Mechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition Prediction

Mechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition PredictionMechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition Prediction

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY FALCUTY OF GEOLOGY AND PETROLEUM ENGINEERING DEPARTMENT OF DRILLING AND PRODUCING PETROLEUM ENGINEERING OFFICE OF INTERNATIONAL STUDY PROGRAM A thesis submitted in accordance with the requirement for the degree of BACHELOR OF ENGINEERING (Petroleum Engineering) Mechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition Prediction INSTRUCTOR: Dr MAI CAO LAN Mr NGUYEN VIET VAN STUDENT’S NAME: VO NHAT LINH CLASS: CC17DK11 STUDENT ID: 1652347 HO CHI MINH CITY September, 2021 SOCIALIST REPUBLIC OF VIETNAM Independence – Freedom – Happiness HO CHI MINH UNIVERSITY OF TECHNOLOGY Faculty of Geology and Petroleum Engineering Department of Drilling and Producing Petroleum Engineering No _/BKĐT FINAL YEAR PROJECT PROPOSAL (This form must be appeared at the first page of the final report of the final year project) Faculty: Geology and Petroleum Engineering Department: Drilling and Producing Petroleum Engineering Student name: VÕ NHẬT LINH Student ID: 1652347 Program: Petroleum Engineering Class: CC17DK11 Topic: “Mechanistic Modeling of Multiphase Flow in Subsea Pipeline of Oil Field X for Wax Deposition Prediction” Expected outcome: • • • • Definition and characteristic of multiphase flow in pipeline Determining criteria for evaluation, selection of pipeline diameter Mathematical calculating the thickness of insulation Establishing model and principle for calculating the wax deposition in pipeline Start date: End date: 01/03/2021 18/09/2021 Adviser Affiliation Responsibility Dr Mai Cao Lan Department of Drilling and Production Petroleum Engineering 100% Mr Nguyen Viet Van Hoang Long – Hoan Vu Joint Operating Company 100% The proposal has been screened by the Head/Deputy Head of the Department Ho Chi Minh City, ……………………… 2021 Head of Department FOR OFFICIAL USE ONLY Faculty: Department: Date of defense: Evaluation grade: Archive place: Advisor Advisor Ph.D Mai Cao Lan M.Sc Nguyen Viet Van ACKNOWNLEDGE I wish to express sincere appreciation to Dr Mai Cao Lan for allowing to pursue this thesis in the “Mechanistic Modeling and Wax Deposition in Multiphase Transportation Pipeline” I am extremely thankful for his personal guidance, assistant and supervision I am most grateful to M.Sc Nguyen Viet Van and his friendly colleague in Hoan Long – Hoang Vu JOC for their unwavering support and understanding throughout this work Their continued support and understanding were instrumental to the success of this wok and greatly appreciated Furthermore, I will like to show gratitude to all petroleum engineers in “Tơi Người Dầu Khí” community for their advice throughout the evaluation and correction for calculation tools Additionally, I will like to show gratitude to Nguyen Huu Nhan, Nguyen Hoai Vu and Nga Duong for their personal guidance and advices over the course of study Finally, my warmest thanks go to my family, who constantly supported me during five years i DECLARATION I declare that the thesis has been composed by myself The thesis is submitted for examination in consideration of graduation of bachelor degree of petroleum engineer in Ho Chi Minh City University of Technology Furthermore, I took reasonable care to ensure that the work is original, and to the best of my knowledge, and has not been taken from other sources except where such work, investigated data have been cited and acknowledge within the text ii ABSTRACT Oil and gas are currently produced through wells and pipelines The far reservoirs are, the restricted in production is Transportation of hydrocarbon components tens of kilometers away or even hundreds of kilometers would rise varieties of situation that a production engineer must concerned Those restrictions may happen in a reservoir, in a production tubing or gathering pipeline One of the main issues is prevent paraffin wax deposition in pipe walls which causes the instabilities, interrupted