phase heat transfer coefficient can be calculated, which can be compared with some correlations found in the
literature. Results show good agreement with the reported data in the literature.
Key-words: two-phase flow, slug flow, heat transfer, convection
1. INTRODUCTION
The study of heat transfer in two-phase flow is a matter of importance as it has many industrial applications. One of
them is the oil transfer in long production lines, where the conduction pipes are exposed to harsh external conditions.
This interaction causes heat exchanges between the two-phase mixture and the surrounding environment. As a result,
the temperature of the fluids will vary along the pipeline producing changes in the in-situ properties of the fluids like
the density or the liquid viscosity, directly related to the pressure drop. In addition, wax deposition or formation of
hydrates can occur, as these processes depend on the thermodynamical equilibrium, which is directly related with the
temperature.
Most of the studies made about two-phase flow heat transfer have as a main concern the calculation of the
convective heat transfer coefficient. The majority of these studies propose correlations found through experimentation.
Kim and Ghajar (2006) propose an accurate correlation as a function of the fluid properties and the rates of flow.
Camargo (1991) propose a mechanistic model for intermittent flow based on the hydrodynamic parameters.
On the other hand, few works have presented temperature simulation through energy balance. One of them is the
unified model presented by Zhang et al (2006) where he finds analytical solutions for the temperature for different two-
phase patterns assuming that the hydrodynamic characteristics of the flow are known.
The intermittent flow pattern occurs in a wide range of flow rates. It is characterized by the alternate succession of
two structures: an aerated slug and an elongated bubble, which constitute a unit cell. Each of the components of the unit
cell has its characteristics changed across the time and space; however it can be modeled as stationary flow if mean time
values are considered.
There are many hydrodynamic models based on the unit cell concept, such as Dukler and Hubbard (1975), Taitel
and Barnea (1990). However, these models are applied just to a unit cell and not to the whole pipe. Besides that, they
ignore the effects of gas compressibility.
In that context, the objective of the present work is to propose a methodology for the calculation of hydrodynamical
and heat transfer parameters of two-phase intermittent flow through a stationary mechanistic model. All the parameters
are calculated for a unit cell and then propagated along the pipe considering the gas compressibility due to pressure and
temperature changes. As the properties of the fluids simulated are far from the saturation region, no phase change
occurs, so that, the liquid can be modeled as incompressible and the gas as perfect gas.
2. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
The problem to be solved considers a two-phase mixture flowing in the intermittent pattern. It flows through a
horizontal pipe with round cross section. The pipe is surrounded by external environmental conditions, which is
physically translated to an external heat transfer coefficient caused by an external flow.
For the development of the present work, first it is presented the governing equations for the hydrodynamics, which
are based in the stationary model of Taitel and Barnea (1990). Then, the equations for the heat transfer are deduced
using energy balance in each of the unit cell components. Then it is explained the calculation process based on the
algebraic model of Freitas et al (2008). The results are validated with data from the literature and discussed properly.
For last, some conclusions are presented.
2. HYDRODYNAMIC MODEL
The hydrodynamic model is based on the stationary one-dimensional model presented by Taitel and Barnea (1990).
The basic assumptions for the model development are: (i) periodic flow, (ii) Newtonian fluids with constant properties,
(iii) guaranteed slug flow pattern at the entrance and exit of the pipe, (iv) ideal gas and incompressible liquid, (v)
uniform distribution of the bubbles along the slug.
The physical model is shown in Figure 1a. The studied control volume is the unit cell, composed by a liquid slug
(length LS) and an elongated bubble (length LB). The unit cell is moving at the translational velocity UT and each of the
components of the unit cell has a different velocity. It can be observed on the cross section of the slug (Figure 1b) that
the liquid wets all the perimeter of the pipe, while in the bubble’s cross section (Figure 1c) the liquid is stratified on the
bottom of the pipe due to gravity.
The concept of superficial velocity (j) is defined as the volumetric flow of the phase per unit of area. Considering
constant gas mass flow rate and the state equation of perfect gases, the superficial velocities of the gas (jG) in two
sections can be expressed as a function of pressure and temperature, as seen on Eq. (1):
• •
mL mG P P
= =
jL ; jG = jGs Gexit
; jG G (1)
ρL A ρG A TG TGexit
•
where m is the mass flow rate, ρ the fluid density, P the pressure and T the temperature and A the pipe cross section
area. Sub-indexes L and G refer to liquid and gas respectively and the sub-index exit indicates the exit of the pipe, taken
as a reference.
