Mathematical modeling of an industrial process of nylon-6,6 polymerization in a two-phase flow tubular reactor

Pipeline solid particle erosion depends on numerous independent variables and fluid flow parameters, making measurements or predictions inherently uncertain. Estimating these uncertainties is challenging due to a limited understanding of variable interactions. Because erosion is a major risk in many industries, accurately estimating uncertainties is vital for both safety and economic viability. This study examines uncertainties in previously gathered erosion measurements under two-phase flow conditions with sand in pipe elbows. The database includes ultrasonic measurements of gas–liquid–sand cases with vertical upward flows through standard elbows. Particle sizes range from 20 to 300 μm, and pipe diameters are 0.0762 and 0.1016 m. Measurements span multiple two-phase flow regimes, including annular, churn, and annular-mist flows. Initial uncertainty bounds were estimated using analytical correlations, yielding upper bounds of 1.45–2.78 times the measurements and lower bounds of 0.18–0.78 times the measurements. Subsequently, machine learning (ML) algorithms, including random forests, LightGBM, XGBoost, and Gaussian processes, were trained on the erosion database. These algorithms used gas and liquid velocities, pipe diameters, and particle sizes as independent variables, along with the initially estimated bounds. Finally, Monte Carlo (MC) simulations examined how uncertainties in these parameters affect the overall uncertainty. The MC-based bounds refined the original estimates, providing an updated range of 1.5–4.00 times the measurements for the upper bound and 0.04–0.78 times for the lower bound. This integrated approach provides a more accurate representation of pipeline erosion uncertainty, improving flow assurance across multiple industries.

. This study investigates the geometric effects of 90-degree vertical elbows and flow configurations in two-phase flow. The study shows that the elbows make a significant effect on the transport characteristics of two-phase flow, which includes the changes in interfacial structures, bubble interaction mechanisms and flow regime transition. The effect of the elbows is characterized for global and local two-phase flow parameters. The global two-phase flow parameters include two-phase pressure, interfacial structures and flow regime transition. In order to characterize the frictional pressure drop and minor loss across the vertical elbows, pressure measurements are obtained across the test section over a wide range of flow conditions in both single-phase and two-phase flow conditions. A two-phase pressure drop correlation analogous to Lockhart-Martinelli correlation is proposed to predict the minor loss across the elbows. A high speed camera is employed to perform extensive flow visualization studies across the elbows in vertical upward, horizontal and vertical downward sections and modified flow regime maps are proposed. It is found that modified flow regime maps immediately downstream of the vertical upward elbow deviate significantly from the conventional flow regime map. A qualitative assessment of the counter-current flow limitation characteristics specific to the current experimental facility is performed. A multi-sensor conductivity probe is used to measure local two-phase flow parameters such as: void fraction, bubble velocity, interfacial area concentration and bubble frequency. The local measurements are obtained for six different flow conditions at ten measurement locations along axial direction of the test section. Both the vertical-upward and vertical-downward elbows have a significant impact on bubble distribution, resulting in, a bimodal distribution along the horizontal radius of the tube cross-section and migration of bubbles towards the inside of the elbow curvatures immediately downstream of the vertical-upward and vertical-downward elbows, respectively. The elbow effect decays further downstream of the elbow and bubbles migrate to more conventional distribution patterns. The axial transport of void fraction and interfacial area concentration shows that the elbows promote bubble disintegration. Preliminary comparisons between the interfacial area transport model and the experimental data for verticalupward and vertical downward section are also presented.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Transient behavior of a molten salt fast reactor under two-phase flow conditions with helium bubbling

A semi-empirical model of two-phase flow of refrigerant-134a through short tube orifices