supply of hydrocarbons to processing platforms and damages of downstream facilities Therefore, it is necessary to develop a flow-assurance model to analyze and calculate optimal conditions for a stable production of gas and hydrocarbon liquids with water in which pressure and temperature profiles are two key factors There are two ways in which are commonly used They are 1) experimentally, through laboratory-sized investigation with appropriate instrumentation to address relative method applied for specific cases (normally called empirical correlation methods) 2) and theoretically, using mathematical equations and models for the flow which is known as mechanistic models Poettmann and Carpenter’s method which is a semi-empirical correlation are quietly used because its reliable and is generally accepted However, a limited range of flow rates and gas-liquid ratios have made this method less used in many cases Furthermore, many empirical models exhibit large discontinuities at flow pattern transitions which may cause convergence problem when are used for the simulation of practical cases Mechanistic models, on the other hands, have become a perfect adjustment due to using mathematical equations based on fundamental laws which can be applied in full range of data with more accurate OLGA, a dynamic multiphase flow simulator, is one of well-known software which is widely used in oil and gas field And in this thesis, a computation program developed in VBA (a coding method created by Microsoft) applied those impressive models to predict the pressure and temperature distribution in seabed pipeline The program was compared with OLGA to correct the workflow and check its validation before it is applied in practical cases In 2014, Thap Minh Thu introduced a model simulating oil and gas transportation pipeline from W2-WHP to X1CPP who applied on fundamental theory of flow assurance and OLGA to propose a suitable inner diameter, insulation materials and analyze factors relating to flow-assurance problems However, Thap Minh Thu only used OLGA to develop model, therefore, a gap between applicability of theorem proposed and practicality was created which might not well understand the fundamental background of OLGA software Hence, in order to overcome this disadvantage, this thesis proposes a comprehensive model based on fundamental theorem for predicting pressure and temperature profiles To this, VBA, a programing language for Excel is used due to its simple and ease of access The program is then compared with OLGA’s sample results to verify the model which would be used to apply in a practical case collected from Thap Minh Thu’s master thesis and shows an impressive accurate Wax deposition problem is one of critical issues that Thap Minh Thu did not mention in his Master thesis To consolidate his contribution, the OLGA software is then used to investigate wax deposition in X field pipelines system by using wax deposition module and his data set The task of this section is determining the paraffin wax thickness consider the maximum allowed inlet pressure to propose a necessary time of pigging operation to prevent blockages All necessary data such as pressure and temperature profiles, wax deposition rate and effects of surrounding environment are also a primary concern to confirm the objective of this section iii TABLE OF CONTENTS ACKNOWNLEDGE i DECLARATION ii ABSTRACT iii TABLE OF CONTENTS iv TABLE OF FIGURES vi TABLE OF TABLES vii INTRODUCTION 1 Background Problem Statement Purpose and Scope Thesis Organization CHAPTER LITERATURE REVIEW Motivation of Study Relevant Researches CHAPTER FUNDAMENTAL BACKGROUND 2.1 PVT Properties Of Oil And Gas 2.1.1 Pseudo-Critical Quantities 2.1.2 Direct Calculation of Compressibility Factor 2.1.3 Gas Density 2.1.4 Gas Formation Volume Factor 10 2.1.5 Gas Viscosity 10 2.1.6 The Bubble-point Pressure 11 2.1.7 Oil Formation Volume Factor 12 2.1.8 Isothermal Compressibility Coefficient of Crude Oil 13 2.1.9 Oil Density 14 2.1.10 Oil Viscosity 14 2.2 Multiphase-Flow In Subsea Pipeline 15 2.2.1 The Main Parameter of Multiphase-Flow 15 