Figure 1. Physical model for intermittent flow (a).Pipe cross sections in the slug (b) and elongated bubble (c).
The mixture velocity (J) in Eq. (2) is defined as the sum of the superficial velocities. In addition it can also be
expressed as a function of the velocities in the slug (ULS and UGS) just as reported by Shoham (2006).
J =+ jG
jL ; J = RLS + U GS (1 − RLS )
U LS (2)
where RLS , RLB and RGB are the volume fraction of liquid in the slug, liquid film and gas bubble respectively.
Expressions for the liquid film and gas bubble velocities are deduced from mass balances for each phase in the
entire unit cell. The reference frame is non-inertial and moves with the velocity of the bubble nose, which is called
translational velocity (UT). Velocities for the liquid film (ULB) and the elongated bubble velocities (UGB) are given
respectively by:
RLS (1 − RLS )
U LB =U T − (U T − U LS ) ; U GB =U T − (U T − U GS ) (3)
RLB RGB
3. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
In order to calculate the bubble length, it is studied the hydrodynamics of the film which is modeled as channel
flow with free surface according to Taitel and Barnea´s model (1990). They present Eq. (4) to define the geometry of
the gas bubble:
S LB S 1 1
τ LB − τ GB GB − τ i Si +
dH LB
=
ALB AGB ALB AGB (4)
dz ' (U − U LS ) RLS dRLB − ρ U − U (U T − U GB )(1 − RLS ) dRLB
− ρ L (U T − U LB ) T G ( T GB )
(1 − RLB )
2 2
RLB dH LB dH LB
where HLB is the liquid height on the film, τ is the shear stress, S the wet perimeter, and LS, LB, GB and i the indexes for
the liquid slug, liquid film, gas bubble and interface respectively.
Eq. (4) is numerically integrated to obtain bubble length (LB), mean gas volume fraction ( RGB ) and equilibrium
liquid film height. Note in Figure 1 that coordinate z’ is positive in the left direction, which is why the slope must be
negative. The integration should start in HLB (z’=0)=D, however, under some flow rate conditions, the slope is positive
in HLB=D. As the integration should not start until that slope is negative, a lower HLB(z’=0) should be selected. Testing
values, it was concluded that a good point to start the integration is HLB(z’=0)=0.9D. Furthermore, a stop condition
should be given; otherwise, an infinite bubble length will be obtained. Thus, unit cell liquid mass balance can be used as
the stop condition:
• LS 1 LB
= U LS ARLS ρ L
mL +
LS + LB LS + LB ∫
0
U LB ARLB ρ L dz (5)
Also, the unit cell frequency (freq) is used, which is defined as freq = UT/(LS+LB). Physically it represents the
inverse of the time of a unit cell passage. Assuming that the frequency is known, the unit cell length in Eq. (5) can be
substituted as a function of the translational velocity UT and the frequency. That way, Eq. (4) is integrated until the
liquid mass balance in Eq. (5) is satisfied. Frequency is calculated with a constitutive equation, presented in section 2.2.
2.1 Pressure Drop and Pressure Gradient
The pressure drop on the unit cell is composed by the sum of the pressure drop on the slug and the film, which can
be calculated using Eq. (6)
τ S
L
B
S LS LS
= τ LS
∆PU + ∫ LB LB dz (6)
A 0
A
where τLS is the shear stress in the slug, τLB the shear stress in the film.
For this work, the pressure distribution in a unit cell is considered linear, so the pressure gradient λU is constant.
P
That way, the pressure can be calculated at any point using a linear equation found by the integration of Eq. (7).
dP ∆P
= λU
U
=
P U
(7)
dz LS + LB
P ( z ) =exit − λU ( L − z )
U P P
(8)
2.2 Constitutive Equations
In order to have the same number of variables and equations, it is necessary to use some constitutive equations. The
first one is used to calculate the translational velocity, which is based on the correlation of Nicklin (1962). In this case,
the coefficients proposed by Bendiksen (1980) are used.