Concerning the two-phase gas-liquid mixtures, the present understanding of even fully developed turbulent flow of a single-phase component in close channels is not completely satisfactory. In the two-phase or multiphase flow of such heterogeneous mixture like gas-liquid many more independent parameters are involved, thereby making this process more complicated and less transparent for understanding, mathematical modeling and simulating or calculating of such parameter like the length pressure gradient. In those mathematical models for calculations and simulations, as well as for interpretation of experimental results, there is a very complicated and random phenomenon of flow patterns, which needs to be quantitatively and accurately, incorporated producing higher accuracy in calculation and description. Unfortunately, nowadays, a method of how to measure flow patterns is not available. Recognizing these challenges this paper will present an approach to incorporate flow pattern phenomenon into the two-phase flow calculation model by (1) developing a mathematical model for pressure losses in two-phase flow based on in-situ parameters, (2) developing and defining a flow pattern coefficient, which incorporates the flow pattern phenomena, and (3) present the developed mathematical model with the incorporation of flow patterns, which demonstrated significant increase of accuracy of calculations based on conducted experimental research on air-water two-phase mixture flow in a horizontal small square channel.Copyright © 2005 by ASME

This paper uses a large amount of available data for liquid holdup in horizontal, two-phase flow to compare and evaluate 12 different correlations and models. In a proposed two-step holdup prediction method, an expected flow pattern is predicted on the basis of the system variables. The correlation recommended for use with that flow regime then is used to predict the holdup. Introduction The complexity of the different flow regimes involved in the flow of gas and liquid mixtures in pipes has greatly hindered the development of theoretical mechanistic flow models. However, because of the industrial importance of these systems, there has been a substantial buildup of knowledge and experience, largely in the form of empirical correlations. As a result, from the pipeline designer's point of view, the task of selecting the "best" or most point of view, the task of selecting the "best" or most appropriate correlations has become rather formidable. A need clearly exists for a periodic critical evaluation of available prediction methods. This paper presents the results of a detailed comparison of correlations and models for holdup in horizontal two-phase flow. The results of other evaluations of flow-pattern maps and pressure-drop prediction methods for two-phase flow pressure-drop prediction methods for two-phase flow in horizontal pipes, and holdup for two-phase flow in inclined pipes are reported elsewhere. The term "liquid holdup" is used in this paper to mean "in-situ liquid volume fraction." Other definitions or measures are often used in the literature to describe the amount of liquid actually contained in a pipe under twophase flow conditions. Previous Evaluations Previous Evaluations In 1964, Dukler et al. published a critical comparison of a number of holdup prediction methods. The correlations of Hoogendoorn, Hughmark, and Lockhart and Martinelli were tested against culled holdup data contained in the AGA/API data bank. On the basis of three statistical parameters, the Hughmark correlation was found to be best parameters, the Hughmark correlation was found to be best over all, but the agreement with the data was generally poor for all the correlations. The parameters used by poor for all the correlations. The parameters used by Dukler et al. were based on percentage errors and thus were very sensitive to incorrect predictions for low measured values of holdup. A recent study by Vohra et al. used the data of Beggs and Eaton to compare the correlations of Beggs and Brill, Dukler, Eaton et al., Guzhov et al., Hughmark, and Lockhart and Martinelli. Perhaps not too surprisingly, they reported that the Eaton et al. correlation was in best agreement with the Eaton data, and the Beggs and Brill correlation was in best agreement with the Beggs data. Over all, the Eaton et al. correlation was found to be in best agreement with the test data, followed by the Beggs and Brill correlation. It should be noted that there were 238 data points from Eaton and only 58 from Beggs used in the study. Approach Followed in This Study In the early stages of this study, it was apparent that there were several aspects of the proposed comparison that should be considered in some detail. The first point concerned the question of bias in the analysis resulting from the test data. Clearly, if a given correlation is based directly on a substantial portion of the test data being used in the comparison, it is given an unfair advantage over the other correlations being tested. JPT P. 1017

Most of nuclear thermal-hydraulic system codes, which adopt a two-fluid model for two-phase flow, use a non-conservative form of momentum equations for numerical convenience. However, it was known that the non-conservative form can cause unphysical momentum convection in some two-phase flow conditions, especially when the two-phase flow was strongly heterogeneous. In order to compensate this defect, a semi-conservative form of momentum equation is suggested, which is non-conservative in time but conservative in space. In this paper, the semi-conservative form of the two-fluid momentum equation was implemented into the multi-dimensional module of the SPACE code. Thereafter, the code was assessed by simulating two multi-dimensional two-phase flow experiments. The results showed that the semi-conservative form has better predicting capability than the non-conservative form.

