Friction factor improved correlations for laminar and turbulent gas–liquid flow in horizontal pipelines

We develop improved correlations for two-phase flow friction factor that consider the effect of the relative velocity of the phases, based on a database that includes 2560 gas–liquid flow experiments in horizontal pipes. The database includes a wide range of operational conditions and fluid properties for two-phase friction factor correlations. We classify the experiments by liquid holdup ranges to obtain composite analytical expressions for two-phase friction factor vs. the Reynolds number by fitting logistic dose curves to the experimental data with. We compute the liquid holdup values used to classify the experimental data using correlations proposed previously. The Reynolds number is based on the mixture velocity and the liquid kinematic viscosity. The Fanning friction factor for gas–liquid is defined in term of the mixture velocity and density. Additionally, we sort the experimental data by flow regime and obtain the two-phase friction factor improved correlations for dispersed bubble, slug, stratified and annular flow for different holdup ranges. We report error estimates for the predicted vs. measured friction factor together with standard deviation for each correlation. The accuracy of the correlations developed in this study is compared with that of other 21 correlations and models widely available in the specialized literature. Since different authors use different definitions for friction factors and Reynolds numbers, we present comparisons of the predicted pressure drop for each and every data point in the database. In most cases our correlations predict the pressure drop with much greater accuracy than those presented by previous authors.     

Investigation of pressure drop in horizontal pipes with different diameters

The pressure drop has a significant importance in multiphase flow systems. In this paper, the effect of the volumetric quality and mixture velocity on pressure drop of gas-liquid flow in horizontal pipes of different diameters are investigated experimentally and numerically. The experimental facility was designed and built to measure the pressure drop in three pipes of 12.70, 19.05 and 25.40 mm. The water and air flow rates can be adjusted to control the mixture velocity and void fraction. The measurements are performed under constant water flow rate (CWF) by adding air to the water and constant total flow rate (CTF) in which the flow rates for both phases are changed to give same CTF. The drift-flux model is also used to predict the pressure drop for same cases. The present data is also compared with a number of empirical models from the literature. The results show that: i) the pressure drop increases with higher volumetric qualities for the cases of constant water flow rate but decreases for higher volumetric qualities of constant total flow rate due to the change in flow pattern. ii) The drift-flux model and homogenous model are the most suitable models for pressure drop prediction.     

Correlations predicting frictional pressure drop and liquid holdup during horizontal gas-liquid pipe flow with a small liquid holdup

Experimental data and correlations available in the literature for the liquid holdup ε_L and the pressure gradient ΔP_TP/L for gas-liquid pipe flow, generally, do not cover the domain 0 < ε_L < 0.06. Reliable pressure-drop correlations for this holdup range are important for calculating flow rates of natural gas, containing traces of condensate. In the present paper attention is focused on reliable measurements of ε_L and ΔP_TPIL values and on the development of a phenomenological model for the liquid-holdup range 0 < ε_L < 0.06. This model is called the “apparent rough surface” model and is referred to as the ARS model. The experimental results presented in this paper refer to air-water and air-water + ethyleneglycol systems with varying transport properties in horizontal straight smooth glass tubes under steady-state conditions. The holdup and pressure gradient values predicted with the ARS model agree satisfactorily with both our experimental results and data obtained from the literature referring to small liquid-holdup values 0 < ε_L < 0.06. Further, it has been shown that in the domain 38 < < 72 mPa m the interfacial tension of the gas-liquid system has no significant effect on the liquid holdup. The pressure gradient, however, increases slightly with decreasing surface tension values.     

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.     

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.     