J > 3.5 → c0 1.20= 0.00
= c1
U T =+ c1 gD →
c0 J (9)
gD < 3.5 → c0 1.05= 0.54
= c1
where g is the gravity acceleration, D the pipe diameter.
The volume fraction of liquid or liquid holdup (RLS), is calculated by the Malnes (1982) correlation:
4. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
1 Fr = J / gD
RLS = 1 − (10)
Bo ( ρ L − ρG ) gD / σ
=
2
83
1+
Fr ⋅ Bo 0.25
where σ is the superficial stress coefficient, Fr is the Froude number and Bo is the Bond number.
A third constitutive equation is needed for the velocity of the dispersed bubbles (UGS) in the liquid slug. Normally,
this velocity is modeled as the superposition of the mixture velocity J and the velocity of a rising bubble in an infinite
stagnant liquid (Omgba, 2006). The effects of the rising velocity are related to the gravity direction; however, as the
pipe is horizontal, these effects can be ignored. That way, it can be assumed that the dispersed bubbles in the slug are
moving with the mixture velocity. (UGS = J).
For the unit cell frequency, Hill and Wood (1990) propose a correlation which uses the flow parameters in stratified
condition. Stratified equivalent flow is found through mass rates of flow using the Taitel and Dukler (1976) model.
0.000761 RL
=freq (U G − U L ) (11)
D 1 − RL STRATIFIED
where the parameters in brackets are evaluated in the stratified condition.
For last, a constitutive equation must be used to calculate the frictional force. The shear stress is expressed as a
function of the Fanning friction factor. Equations used are presented in Table 1.
Table 1. Constitutive equations for the frictional force (1)
Hydraulic Reynolds
Friction Factor (f) Shear Stress (τ)
Diameter (DH) Number (Re)
RF A ρ F U F DF ε 106 f F ρ FU F
2
DHF = 4 Re F = = 0.001375 1 + 2 ⋅104 +
fF τF =
S LF µF
D Re F
2
(1)
: The index F changes according to the element evaluated, being LS for liquid slug, LB for liquid film and
GB for elongated gas bubble.
All correlations in this section can be found in Omgba (2006)
3. HEAT TRANSFER MODEL
The development of the heat transfer model is based on the first law of thermodynamics for control volumes in
stationary regime in the absence and viscous work, as according to White (2003) it is rarely important.
•
1
Q = ∫ i + U 2 + e p ρVr ⋅ d A (12)
SC
2
•
where i is the specific enthalpy, Vr is the relative velocity and Q is the heat externally transferred to the flow and ep
the potential energy. The mechanism of heat transfer is the forced convection. In addition, kinetic energy is small
compared to the enthalpy and variation in potential energy is null due to the horizontal position of the pipe.
Considering the hypotheses explained before, Eq. (12) is applied to the three differential control volumes specified
in Figure 2: the elongated bubble, the slug and the liquid film. Despite the heat transfer, the state of the fluids is far from
the saturation region, which is why no phase change occurs. In addition, enthalpy can be expressed as the product of
specific heat (CL) and temperature for incompressible liquids and specific heat at constant pressure (CpG) times
temperature for ideal gases (Moran and Shapiro, 2006)
Figure 2. Stationary balance of energy on a unit cell
5. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
Eq. (12) is applied to the liquid slug, considering that the only source of heat is the contact with the duct wall. Heat
can be expressed by the Newton’s law of cooling. Besides that, the term ρVr ⋅ d A applied to the slug physically
represents the mass flow rate of the liquid. Considering the definition of derivative in the infinitesimal control volume,
the following differential equation can be deduced:
• dT
mL= hLS S LS (T0 − TLS )
CL LS G
(13)
dz
In the case of the elongated bubble and the liquid film, besides the wall contact heat, there is a heat exchange
between the liquid and the gas in the interface. As the heat lost by one phase is gained by the other, the heat transfer
equations for the elongated bubble and the liquid film are respectively given by:
• dTGB
mG CpG = hSG SGB (T0 − TGB ) − hi Si (TGB − TLB )
G
(14)
dz
• dTLB
mL CL = hLB S LB (T0 − TLB ) + hi Si (TGB − TLB )
G
(15)
dz
Solution for the slug temperature, Eq. (13), is easily obtained by direct integration. For convenience, the coordinate
system is set to zero for each unit cell at the beginning of the slug. It is assumed that the temperature of the dispersed
gas bubbles in the slug is the same as in the liquid. The boundary conditions are the following:
• In z=0, TLS = TLS0 for the slug. The integration is from z=0 to z=LS
• In z=0, TLB=TLB0=TLS(z=LS) for the liquid film. The integration is from z=0 to z=LB
• In z=0, TGB=TGB0=TLS(z=LS). The integration is from z=0 to z=LB
On the other hand, equations (14) and (15) constitute a differential equations system, which solution is obtained by
mathematical analysis. Finally, explicit expressions for the temperature of each component of the unit cell are found.