2.2.2 Flow Regimes in Pipeline 16 2.2.3 Pressure Drop Along Multiphase Flow Pipeline 17 2.2.4 Mechanistic Model for Predicting Flow Regimes and Pressure Distribution 20 2.2.5 Heat Transfer in Pipes 36 2.2.6 Temperature Prediction Along Multiphase-Flow Pipeline 40 2.3 Basic Concepts of Wax Deposition and Wax Deposition Mechanisms 41 2.3.1 Cloud Point or Wax Appearance Temperature (WAT) 41 2.3.2 Pour Point 41 iv TABLE OF CONTENTS 2.3.3 Mechanisms of Wax Deposition 41 CHAPTER PREDICTING TEMPERATURE AND PRESSURE FOR SEABED PIPELINE IN X FIELD 43 3.1 Computational Workflow for Pressure and Temperature Prediction 43 3.2 Validation of Computational Workflow 45 3.2.1 Preparation 45 3.2.2 Results and Discussions 47 3.3 Application of The Computational Workflow for The Gathering Pipeline in Field X 49 3.3.1 Background 49 3.3.2 Data Preparation 49 3.3.3 Results and Discussion 53 CHAPTER WAX DEPOSITION MODELING FOR SEABED PIPELINE IN X FIELD 57 4.1 Factors Affecting The Wax Deposition 57 4.1.1 Temperature Difference And Cooling Rate 57 4.1.2 Crude Oil Composition 57 4.1.3 Flow Rate 57 4.1.4 Pressure 58 4.1.5 Pipe Surface Properties 58 4.2 Wax Control Strategies For The Field 58 4.3 Computational Workflow for Deposited Wax Prediction 59 4.4 Application of OLGA for Wax Thickness Prediction in The Gathering Pipeline, Field X 62 4.4.1 Construction of OLGA Wax Module for Wax Thickness Prediction 62 4.4.2 Methodology of Wax Deposition Simulation 62 4.4.2 Results and Discussions 63 CONCLUSIONS AND RECOMMENDATIONS 66 Conclusions 66 Recommendations 66 NOMENCLATURE 67 REFERENCE 69 APPENDIX 71 Appendix A Newton-Raphson Iterative Procedure and the Existence of the Solution For Slug Flow Model 71 Appendix B Stratified Flow Model 72 Appendix C The Derivation Of Equations For Temperature Prediction 75 v TABLE OF FIGURES Figure 2-1 Gas-solubility pressure diagram [24] 12 Figure 2-2 Oil formation volume factor versus pressure [24] 12 Figure 2-3 Major flow patterns in horizontal flow [12] 17 Figure 2-4 Major flow patterns in vertical flow [13] 17 Figure 2-5 Control volume and relevant variable describe a system for flow through a pipe section [9] 19 Figure 2-6 Workflow of flow regime prediction 25 Figure 2-7 Physical model for bubble flow [11] 27 Figure 2-8 Equilibrium stratified flow 29 Figure 2-9 Physical model for slug flow [11] 33 Figure 2-10 Physical model for annular flow [11] 35 Figure 2-11 A cylinder with conduction surface condition [18] 36 Figure 2-12 Composite hollow cylinder with convection both surface: a) Temperature distribution and b) Equivalent thermal circuit [18] 39 Figure 3-1 An illustration of computational workflow presented in Figure 3-2 in which the pressure and temperature is calculated in each incremental length of pipe 43 Figure 3-2 The flow chart for the pressure and temperature prediction 44 Figure 3-3 Flow chart for inner diameter selection when pressure and temperature is predicted 45 Figure 3-4 Pipeline profile from OLGA_sample_case 46 Figure 3-5 Temperature distribution predicted by OLGA and computational program along 400-m pipeline 48 Figure 3-6 Pressure distribution predicted by OLGA and computational program along 400-m pipeline 48 Figure 3-7 Pipeline Profile between X2-WHP to X1-CPP, X field 51 Figure 3-8 A PT phase diagram of hydrocarbon components generated by Multiflash software 51 Figure 3-9 Temperature distribution comparison between TMT’s data and calculation 54 Figure 3-10 Pressure distribution comparison between TMT’s data and calculation 54 Figure 4-1 Summary of general wax deposition and control methodologies 58 Figure 4-2 The flow chart for wax thickness prediction in seabed pipeline 61 Figure 4-3 An illustration of Figure 4-2 about application of wax deposition model to calculate the wax thickness in seabed pipeline over a period of time ∆𝑡 62 Figure 4-4 Schematic flow line diagram in the OLGA software 63 Figure 4-5 Pressure profiles in pipeline after 50 days in X field’s pipeline 63 Figure 4-6 Temperature profiles in X field’s pipeline after 50 days 64 Figure 4-7 Thickness of wax deposition in X field’s pipeline after 50 days 64 