Composite power law holdup correlations in horizontal pipes

A wide range of experimental holdup data, from different sources, are analyzed based on a theoretical model proposed in this work to evaluate the holdup in horizontal pipes. 2276 gas–liquid flow experiments in horizontal pipelines with a wide range of operational conditions and fluid properties are included in the database. The experiments are classified by mixture Reynolds number ranges and composite analytical expressions for the relationship between the liquid holdup and no-slip liquid holdup vs. the gas–liquid volumetric flow rate are obtained by fitting the data with logistic dose curves. The Reynolds number appropriate to classify the experimental data for gas–liquid flows in horizontal pipes is based on the mixture velocity and the liquid kinematic viscosity. Composite power law holdup correlations for flows sorted by flow pattern are also obtained. Error estimates for the predicted vs. measured holdup correlations together with standard deviation for each correlation are presented. The accuracy of the correlations developed in this study is compared with the accuracy of 26 previous correlations and models in the literature. Our correlations predict the liquid holdup in horizontal pipes with much greater accuracy than those presented by previous authors.     

An improved semi-empirical friction model for gas-liquid two-phase flow in horizontal and near horizontal pipes

Pressure drop and liquid hold-up are two very important fluid flow parameters in design and control of multiphase flow pipelines. Friction factors play an important role in the accurate calculation of pressure drop. Various empirical and semi-empirical closure relations exist in the literature to calculate the liquid-wall, gas-wall and interfacial friction in two-phase pipe flow.However most of them are empirical correlations found under special experimental conditions. In this paper by modification of a friction model available in the literature, an improved semiempirical model is proposed. The proposed model is incorporated in the two-fluid correlations under equilibrium conditions and solved. Pressure gradient and velocity profiles are validated against experimental data. Using the improved model, the pressure gradient deviation from experiments diminishes by about 3%; the no-slip condition at the interface is satisfied and the velocity profile is predicted in better agreement with the experimental data.     

Heat transfer measurements and correlations for air–water flow of different flow patterns in a horizontal pipe

Heat transfer coefficients were measured and new correlations were developed for two-phase, two-component (air and water) heat transfer in a horizontal pipe for different flow patterns. Flow patterns were observed in a transparent circular pipe using an air–water mixture. Visual identification of the flow patterns was supplemented with photographic data, and the results were plotted on the flow regime map proposed by Taitel and Dukler and agreed quite well with each other. A two-phase heat transfer experimental setup was built for this study and a total of 150 two-phase heat transfer data with different flow patterns were obtained under a uniform wall heat flux boundary condition. For these data, the superficial Reynolds number ranged from 640 to 35,500 for the liquid and from 540 to 21,200 for the gas. Our previously developed robust two-phase heat transfer correlation for a vertical pipe with modified constants predicted the horizontal pipe air–water heat transfer experimental data with very good accuracy. Overall the proposed correlations predicted the data with a mean deviation of 1.0% and an rms deviation of 12%.     

Distribution of the velocity profiles of the liquid phase in a gas-liquid flow with small gas contents

Artilce [1] gives the results of measurement of the friction at the wall of a channel under bubble conditions, in a wide range of Reynolds numbers. It is shown that the concept of laminar flow conditions has no meaning when it is applied to the flow of a two-phase mixture, since, even with very small Reynolds numbers, the level of the pulsations of the friction is high, and the spectrum of the pulsations of the friction is continuous. In this case, the mean friction is much greater than the calculated; here the value of the resistance coefficient is not a single-valued function of the Reynolds number. The present article gives the results of measurement of the velocity profiles of the liquid phase, carried out using an electro diffusion method. It is shown that, with Reynolds numbers corresponding to turbulent flow conditions, the profile of the velocity in a two-phase mixture is close to turbulent and does not depend on the gas content.     

A model for the flow of solid particles in gases

The flow of solid particles in air streams involves a great deal of variables and complex phenomena, difficult to analyse. In practice the flow quantities in gas-solid flows are predicted by the use of empirical correlations of data or semi-empirical methods. The predictive power of these methods varies substantially between different systems. This paper presents an analytical approach to the subject of gas-solid flows, based on a turbulent model. The mixture is modeled as a variable density fluid flowing in a duct; the equations for the Reynolds stress incorporate the variation of velocity and density together, and yield the velocity profile of the flow and average quantities of interest such as the mass flux, the friction factor, the average density and average areas occupied by each phase. The predicted values for the friction factor are compared with known correlations emanating from experimental data. It is found that there is a very good agreement between the predicted values and the experimental correlations.     