For the slug, the expression is:
hG S
TLS =TLS 0 − (TLS 0 − TLSi ) exp − LS LS z (16)
•
CpL mL
For the liquid film and elongated bubble:
TLB =ϕ LB1 exp ( r1 z ) + ϕ LB 2 exp ( r2 z ) + ϕ LB 3 (17) TGB =ϕGB1 exp ( r1 z ) + ϕGB 2 exp ( r2 z ) + ϕGB 3 (18)
which constants are specified on Table 2.
Table 2: Constants for the analytic solution of the film and bubble temperatures.
G G G G
hLB S LB hi Si hi Si hi Si hGB SGB hi Si hLB S LB hGB SGB
=a1 •
+ •
a2 = •
b1 = • =b2 •
+ •
c1 = •
T0 c2 = •
T0
mL CpL mL CpL mG CpG mL CpL mG CpG mG CpG mL CpL mG CpG
− ( a1 + b2 ) − ( a1 + b2 )
2
− 4 ( a1b2 − a2 b1 ) c1a2 + c2 a1 − ( a1 + b2 ) + ( a1 + b2 )
2
− 4 ( a1b2 − a2 b1 )
r1 = ϕGB 3 = r2 =
2 a1b2 − a2 b1 2
a T − ( r + b )(T − ϕ ) + c − b ϕ a2TLB 0 − ( r1 + b2 )(TGB 0 − ϕGB 3 ) + c2 − b2ϕGB 3
ϕGB1 = 2 2 GB 3 − ϕGB 3
TGB 0 − 2 LB 0 1 2 GB 0 GB 3 ϕGB 2 =
r2 − r1 r2 − r1
ϕGB1r1 + b2ϕGB1 ϕGB 2 r2 + b2ϕGB 2 1
ϕ LB1 = ϕ LB 2 = ϕ LB 3
= ( b2ϕGB 3 − c2 )
a2 a2 a2
3.1 Heat Transfer Coefficient
As presented before, the duct is surrounded by an external cooling flow. For the heat transfer coefficient, it must be
considered the external convection of weather conditions, the conduction in the duct thickness and the internal
6. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
convection between the duct wall and the two-phase mixture. That way, it is used the concept of global heat transfer
coefficient, based on thermal resistances (Incropera, 2008):
1
hF =
G
(19)
1 D D D
+ Ln e +
hF 2kc D De h0
where kc is the thermal conductivity of the pipe material, De is the external diameter, h0 is the convective coefficient of
the external flow and hF is the heat transfer coefficient for each component of the unit cell. hF is calculated as a one-
phase coefficient. According to the experimental studies of Lima (2009), the expression that adjusts better to the one-
phase flow behavior is the Gnielinski (1976) correlation:
( f / 8)( Re F − 1000 ) PrF kF −2
hF = F where f F =0, 079 ⋅ Ln ( Re F ) − 1, 64
(20)
1 + 12, 7 ( f F / 8 ) ( PrF / 3 − 1) DHF
0,5 2
This correlation has a better adjustment to the experimental data because it shows an explicit dependence on the
friction factor, which has considerable influence on the heat transfer coefficient (Bejan, 2004). In the case of the liquid
slug, in order to simulate the higher turbulence occurring in this region, the heat transfer coefficient hLS is increased by
30% (Camargo, 1991).