Two-phase discharge from a stratified region through two horizontal branches with centerlines falling on an inclined plane

The main objective of this study was to obtain new experimental data for conditions not previously tested for discharging two-phase flow through two 6.35 mm diameter branches with centerlines falling in an inclined plane. The present results are relevant to many industrial applications including headers and manifolds, multichannel heat exchangers and small breaks in horizontal pipes. In the experimental investigation, the critical heights for the onsets of liquid and gas entrainment (OLE and OGE, respectively) were obtained, analyzed and correlated for two different branch spacings and two different angles between the branches. For each combination of branch spacing and angle between the branches, a wide range of Froude numbers was used. Two-phase mass flow rate and quality results were also obtained and analyzed for a range of interface heights for 16 different combinations of branch spacing, inclination angle, test section pressure and pressure drop across each branch. New empirical correlations were developed to predict the dimensionless mass flow rate and quality. The new correlations show good agreement with the present data and with previous correlations.     

Assessment of pressure drop in conical spouted beds of biomass by artificial neural networks and comparison with empirical correlations

Pressure drop is an essential parameter in the operation of conical spouted beds (CSB) and depends on its geometric factors and materials used. Irregular materials, like biomass, are complex to treat and, unlike other gas–solid contact methods, CSB turn out to be a suitable technology for their treatment. Artificial neural networks were used in this study for the prediction of operating and peak pressure drops, and their performance has been compared with that of empirical correlations reported in the literature. Accordingly, a multi-layer perceptron network with backward propagation was used due to its ability to model non-linear multivariate systems. The fitting of the experimental data of both operating and peak pressure drop was significantly better than those reported in the literature, specifically in the case of the peak pressure drop, with R² being 0.92. Therefore, artificial neural networks have been proven suitable for the prediction of pressure drop in CSB.     

Effect of diameter on two-phase pressure drop in narrow tubes

The effect of tube diameter on two-phase frictional pressure drop was investigated in circular tubes with inner diameters of 0.6, 1.2, 1.7, 2.6 and 3.4 mm using air and water. The gas and liquid superficial velocity ranges were 0.01-50 m/s and 0.01-3 m/s, respectively. The gas and liquid flow rates were measured and the two-phase flow pattern images were recorded using high-speed CMOS camera. Unique flow patterns were observed for smaller tube diameters. Pressure drop was measured and compared with various existing models such as homogeneous model and Lockhart-Martinelli model. It appears that the dominant effect of surface tension shrinking the flow stratification in the annular regime is important. It was found that existing models are inadequate in predicting the pressure drop for all the flow regimes visualized. Based on the analysis of present experimental frictional pressure drop data a correlation is proposed for predicting Chisholm parameter “C” in slug annular flow pattern. For all other flow regimes Chisholm’s original correlation appears to be adequate except the bubbly flow regime where homogeneous model works well. The modification results in overall mean deviation of pressure drop within 25% for all tube diameters considered. This approach of flow regime based modification of liquid gas interaction parameter appears to be the key to pressure drop prediction in narrow tubes.     

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.     

Determination of slug permeability factor for pressure drop prediction of slug flow pneumatic conveying

Current models for pressure drop prediction of slug flow pneumatic conveying in a horizontal pipeline system assume some type of steady state conditions for prediction,which limits their capability for increased predictive accuracy relative to experimental data.This is partly because of the nature of slug flow pneumatic conveying system,which,as a dynamic system,never becomes stable.By utilising conservation of mass (airflow),a dynamic pressure analysis model is proposed on the basis of the derivative of the upstream pressure behaviour.The rate of air permeation through slug,one of the important factors in the conservation model,is expressed as a function of a slug permeability factor.Other factors such as slug velocity,slug length and the fraction of stationary layer were also considered.Several test materials were conveyed in single-slug tests to verify the proposed pressure drop model,showing good agreement between the model and experimental results.     

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.     

Determination of slug permeability factor for pressure drop prediction of slug flow pneumatic conveying

Current models for pressure drop prediction of slug flow pneumatic conveying in a horizontal pipeline system assume some type of steady state conditions for prediction, which limits their capability for increased predictive accuracy relative to experimental data. This is partly because of the nature of slug flow pneumatic conveying system, which, as a dynamic system, never becomes stable. By utilising conservation of mass （airflow）, a dynamic pressure analysis model is proposed on the basis of the derivative of the upstream pressure behaviour. The rate of air permeation through slug, one of the important factors in the conservation model, is expressed as a function of a slug permeability factor. Other factors such as slug velocity, slug length and the fraction of stationary layer were also considered. Several test materials were conveyed in single-slug tests to verify the proposed pressure drop model, showing good agreement between the model and experimental results.     