Forced convective flow and heat transfer of upward cocurrent air–water slug flow in vertical plain and swirl tubes

This experimental study comparatively examined the two-phase flow structures, pressured drops and heat transfer performances for the cocurrent air–water slug flows in the vertical tubes with and without the spiky twisted tape insert. The two-phase flow structures in the plain and swirl tubes were imaged using the computerized high frame-rate videography with the Taylor bubble velocity measured. Superficial liquid Reynolds number (Re_L) and air-to-water mass flow ratio (AW), which were respectively in the ranges of 4000–10000 and 0.003–0.02 were selected as the controlling parameters to specify the flow condition and derive the heat transfer correlations. Tube-wise averaged void fraction and Taylor bubble velocity were well correlated by the modified drift flux models for both plain and swirl tubes at the slug flow condition. A set of selected data obtained from the plain and swirl tubes was comparatively examined to highlight the impacts of the spiky twisted tape on the air–water interfacial structure and the pressure drop and heat transfer performances. Empirical heat transfer correlations that permitted the evaluation of individual and interdependent Re_L and AW impacts on heat transfer in the developed flow regions of the plain and swirl tubes at the slug flow condition were derived.     

Foam flow in vertical gas wells under liquid loading: Critical velocity and pressure drop prediction

Foam lift is one of the most cost effective methodologies for unloading gas wells. The surfactants are either injected intermittently or continuously to lift the liquid to the surface. By reducing the gravitational gradient and increasing the frictional gradient, the critical velocity at which liquid loading occurs is shifted to lower gas velocities. Currently, we do not have a methodology to predict the critical velocity (at the transition boundary of annular and intermittent flow) and the pressure drop under foam flow conditions.To address this, we measured several foam flow characteristics in both small scale and large scale facilities. Small scale facility involved measurement of foam carryover capacity as a function of time and surfactant concentration. Large scale facility involved measurement of liquid holdup, pressure drop, fraction of gas trapped in foam and foam holdup in 40-ft 2-in. and 4-in. tubing.We developed closure relationships for liquid hold up, foam holdup, fraction of gas trapped in the foam and interfacial friction factor by combining the small scale data with the data collected in the large scale experiments. These closure relationships are applicable to four different surfactants tested. A new transition criterion was developed and successfully used to predict onset of liquid loading under foam flow. Using a force balance over the gas core in annular flow, we developed a new procedure to calculate the pressure drop under foam flow conditions. We compared our model results with actual measurements in the large scale facility. Our model was reasonably able to predict the pressure drop within ±30%. The reason for such a large variance is that the small scale facility was not able to capture all the characteristics of the foam which were observed in the large scale facility. It is very difficult to reproduce the foam characteristics exactly in two different experiments. This is discussed further in this paper.The procedure developed is the only one currently available to calculate the pressure drop under the foam flow conditions using the small scale data. It is superior to conventional annular flow pressure drop prediction models which are currently available in the literature.     

Algebraic turbulence modeling in adiabatic gas–liquid annular two-phase flow

The study considers algebraic turbulence modeling in adiabatic gas–liquid annular two-phase flow. After reviewing the existing literature, two new algebraic turbulence models are proposed for both the liquid film and the droplet laden gas core of annular two-phase flow. Both turbulence models are calibrated with experimental data taken from the open literature and their performance critically assessed. Although the proposed turbulence models reproduce the key parameters of annular flow well (average liquid film thickness and pressure gradient) and the predicted velocity profiles for the core flow compare favorably with available core flow velocity measurements, a more accurate experimental database is required to further improve the models accuracy and range of applicability.     

Choking and shock phenomena in a single component two-phase flow including vibrational effects

An analytical and experimental investigation including vibratory effects of flashing flow in a tube with a sharp edged entrance is presented. A free streamline flow model is applied to predict choking in single-component two-phase flow. By identifying three separate regimes (i.e. jet flow, two-phase homogeneous flow, and single-phase liquid flow) in the flashing flow system, an expression is obtained for the prediction of the minimum stagnation pressure loss under choked flow conditions. A normal shock located between the flashing two-phase mixture and the single-phase liquid was experimentally observed. The location of the shock is predicted as a function of the stagnation pressure drop across the tube. The analytical predictions are verified by experimental data.     

Friction factor and holdup studies for lubricated pipelining—I. Experiments and correlations

Results from new experiments on the lubricated pipelining of emulsified waxy crude oil and No. 6 fuel oil are presented and compared with other sources of literature. A correlation formula which estimates the holdup fraction is introduced and evaluated for all available experimental data. A simple theory is given which is based on the concentric core-annular flow model and leads to a Reynolds number and friction factor which reduce a large body of experimental data onto one curve; with the best results in the high Reynolds number flow regime.     