3.2 Two-Phase Heat Transfer Coefficient
Once the temperatures are known in a unit cell, it is possible to calculate a two heat transfer coefficient for the
internal convection based on the Newton’s cooling law. It is calculated one coefficient for each unit cell, considering
the temperature drop from z=0 to z=LS+LB=LU
• LS LB
=Q ∫h
G
LS S LS (TLS − T0 ) dz + ∫ h
G
LB S LB (TLB − T0 ) + hGB SGB (TGB − T0 ) dz
G
•
0 0
Q
hTP =
π D.LU .∆T
(T − T ) − (TLSe − Twe ) (21)
∆T = LBs ws
T −T
Ln LBs ws
TLSe − Twe
•
where Q is the total heat transferred to the fluids, ∆T is the logarithmic mean temperature difference, TLBs and TLSe are
the mean temperatures at the exit and entrance of the unit cell respectively, Tws and Twe are the mean temperatures on the
internal wall at the exit and entrance. Eq. (21) calculates the internal convection coefficient between the two-phase
mixture and the duct wall. Temperature in the internal wall is given by Eq. (22):
hG D
Tw = F (TF − T0 )
TF − (22)
hF De
3.3 Unit cell mean temperature
It is necessary to obtain a unit cell mean temperature in order to compare the results of the model with the
experimental data. Zhang et al (2006) calculated this temperature as a function of the liquid temperatures. In addition,
using the concept of frequency and translational velocity, the unit cell temperature is calculated based on the lengths:
tF tF LB LS
=TU
∫= ∫
0
T dt + ∫ T dt
LB 0 LS 0
TLB dz + ∫ TLS dz
0
(23)
( LS + LB ) / U T LS + LB
The fact of using just the liquid temperature is based on the thermal capacity (ρCp) being considerably greater in
the liquid than in the gas. For example, considering water (ρ=1000 kg/m3, Cp=4180 J/kg.K) and air (ρ=1.2 kg/m3,
Cp=1003 J/kg.K) at normal conditions, the thermal capacity of the water is almost 4000 times greater than the thermal
capacity of the gas.
7. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
3.4 Temperature Gradient
Just as the assumption for the pressure, in temperature is also introduced the concept of temperature gradient in a
unit cell. It is simply defined as the difference between the temperature in the entrance of the slug and the temperature
at the exit of the liquid film divided by the total unit cell length. It is assumed that this gradient is constant in a unit cell,
which is why the mean unit cell temperature has a linear tendency.
dTU = 0=TLS z − TLB z
λU = = −
T LB
(24)
dz LS + LB
TU ( z ) Tentrance − λU z
= T
(25)
4. CALCULATION PROCEDURE
The calculation procedure is based on the algebraic hydrodynamic model of Freitas et al (2008). The solution starts
with the determination of the flow hydrodynamics through the bubble design and pressure drop equations in (4) and (6)
respectively. Then, from the hydrodynamics results, the heat transfer equations are applied to obtain the temperature
profile. Input data necessary for the solution of the problem is the following: properties of the fluids, superficial
velocities of each of the fluids at the exit of the pipe (or mass flow rates), pressure at the exit (Pexit), fluid temperature at
the entrance and external conditions (temperature and external heat transfer coefficient). The calculation procedure has
two stages: calculation of the unit cell at the entrance and propagation along the pipe.
STAGE 1: Properties at the pipe entrance (z = 0).
a) Estimate a pressure gradient and temperature gradient at the pipe entrance ( λU and λU ) and assume a pressure
P T
at the entrance.
b) Calculate the following flow parameters at the pipe entrance: jG [Eq. (1)], J, UT [Eq. (9)], RLS [Eq. (10)], ULS
[Eq. (2)].
c) The bubble length LB is obtained by the numerical integration of Eq. (4). Integrate until Eq. (5) is satisfied.
d) Calculate the liquid slug length LS and the gas fraction on the bubble RGB,
e) Calculate the pressure drop through Eq. (6) and the new pressure gradient through (7)
f) Assume a TLS(z=0), calculate the temperatures distribution using equations (16), (17) and (18). Calculate the
mean temperature in the unit cell using (23) and the temperature gradient using (24). Recalculate the TLS(z=0)
and repeat the process until the mean temperature calculated converge with the temperature at the entrance
given as input.
g) Using the new pressure and temperature gradients, recalculate the parameters until the gradients converge.