Figure 4-8 Deposited wax thickness after 50 days in PIPE-50 and PIPE-51 section, X field’s pipeline 65 Figure 4-9 Temperature at the pipe wall of seabed pipeline in X field 65 vi TABLE OF TABLES Table 2-1 Important Dimensionless Group in convection heat transfer 37 Table 2-2 Constants for the Hilpert Correlation for Circular (Pr≥0.7) and Noncircular (Gases only) Cylinders 38 Table 3-1 Bathymetry data for the production pipeline from OLGA_sample_case 46 Table 3-2 Input parameter for OLGA_sample_case 46 Table 3-3 Pipeline details for OLGA_sample_case 47 Table 3-4 Material properties for OLGA_sample_case 47 Table 3-5 Material/Coating thickness for OLGA_sample_case 47 Table 3-6 Statistical analysis of results compared between OLGA_sample_case and computational program 47 Table 3-7 Major programming verification activities 48 Table 3-8 Bathymetry Data for the Production Pipeline from the X2-WHP to the X1-CPP, X field 49 Table 3-9 Hydrocarbon components of oil and gas gathering in seabed pipeline, X field 51 Table 3-10 Critical parameters which require for gathering oil and gas in pipeline, X field 52 Table 3-11 Pipeline details to gathering oil and gas in X field 52 Table 3-12 Material properties of X field’s pipeline 52 Table 3-13 Material/Coating thickness of X field’s pipeline 53 Table 3-14 Seawater temperature - Surface 53 Table 3-15 Seawater temperature - Seabed 53 Table 3-16 Air temperature 53 Table 3-17 Air velocity 53 Table 3-18 Other parameters 53 Table 3-19 Production data 53 Table 3-20 Results of temperature distribution prediction calculated by Thap Minh Thu and computational program in pipeline from X2-WHP to X1-CPP 54 Table 3-21 Results of pressure distribution prediction calculated by Thap Minh Thu and computational program in pipeline from X2-WHP to X1-CPP 55 Table 4-1 𝑁𝑆𝑅 expressed for each flow pattern proposed in Matzain model 59 vii INTRODUCTION BACKGROUND Over recent decades, large amounts of oil and gas are found in extreme conditions as they are buried at thousand meters deep with special reservoir characteristics This situation raises new challenges within the field of petroleum exploration, production, and transportation Low reservoir pressure, low seabed temperature, or far from shoreland are some of the field challenges In Vietnam, the substantial development of the petroleum industry has been made a great contribution to the construction of the country Many reservoirs with large potential reserves have been producing and exploring Besides, some small reservoirs are also considered and produced but they are hard to be developed separately due to small scale and low recovery enhance In these cases, therefore, the development option by connecting with the existing reservoir is often chosen because of the possibility of higher economic efficiency than the independent development option PROBLEM STATEMENT The main objective in the transportation of oil and gas is to ensure efficient production, profitability, and safety One of the biggest challenges preventing those factors which occur flow instabilities in the seabed pipeline system is wax deposition on the tube walls due to heat transfer from the environment Hence, it is very important to understand the flow behaviors, fluid properties as well as in-situ performance in the intra-reservoir area, and connection between neighboring reservoirs to propose a proper production plan To achieve that, a thorough interpretation of current system conditions such as pressure and temperature is regularly evaluated Moreover, modeling wax deposition is a complex task that involves several disciplines, such as, thermodynamics, phase equilibrium, mass/heat transfer, and fluid mechanics The precipitation and deposition rate can significantly influence to stabilization of the whole system and the economy of the field because operational and remedial costs are increased on account of product reduction Therefore, the ability to accurately predict wax deposition is an invaluable tool that would help in the design stages of the pipeline, as well as, in the scheduling of interventions PURPOSE AND SCOPE Successful