Effect of void fraction and friction factor models on the prediction of pressure drop of R134a during downward condensation in a vertical tube

The theoretical flow models of homogeneous and separated flow are applied to in-tube condensation to predict the pressure drop characteristics of R134a. The homogeneous flow model is modified by ten different dynamic viscosity correlations and various alternative correlations of total, frictional and momentum pressure drops to take account of the partial condensation inside the tube. Numerical analyses were performed to determine the average and local homogeneous wall shear stresses and friction factors by means of a CFD program. The equivalent Reynolds number model is modified by six different two-phase friction factors to determine the total condensation pressure drop in the separated flow model. The refrigerant side total pressure drops, frictional pressure drops, friction factors and wall shear stresses are determined within a ±30% error band. The importance of using the alternative total, momentum and frictional pressure drop correlations for the homogeneous flow model is also shown.     

A separated flow model for predicting two-phase pressure drop and evaporative heat transfer for vertical annular flow

A separated flow model has been developed that is applicable to vertical annular two-phase flow in the purely convective heat transfer regime. Conservation of mass, momentum, and energy are used to solve for the liquid film thickness, pressure drop, and heat transfer coefficient. Closure relationships are specified for the interfacial friction factor, liquid film eddy-viscosity, turbulent Prandtl number, and entrainment rate. Although separated flow models have been reported previously, their use has been limited, because they were tested over a limited range of flow and thermal conditions. The unique feature of this model is that it has been tested and calibrated against a vast array of two-phase pressure drop and heat transfer data, which include upflow, downflow, and microgravity flow conditions. The agreements between the measured and predicted pressure drops and heat transfer coefficients are, on average, better or comparable to the most reliable empirical correlations. This separated flow model is demonstrated to be a reliable and practical predictive tool for computing two-phase pressure drop and heat transfer rates. All of the datasets have been obtained from the open literature.     

Two-phase two-component interface drag coefficients in separated phase flows

This work examines the behavior of the interface friction factor or drag coefficient as a means for extending the modeling of separated two-phase flows through the separate consideration of each phase. The model development of this work builds primarily upon the work of Carofano & McManus (1969), Wallis (1970) and Smith (1968). A one-dimensional flow model was developed for the case of vertical upward annular fiow of an air-water mixture with droplet entrainment. The model was developed for the investigation of accelerating flows in a nozzle but is utilized in this study for the investigation of momentum transport occurring in non-accelerating flows. This study presents experimental data showing the behaviour of the flow pressure drop occurring at various flow qualities and gas velocities. Also presented are empirical results for values of the air-water interface drag coefficient as a function of flow quality and gas core Reynolds number. The drag coefficient variation is compared to a previous correlation developed by Wallis 1969).     

The prediction of stratified two-phase flow with a two-equation model of turbulence

A two-equation model is applied to a stratified two-phase flow system to predict turbulent transport mechanisms in both phases.In the present analysis, the effects of interfacial waves on the flow field are formulated in terms of boundary conditions for the gas-liquid interface. For the gas phase, the wavy interface has such flow separation effects as a rough surface in a single-phase flow. While for the liquid phase, the waves generate turbulant energy which is transported progressively toward a lower wall region. The analytical results are in good agreement with available data regarding pressure drop, holdup and velocity profiles.     

Wave behavior in horizontal annular air–water flow

In this study, non-intrusive pressure drop, liquid base film thickness distribution, and wave behavior measurements have been obtained for 206 horizontal annular two-phase (air–water) flow conditions in 8.8, 15.1, and 26.3 mm ID tubes. Reliable wave velocity measurements are available for 185 of these flow conditions, while 131 flow conditions allow for reliable wave frequency measurements. The wave velocity is found to be predicted to within 9% by gas friction velocity and 6% by an optimized correlation of similar structure. Wave frequency can also be predicted with a simple correlation to within 5% for the 8.8 and 15.1 mm tubes, but a separate relation is required to explain 26.3 mm frequency data. The differences in wave behavior between the annular and wavy-annular/wavy regimes are also discussed.     