This work experimentally investigates the impact of gas bubbles on the thermal and acoustic energy prorogation within the wellbore, at various water and air flow rates using Distributed Acoustic Sensing (DAS). This is the first study that experimentally investigates the thermal and acoustic propagation at both single and two-phase flow conditions using DAS. Our results will improve the production logging algorithms especially at multiphase flow conditions, benefit the detection of gas leakage in the vertical section of a wellbore, as well as pipeline flow monitoring. Distributed Fiber-Optic Sensing (DFOS) based production logging has drawn much attention in recent years. Comparing to conventional production logging tools, fiber-optic cables can endure much harsher borehole environments and can be deployed at a lower cost. This work presents an experimental study of using DAS to track the thermal slugging in single-phase (liquid) and two-phase (gas and liquid) flow within a vertical wellbore to estimate flow velocity and characterize multiphase flow behavior. A vertical flow loop is constructed for this research, which consists of a 7-m long transparent polyvinyl chloride (PVC) test section with a 1-inch pipe inner diameter. A single-mode optic fiber with thin plastic coating is wrapped evenly around the PVC pipe, with a fiber-to-pipe length ratio of 10.7. Tap water and compressed air are used as the testing fluids. The water and air are injected at the bottom of the test section, similar to field conditions where oil and gas mixture flows into the wellbore from the bottom. The water is directly supplied from the building water system without using an additional pump to minimize the unrelated acoustic noise. Air is supplied from an air tank charged by the building compressor with a maximum pressure of 80 psi. A peristaltic pump is used to inject a small amount of hot water (< 2% of the minimum water volume from the inlet) to generate thermal slugs at the bottom of the test section. For the case of single-phase water flow, the velocity of the thermal slugging signal is similar to the actual water velocity, as expected. For multiphase flow, the thermal signal looks almost identical with that for single-phase, although the bubble velocity is much higher than the water velocity. This observation indicates that gas, with low thermal capacity and in its bubble form, cannot carry enough heat to perturb the thermal slugs in the water severely. However, a detailed analysis of the thermal slugging velocity indicates a small increase with an increase of air flow rate. We interpret that the thermal slugging velocity is associated with the actual water velocity. The existence of the gas bubble decreases the effective water holdup (cross-sectional area occupied by the water phase) in the pipe. We find that, with a constant water volumetric rate, smaller holdup leads to higher in-situ water velocity, thus higher thermal slugging velocity. The decay of the thermal slugging signal is also analyzed. The signal decay is due to the heat exchange between the fluid and the surrounding, as well as between the warmer and cooler fluid within the pipe. We observe a faster signal decay associated with a higher bubble rate, which indicates a faster heat exchange rate with the existence of gas bubbles. Multiple physical processes may cause this correlation. First, as the gas bubbles travel through the water with a higher velocity, they generate local turbulence in the water phase and accelerate the heat exchange within the water. Another possibility is that the gas, although with much smaller thermal capacity, carries heat from the warmer section to the cooler section, therefore accelerating the thermal equilibrium process. The comparison between the single-phase water and two-phase air-water experimental results indicate that the gas bubbles generate acoustic energy as they move through the pipe. Even in the low-frequency DAS data band (<0.5 Hz), it appears that the higher background noise-level is associated with the rising bubbles. Detailed analysis of the DAS data indicates individual bubbles can be traceable if they are separated more than a gauge length.