development of the seabed hydrocarbon transportation project requires engineering designs of the facilities that can handle the flow assurance challenges such as wax deposit problems A robust can only be obtained with an accurate prediction of pressure and temperature, which are two factors causing wax precipitation To accomplish this, physical approaches are used to predict the drop of pressure and temperature distribution in multiphase flow along the pipeline and defines potential wax problems, and estimates the pigging frequency In this thesis, mechanistic models for multiphase flow are used to predict pressure drop and flow behavior in pipes Those models are based on Petalas and Aziz’s method [20] which are proved that can take into account a wide range of data and all types of fluids and conditions where exhibit large discontinuities that empirical models often show their limits On the other hand, it was found that their mechanistic model for predicting flow patterns could improve the capability of calculating pressure gradients along the pipeline These new empirical correlations were applied to all fluid properties and pipe geometries Alves et al [2] proposed a method that improved Ramey’s model to predict the flowing temperature distribution in pipes This method shows a closure relationship of pressure and temperature in multiphase flow which takes into account thermodynamic laws and the Joule-Thomson effect and adapts changes in well or pipeline deviation and variable thermal properties The use of unified equations could obtain the entire range of inclination angles make this method applicable in pipelines and wellbores In the second part of the thesis, the main issue of transporting gas and hydrocarbon fluids paraffin components Matzain develop a wax deposition model based on mechanisms that are dominant factors for the accumulation of waxy crystal This model shows that wax deposition depends on flow patterns which the build-up trends for deposition are similar to the observation of flow regimes The Thesis Objectives CHAPTER WAX DEPOSITION MODELING FOR SEABED PIPELINE IN X FIELD 85 10 Days Temperature (C) 20 Days 80 30 Days 40 Days 50 Days 75 70 65 1000 2000 3000 4000 5000 6000 7000 8000 Length (m) Figure 4-6 Temperature profiles in X field’s pipeline after 50 days Deposition occurs at end of the pipeline results from both dissolved and precipitated waxy crystals transported by the fluid Molecular diffusion is a dominant factor at the high temperature and heat flux whereas shear dispersion is at lower temperatures and low heat rates The temperature drop below WAT is followed by an increase in the wax deposition rate Therefore, the paraffin wax tends to deposit at the end of the pipeline where the size distribution of the precipitated waxy particles increases with decreasing temperature In addition, the formation of slugs at the riser makes the fluid cannot be continuously produced out and cause a chaotic medium at the perpendicular angle of the pipeline that peels off any waxy crystals precipitated 90 10 Days Wax thickness (mm) 80 20 Days 70 30 Days 60 40 Days 50 50 Days 40 30 20 10 0 1000 2000 3000 4000 5000 6000 7000 8000 Length (m) Figure 4-7 Thickness of wax deposition in X field’s pipeline after 50 days Figure 4-8 and Figure 4-9 show how waxy particles deposit in PIPE-50 and PIPE-51 sections after 50 days The wax thickness is zero at the beginning On the day of 30, the thickness of wax in the PIPE-50 section is equal to a half inch which reduces the inner diameter of this pipeline section to in However, this trend rapidly changes to an increase in thickness After 40 days, the thickness of deposited wax takes into account over 30% of inner diameter which causes the backpressure at WHP to be about 242.7 psia, just less than psia of recommendation The temperature distribution along the wall pipeline is one of the critical factors which cause wax deposition becomes a primary flow assurance problem The lower the temperature at the pipe wall is, the faster the wax precipitation rate is which results in the steep slopes in the wax thickness, as