STAGE 2: Propagation of the properties along the pipe
a) With the gradients found at the entrance, calculate the pressure and unit cell mean temperatures on the other
points along the pipe using equations (8) and (25) respectively.
b) Calculate the unit cells in the evaluation points through b), c), d), e), f) and g) on Stage 1. That way, new
gradients are obtained for each point.
c) Calculate the flow parameters with b), c), d), e), f) and g) using the new gradients found in b). New pressure
and temperature gradients are calculated.
d) Repeat c) until the gradients converge with 0.01%.
5. RESULTS
The proposed model is applied to an air-water flow cooled by an external water flow. Results are compared with
experimental studies obtained from Lima (2009). In this experiment, the pressure and temperature were measured at the
entrance and at the exit of the pipe. In addition, there was a transparent section of the pipe to visualize the flow pattern
and to measure the geometric characteristics of the bubbles. The characteristics of one specific test of each study are
specified in the table 3.
Table 3. Input values for the simulations
Air and Water Flow
Pipe Length (m) 6.07 Exit pressure (kPa) 171.0
Pipe diameter (mm) 52.00 External fluid temperature (K) 282.4
Exit gas superficial velocity (m/s) 0.2825 Entrance two-phase temperature (K) 307.7
Liquid superficial velocity (m/s) 1.378
8. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
The results of the simulations using the proposed model are presented in the Table 4. Good agreement for the
pressure drop and the translational velocity are observed. However, there is a considerable deviation for the slug and
bubble length. Probably this occurred due to the stationary flow hypothesis, as the bubble length is directly related to
the intermittency. Another reason for this to happen is that a stationary model cannot predict unsteady phenomena like
coalescence. Just as one-phase flow, the pressure drop is directly influenced by the pipe’s diameter; the smaller the
diameter is the grater the pressure drop.
Table 4. Results of the simulations
Lima Error
Air and Water Model
(2009) (%)
Bubble Length at the exit (m) 0.340 0.213 -37.35
Slug Length at the exit (m) 0.82 1.018 24.14
Translational Velocity (m/s) 2.10 2.15 2.38
Pressure Drop (kPa) 2.718 2.470 9.10
Exit two-phase temperature (K) 304.60 304.24 -0.11
Heat Transfer Coefficient (w/m2K) 7366 7121 -3.32
In Figure 3a it is observed the bubble geometry. The bubble presents a curved profile at the nose and a quasi linear
profile at the tail. Due to the high liquid superficial velocity, a low gas volume fraction is observed, as the liquid-gas
interface is over the center of the pipe. In Figure 3b, it is observed the pressure distribution along the pipe considering
six evaluation points. Pressure presents a linear behavior. Also, it is observed that the model tends to slightly
underestimate the pressure drop.
In Figure 4a it is observed the temperature distribution along the pipe. The calculated mean unit cell temperature
shows good agreement with the experimental data, which validates the expression in Eq. (23). It is observed that the
mean temperature of the gas bubble is always considerably lower than the whole unit cell. In Figure 4b it is presented
the temperature profile of the unit cell at the exit. Despite the temperature function being a exponential, a linear
behavior is observed which also validates the hypothesis in Eq. (25). The high temperature drop in the bubble region
shows the influence of its low thermal capacity. This means that the gas requires less heat to modify its temperature,
which is why despite the same amount of heat being provided, the gas temperature decreases more.
Lima (2009) reported 23 more tests with different rates of flow which are also tested in the model. On Figure 5a and
5b the gauge pressure at the entrance and the two-phase heat transfer coefficient are plotted against the experimental
data. As seen on Figure 5a and 5b the calculated parameters gave good predictions. More than 92% of the data is
confined in the 25% error range.
1 174
a) b)
0.8
173
0.6
P [kPa]
HLB/D
172
0.4
171 Present Work
0.2 Lima (2009)
0 170
4 3 2 1 0 0 30 60 90 120
LB/D L/D
Figure 3. a) Geometric design of the exit bubble. b) Pressure distribution along the pipe length.
9. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
308 310
a) b)
306 308
T [K]
T [K]
304 306
302 Unit Cell Mean Temperature 304 Liquid Temperature
Gas Bubble Mean Temperature
Gas Temperature
Lima (2009)
300 302
0 30 60 90 120 0 5 10 15 20 25
L/D L/D
Figure 4. a) Mean temperatures along the pipe. b) Distribution of the temperatures along the entrance unit cell.
100 10000
a) hTP - Calculated [w/m2K] b)
80 8000 +25%
P - Calculated [kPa]
+10%
60 6000
-10%
-25%
40 4000
20 2000
Lima (2009) Lima (2009)
0 0
0 20 40 60 80 100 0 2000 4000 6000 8000 10000
P - Experimental [kPa] hTP - Experimental [w/m2K]
Figure 5. a) Calculated vs Experimental Gauge pressure at the entrance of the pipe. b) Calculated vs experimental two-
phase heat transfer coefficient.
6. CONCLUSIONS
Hydrodynamics and heat transfer simulation on intermittent flow was presented. The model is based on stationary
balances of momentum and energy. It calculates the parameters for a unit cell and then propagates the results along the
pipe. As the model is composed just by algebraic equations, its implementation is easy and its low computing time turns
it a powerful tool to predict velocities, pressure drop and temperatures.
It is observed that the pressure has a quasi linear behavior; however, the pressure gradient is different for each unit
cell. From the temperature profile, it is observed that temperature of the liquid phase is dominant. Although the equation
for the liquid temperature is an exponential function, its distribution is quasi linear due to its high specific heat. As seen
on the results, the hypothesis of constant pressure and temperatures gradients is correct.
Good agreement with experimental data is observed for the pressure drop, the temperatures and the heat transfer
coefficient. However, it fails to predict bubble and slug lengths as they depend on the intermittency at the entrance. As
the model considers periodic unit cells, the unsteady effects of intermittency and coalescence are ignored.
7. REFERENCES
Bejan, A., 1995, “Convection Heat Transfer”, 2a ed., John Wiley & Sons, Inc.
Camargo, R. M. T. (1991) “Hidrodinâmica e Transferência de calor no Escoamento Intermitente Horizontal”,
Universidade Estadual de Campinas, Brazil, Master thesis, 142 p.
Dukler A. E. e Hubbard M. G., 1975, “A model for gas-liquid slug flow in horizontal and near horizontal tubes”. Ind.
Eng. Chem. Fundam., 14 (4), pp. 337-347.
10. II Seminário do PPGEM, 10 e 11 de Dezembro 2009, Curitiba-Paraná
Freitas, F. G., Rodrigues, H. T., Morales R. E. M., Mazza, R., Rosa, E., 2008, “Algebraic Model for Bubble Tracking in
Horizontal Gás-Liquid Slug Flow” Heat 2008, Fifth International Conference on Transport Phenomena in
Multiphase Systems.
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Horizontal” São Paulo: Universidade Estadual de Campinas, Brazil, Master Thesis 135 p.
Kim, J. and Ghajar, A., 2006, “A general heat transfer correlation for non-boiling gas-liquid flow with different flow
patterns in horizontal pipes”, International Journal of Multiphase flow 32, pp 447-465.
Moran, M. e Shapiro, H., 2006, “Fundamentals of Engineering Thermodynamics”, 5th Ed. John Wiley & Sons, Inc.
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Regimes)” Cranfield University, PhD Thesis.
Shoham, O., 2006, “Mechanistic Modeling of Gas-Liquid Two-phase Flow In Pipes” Society of Petroleum Engineers,
Richardson, TX, 396p.
Taitel, Y. and Barnea, D., 1990, "Two phase slug flow", Advances in Heat Transfer, Hartnett J.P. and Irvine Jr. T.F. ed.,
vol. 20, 83-132, Academic Press (1990).
Taitel, Y. and Dukler, A. E., 1976 “A Model for Predicting Flow Regime Transitions in Horizontal and near Horizontal
Gas-Liquid Flow”, AIChE J., 22, 47-55.
White, F., 2003, “Fluid Mechanics”, Ed. Mc Graw-Hill, 5th edition.
Zhang, H., Wang, Q., Sarica, C. e Brill, J. P., 2006, “Unified Model of Heat Transfer in Gas-Liquid Pipe Flow” Society
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