Correlation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow

Experimental data for gas holdup in liquid slugs are reported for two different pipe sizes (2.58 cm and 5.12 cm I.D.). A simple empirical correlation is developed and is shown to be a significant improvement over the only other published correlation proposed by Hubbard (1965). The results of this investigation are important for the development of a mechanistic model for the prediction of pressure drop and holdup for slug flow in pipes.     

Experimental investigation of ice slurry flow pressure drop in horizontal tubes

Pressure drop behaviour of ice slurry based on ethanol–water mixture in circular horizontal tubes has been experimentally investigated. The secondary fluid was prepared by mixing ethyl alcohol and water to obtain initial alcohol concentration of 10.3% (initial freezing temperature ?4.4 °C). The pressure drop tests were conducted to cover laminar and slightly turbulent flow with ice mass fraction varying from 0% to 30% depending on test conditions. Results from flow tests reveal much higher pressure drop for higher ice concentrations and higher velocities in comparison to the single phase flow. However for ice concentrations of 15% and higher, certain velocity exists at which ice slurry pressure drop is same or even lower than for single phase flow. It seems that higher ice concentration delay flow pattern transition moment (from laminar to turbulent) toward higher velocities. In addition experimental results for pressure drop were compared to the analytical results, based on Poiseulle and Buckingham–Reiner models for laminar flow, Blasius, Darby and Melson, Dodge and Metzner, Steffe and Tomita for turbulent region and general correlation of Kitanovski which is valid for both flow regimes. For laminar flow and low buoyancy numbers Buckingham–Reiner method gives good agreement with experimental results while for turbulent flow best fit is provided with Dodge–Metzner and Tomita methods.Furthermore, for transport purposes it has been shown that ice mass fraction of 20% offers best ratio of ice slurry transport capability and required pumping power.     

Horizontal laminar flow of coarse nearly-neutrally buoyant particles in non-Newtonian conveying fluids: CFD and PEPT experiments compared

The horizontal flow of coarse particle suspensions in non-Newtonian carrier fluids was numerically simulated using an Eulerian–Eulerian CFD model. This study was concerned with nearly-neutrally buoyant particles of 5 and 10 mm diameter conveyed by fluids of Ellis rheology in laminar flow, in a 45 mm diameter pipe at concentrations up to 41% v/v. CFD predictions of solid phase velocity profiles and passage times were compared to experimental data obtained by a Positron Emission Particle Tracking (PEPT) technique and Hall effect sensors, and a very good agreement was obtained considering the complexity of the flows studied. CFD predictions of solid–liquid pressure drop were compared to a number of relevant correlations gleaned from the literature. Only one of them showed a good agreement over the whole range of conditions studied. Other correlations generally showed large deviations from CFD, and their limitations in predicting the influence of solids concentration and particle size have been demonstrated. Overall, it emerged that for the flows studied, CFD was capable of giving predictions of pressure drop which were probably better and more reliable than the correlations available in the literature.     

Experimental two-phase pressure gradients during evaporation of pure and mixed refrigerants in a smooth horizontal tube. Comparison with correlations

The paper reports the results of an experimental study on pressure drop during horizontal flow boiling of refrigerants R22, R507, R404A, R134a, R407C and R410A. The test section is a smooth, horizontal, stainless steel tube (6 mm I.D., 6 m length) uniformly heated by Joule effect. The experimental tests are carried out at an almost constant evaporating pressure of 7.0 bar varying the mass flow rate in the range 280–1,080 kg/m² s. The experimental comparison clearly shown that the pressure drop of R22 is significantly higher as compared to all the other fluids. The results are compared against well-known pressure drop prediction methods. The available correlations can be used for both pure fluids and mixtures with no corrective factors, provided the mixture properties are evaluated at local compositions. The Chawla friction correlation is the best-fitting of our experimental data in combination with the heterogeneous momentum pressure drop model on the basis of the Rouhani-Axelsson void fraction correlation.     