Investigation of pressure wave propagation and attenuation characteristics in wellbore gas-liquid two-phase flow

Theoretical Study of Flow Phenomena in the Constricted Tubes

The effect of free gas on the Electrical Submersible Pump (ESP) performance is well known. At a constant rotational speed and constant liquid flow rate, small amount of gas causes a mild head reduction when compared to the single phase liquid head. However, at higher gas rates, a drastic reduction in the head is observed. This critical condition, known as surging point, is a combination of liquid and gas flow rates that cause a maximum in the head performance curve. The first derivative of the head with respect to the liquid flow rate change sign as the liquid flow rate crosses the surging point. In several works on ESP two-phase flow performance, production conditions to the left of the surging region are described or reported as unstable operational conditions.This paper reviews basic concepts on stability of dynamical systems and shows through simulation that ESP oscillatory behavior may result from two-phase flow conditions. A specific drift flux computation code was developed to simulate the dynamic behavior of ESP wells producing without packer. The code is also able to capture the classical casing-heading phenomenon in natural flowing wells.

The main goal of the present work is to establish the tools of the analysis and performance prediction of an axial pump stage under two-phase flow presence of liquid and gas. This knowledge is very important for different applications, for example in the oil industry. The transport of two-phase flow (oil and gas) that comes from the well implies the utilization of separation and treatment facilities before they are pumping. It means that a lot of economical resources are involved in this kind of industrial operation. Therefore, depending on the function optimization of this type of two-phase pumps, it would permit the substitution of the traditional expensive facilities in addition with energy cost savings. In order to predict the fluid dynamics characteristics of an axial pump stage under two-phase flow conditions and in view to improve its performance, the present research will describe a multi-fluid model in order to solve the momentum equations (Navier-Stokes) coupled with the continuity equation. Here, we will take into account the viscosity of the liquid phase and the compressibility of the gas phase, using the CFD simulator: CFX 4.0. Finally, an experimental facility was designed and built to test one axial pump of two stages. Therefore, experimental data is shown in order to validate the previous numerical results obtained.