Modeling study of three-phase low liquid loading flow in horizontal pipes

A theoretical study is conducted to model the flow characteristics of three-phase stratified wavy flow in horizontal pipelines with a focus on the low liquid loading condition, which is commonly observed in wet gas pipelines. The model predictions are compared to the experimental data of Karami et al. (2016a, b). These experiments were conducted with water or 51 wt% of MEG in the aqueous phase, and inlet aqueous phase fraction values from 0 to100%.Modeling of three-phase flow can be described as a combination of two-phase gas-liquid flow modeling, and a liquid phase oil-water mixing modeling. A mechanistic model is proposed to predict flow characteristics of three-phase stratified wavy flow in pipeline. For the gas-liquid interactions, Watson's (1989) combined momentum balance equation derivation was applied. However, the calculation procedure was reversed, and the wave celerity was assumed as an input, while interfacial friction factor was one of the model's outputs. The liquid-liquid interactions were modeled using a simple energy balance equation and shift in liquid phase center of gravity calculations. The liquid phases can be separated, partially mixed, or fully mixed. The bottom aqueous film velocity was calculated using the law of the wall formulation, and was used to calculate the flowing aqueous phase fraction.The model predictions of different flow characteristics for two and/or three-phase flows were compared with available experimental data. The pressure gradient, wave amplitude, and aqueous phase fraction predictions were in good agreement with the experimental data. However, the liquid holdup predictions were slightly under-predicted by the model. Overall, an acceptable agreement was observed for all cases.Most of the common multiphase stratified flow models are developed with the assumption of steady-state conditions and with constant interfacial friction factor value. This study proposes a novel method to model stratified flow. The predictions are in acceptable agreement with experimental data conducted under stratified wavy flow pattern conditions.     

Non-newtonian liquid-air stratified flow through horizontal tubes—ii

Non-Newtonian liquid gas stratified flow data were obtained using 0.052 and 0.025 m dia horizontal circular ducts. Unless the liquid velocity was very low, the flow pattern generally observed was non-uniform stratified flow having an interfacial level gradient between the two phases. The Heywood-Charles model is valid for predicting the pressure drop and liquid holdup in pseudoplastic (shear thinning) non-Newtonian liquid-gas uniform stratified flow. Two-phase drag reduction, which is predicted by the Heywood-Charles model did not occur because there was a transition to semi-slug flow before the model criteria were reached. Interfacial liquid and gas shear stresses were compared.     

The turbulent flow of non-Newtonian fluids in the absence of anomalous wall effects

For Newtonian fluids, the engineering predictions for pressure drop in turbulent pipe flow are well established. However, in the case of non-Newtonian liquids, only a few design techniques have been proposed and these do not share a common basis with the approach for Newtonian systems. This present work attempts to provide a common basis for both Newtonian and non-Newtonian systems in situations where anomalous wall effects are absent. Previously published experimental data suggest that if the Reynolds number is calculated on the basis of the apparent viscosity at the wall then the standard Newtonian correlations can be used for the prediction of pressure drop. The use of the wall viscosity in defining the Reynolds number also serves as a test for anomalous behaviour. Any departure of the experimental data from the Newtonian turbulent friction factor correlation indicates anomalous behaviour.     

Calculation of the thermodynamic sound velocity in two-phase multicomponent fluids

The Peng-Robinson and Soave-Redlich-Kwong equations of state have been used to calculate the thermodynamic speed of sound in single-phase fluids consisting of pure components and mixtures, and in the two-phase region of a multicomponent mixture. The calculated velocities for single-phase fluids were compared with available data for methane, ethane, propane and a binary mixture of benzene in hexane. The basic trends observed in the data were predicted and in general, the comparison between theory and data was good.Values calculated for the multicomponent mixture along an isobar showed a sudden drop in the thermodynamic sound velocity when the dew point temperature was reached. Inside the two-phase region the velocity decreased along an isobar with a lowering of temperature and increased suddenly when the other dew point or bubble point temperature was reached. At temperatures below the bubble or second dew point, the fluid behaved as a liquid, i.e. sound velocity increased with temperature. Similar overall behavior was predicted when the assumption of “frozen” sound velocity was employed except that a sudden drop did not occur at the dew point temperature of the mixture.     

Atomized non-equilibrium two-phase flow in mesochannels: Momentum analysis

Two-phase pressure drop measurements are very difficult to make while the fluid is in non-equilibrium condition, i.e. while phase change is taking place. This is further complicated when an atomized liquid is introduced in the system at much higher velocity than other components such as liquid layer, vapor core, and entrained droplets. The purpose of this paper is to develop a model to predict the two-phase pressure characteristics in a mesochannel under various heat flux and liquid atomization conditions. This model includes the momentum effects of liquid droplets from entrainment and atomization. To verify the model, an in-house experimental setup consisting of a series of converging mesochannels, an atomization facility and a heat source was developed. The two-phase pressure of boiling PF5050 was measured along the wall of a mesochannel. The one-dimensional model shows good agreement with the experimental data. The effects of channel wall angle, droplet velocity and spray mass fraction on two-phase pressure characteristics are predicted. Numerical results show that an optimal spray cooling unit can be designed by optimizing channel wall angle and droplet velocity.