Drift-flux correlation for gas-liquid two-phase flow in a horizontal pipe

A drift-flux correlation has been often used to predict void fraction of gas-liquid two-phase flow in a horizontal channel due to its simplicity and practicality. The drift-flux correlation includes two important drift-flux parameters, namely, the distribution parameter and void-fraction-weighted-mean drift velocity. In this study, an extensive literature survey for horizontal two-phase flow is conducted to establish void fraction database and to acquire existing drift-flux correlations. A total of 566 data is collected from 12 data sources and 4 flow-regime-dependent and 1 flow-regime-independent drift-flux correlations are identified. The predictive capability of the existing drift-flux correlations is assessed using the collected data. It is pointed out that the drift velocity determined by a regression analysis may include a significant error due to a compensation error between distribution parameter and drift velocity. In this study, a simple flow-regime-independent drift-flux correlation is developed. In the modeling approach, the void-fraction-weighted mean drift velocity is approximated to be 0 m/s, whereas the distribution parameter is given as a simple function of the ratio of non-dimensional superficial gas velocity to non-dimensional mixture volumetric flux. The newly developed correlation shows an excellent predictive capability of void fraction for horizontal two-phase flow. Mean absolute error (or bias), standard deviation (random error), mean relative deviation and mean absolute relative deviation of the correlation are 0.0487, 0.0985, 0.0758 and 0.206, respectively. The prediction accuracy of the correlation is similar to the correlation of Chexal et al. (1991), which was formulated based on the drift-flux parameters by means of many cascading constitutive relationships with numerous empirical parameters.     

Frictional pressure drop during vapour–liquid flow in minichannels: Modelling and experimental evaluation

Condensation in minichannels is widely used in air-cooled condensers for the automotive and air-conditioning industry, in heat pipes and other applications for system thermal control. The knowledge of pressure drops in such small channels is important in order to optimize heat transfer surfaces. This paper presents a model for calculation of the frictional pressure gradient during condensation or adiabatic liquid–gas flow inside minichannels with different surface roughness. In order to account for the effects of surface roughness, new experimental frictional pressure gradient data associated to single-phase flow and adiabatic two-phase flow of R134a inside a single horizontal mini tube with rough wall has been used in the modelling. It is a Friedel (1979) [Friedel, L., 1979. Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow. In: Proceedings of the European Two-Phase Flow Group Meeting, Ispra, Paper E2] based model and it takes into account mass velocity, vapor quality, fluid properties, reduced pressure, tube diameter, entrainment ratio and surface roughness. With respect to the flow pattern prediction capability, it has been built for shear dominated flow regimes inside pipes, thus, annular, annular-mist and mist flow are here predicted. However, the suggested procedure is extended to the intermittent flow in minichannels and it is also applied with success to horizontal macro tubes.     

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.     

A mathematical model for pressure drop of two-phase dry-plug flow in circular mini/micro channels

Dry-plug flow is a variation of two-phase plug flow that occurs in small scale channels and refers to the dry wall conditions at gas portions of the flow regime. Previous experimental studies found a significant increase in pressure drop in this flow regime from the two-phase wet-plug flow regime. In this work an analytical model for the pressure drop of this flow regime is developed and phenomena that influence pressure drop are examined. Unlike previous models, the proposed model seems to be applicable to a wide range of capillary numbers and give good estimations for all static contact angles. Contact angle hysteresis turned out to play a major role in inducing pressure drop in this flow regime. Model’s predictions were in good agreement with previous experimental data. Finally the model is applied to very low capillary number region and pressure drop predictions for this region are presented.