This study examines the displacement of non-Newtonian fluid invasion in naturally fractured reservoirs (NFR) by employing the evaluated diffusivity solution proposed by Keshavarz, M., & Moreno, R. B. Z. L. (2023)1,for a comprehensive quantitative analysis. The solution is capable of considering not only drilling fluid pseudo-plasticity throughout the fractured but also the matrix system under constant wellbore pressure assumptions. The effects of non-Newtonian fluid parameters and NFR characteristics on fluid flow, leak-off phenomenon, and cumulative volume are examined. The developed solution generates type-curves for quantitative analysis, it validated by field data of loss measurements in a fractured well in the Gulf of Mexico. Parametric analysis reveals that drilling fluid pseudo-plasticity significantly impacts cumulative volume and ROI (radius of invasion) in NFRs with higher differential pressure, leak-off (particularly at earlier times), and larger fracture apertures. Additionally, the procedure allows operators to determine the ROI for equivalent starting and ending times of the transient period. The proposed solution, applicable to well-completion, enhanced oil recovery (EOR) processes, NFR characterization, and drilling operations, facilitates quantitative analysis of reservoir properties and cumulative volume in NFRs. It also assists operators in optimizing drilling fluid rheology in each period, reducing mud loss or enhancing volumetric sweep efficiency in polymer flooding cases. INTRODUCTION Non-Newtonian fluid flow through naturally fractured reservoirs (NFR) has received considerable attention over the last decades in both experimental and modeling research (Yi, 2004; Adenuga et al., 2019), as it controls important technological processes in petroleum and geotechnical engineering. The viscosity of such fluids has a non-linear relationship between shear stress and shear rate. Polymer solutions, heavy (waxy) crude oil, and drilling fluid are common examples of such fluids in petroleum; thus, they have been used for many years as fracturing agents, drilling fluids, and in some enhanced oil recovery processes (Ikoku & Ramey, 1979; Odeh & Yang, 1979; Garcia-Pastrana et al., 2017; Elkatatny et al., 2020; Albattat & Hoteit, 2021; Albattat et al., 2022; Pang et al., 2022). Even though their application is not novel, the subject of transient flow of non-Newtonian fluids in NFR is critical for improving the accuracy of total cumulative volume measurement and well testing (Ikoku & Ramey, 1979; Escobar et al., 2011; Dokhani et al., 2020). When fluids, flow through porous rocks, pressure behavior is affected by fluid rheology, reservoir architecture, reservoir type, and wellbore condition. NFR has been idealized as double porosity reservoir. The governing equation of the fluid for this type of reservoir is modeled considering flow regime and the geometry of the reservoir. While well test analysis involving the flow of Newtonian and non-Newtonian fluids in single and double-porosity reservoirs under wellbore constant rate has been extensively studied (Odeh & Yang, 1979; Da Prat, 1990; Dake, 2001; Escobar et al., 2011), none of these studies considered wellbore condition constant pressure, to measure non-Newtonian fluid rate and cumulative volume. The basic partial differential equation for newtonian fluid flow in NFRs are originally presented and solved under wellbore constant rate by Warren & Root et al (1963). The case was then studied and solved by Da Prat (1990), under constant wellbore producing pressure. Ikoku & Ramey et al. (1979) proposed a model characterizing homogeneous reservoirs with presence of a non-Newtonian (pseudoplastic) fluid. Olarewaju et al. (1992), was the first to develop an analytical solution for the transient behavior of dual-porosity formations containing a non-Newtonian pseudoplastic fluid. More than a decade later, Escobar et al. (2011) developed and successfully tested a methodology (based on solution presented by Olarewaju et al. 1992) for the interpretation of pressure tests in heterogeneous formations bearing a non-Newtonian fluid, based on the pressure and pressure derivative plot; as a result, they presented new equations for estimation of NFR parameters, which was later modified by Garcia-Pastrana et al (2017). Therefore, the focus of this research is on developing the applicability of evaluated solution presented by Keshavarz, M., & Moreno, R. B. Z. L. (2023) for the diffusivity model, which is solved under wellbore constant pressure and is applicable for cumulative fluid measurement. This evaluated solution has been applied as reference model due to considering dimensionless matrix contribution in pseudosteady-state inter-porosity transfer as the assumption of the dual porosity model. To the best of the Author's knowledge, no study has utilized an evaluated non-Newtonian dual porosity model to measure cumulative volume through NFR while taking leak-off into account. In this regard, evaluated dimensionless rate and cumulative volume solutions in Laplace domain were applied for radial flow of non-Newtonian pseudoplastic fluids in infinite dual-porosity reservoirs. The solutions were numerically inverted by Stehfest et al (1970) algorithm, followed by the formulation of dimensionless terms and generation of type curves to facilitate quantification of parameters in real time.

Two-phase cross flow over heat exchanger tubes creates vibrations which contribute greatly to the wear on the tubes. Fluidelastic instability is a major mechanism by which tubes can fail. In this work, the fluidelastic instability of a tube placed in an array subjected to two-phase cross flow has been studied. For the determination of fluidelastic instability, a triangular tube array was used. The tubes were made of acrylic and were 2.2 cm or 2.37 cm in diameter and 20 cm in length. Eighteen tubes and 4 half tubes formed 5 rows with a pitch to diameter ratio of 1.4. All of the tubes except the test tube were rigidly supported at the text section wall. The test tube was flexibly supported with two cantilever beams. By installing cantilever beams horizontally and vertically, drag and lift direction tube vibration were studied. Parameters of tube mass, structural stiffness, natural frequency, and pitch to diameter ratio were varied. The drag coefficients on a rigidly held tube in an array subjected to two-phase cross flow were measured. The tube in an array was located at displaced positions as well as at the normal position in order to study the variation of fluid force as the tube vibrates. In the experiments, gap Reynolds numbers up to 1 x 105 were obtained, while void fraction was varied from zero to 0.5. The drag coefficients in two-phase flow are much higher than those in single phase flow. The ratio of two-phase to single phase drag coefficient decreases as Reynolds number increases. The drag coefficient on a tube in an array increases as the tube is displaced in the direction of flow. The drag coefficient increases rapidly when the tube is displaced more than a certain critical distance.