shown in Figure 4-8 64 CHAPTER WAX DEPOSITION MODELING FOR SEABED PIPELINE IN X FIELD PIPE-50 100 Wax thickness (mm) PIPE-51 80 60 40 20 0 10 20 30 40 50 Time (day) Figure 4-8 Deposited wax thickness after 50 days in PIPE -50 and PIPE-51 section, X field’s pipeline 70 Temperature (C) 60 50 40 30 WAT 20 Inner Wall Surface of PIPE-50 10 Inner Wall Surface of PIPE-51 0 10 15 20 25 30 35 40 45 50 Time (day) Figure 4-9 Temperature at the pipe wall of seabed pipeline in X field Flow assurance aims to make sure gas and hydrocarbon fluids keep flowing under reliable parameters In this thesis, according to the dataset collected from Thap Minh Thu’s thesis, there are two criteria could be selected to propose an intervention method The first one is the temperature distribution alone the pipe walls The second one is the minimum pressure at the downstream After 30 days, the pressure at WHP must be 250 psia to transport fluid to the CPP However, this high pressure would cause damage to the upstream facilities, such as pumps or valves Moreover, the slope of temperature distribution at the pipe walls increases after 30 days that leads to increase in wax deposition rate Therefore, in case of X field’s pipeline that Thap Minh Thu has been studied, in order to prevent any damage to upstream equipment and avoid pig getting stuck when deposited wax is too thick, a scheduled pigging program is about 30 days to optimize the production 65 CONCLUSIONS AND RECOMMENDATIONS CONCLUSIONS The application of the mechanistic model is a complex task that involves several disciplines, such as thermodynamics, fundamental laws of conservation, heat transfer, and fluid mechanics However, a comprehensive workflow of mechanistic models in predicting pressure and temperature has been developed with acceptable results that could promise in its application to other cases To be able to develop workflows for VBA coding, a comprehensive procedure has been reported which is divided into two main sections: 1) A mechanistic model for gas and hydrocarbon fluid has been summarized 2) The results were compared with OLGA’s sample cases to verify and validate the computation methods Once the correcting process is completed, the data related to the X field was collected from Thap Minh Thu’s Master Thesis to predict the pressure and temperature distribution in a gathering seabed pipeline from X2-WHP to X1-CPP, X field A wax deposition model was also simulated to propose a schedule of interventions The calculation and simulation results are summarized as follows: 1) Pressure and temperature profiles: During transporting gas and hydrocarbon fluid with water, it is important to keep the pressure at the outlet under desirable values to avoid severe damages to downstream facilities Hence, the backpressure from WHP is investigated by setting operating pressure to enter X1-CPP The results presented in Table 3‑20 and Table 3‑21 showed acceptable error values at both pressure and temperature prediction The pressure at the X2-WHP is about 217 psia and the temperature at X1-CPP is 70.6℃ which satisfy the design requirement 2) Wax deposition investigation: Wax deposition in pipelines is one of the most critical problems faced by the petroleum industry in offshore operations Significant losses of capital have been caused by wax deposition problems, due to the high costs of prevention, reduced production, and increased pumping power The ability to accurately predict wax deposition is an invaluable tool that would help in the design stages of the pipeline, as well as, in the scheduling of interventions Therefore, in the Thap Minh Thu’s investigated case, the pigging frequency is 30 days based upon wax deposition modeling which is created by OLGA to avoid damage for producing and gathering facilities and optimizing the efficiency RECOMMENDATIONS The impressive results calculated from EXCEL VBA flatform show potential consideration of other subsea pipeline systems Moreover, the convenience and applicability of EXCEL allow students to study and investigate the flow behavior in wells and pipelines Pigging is a common way to remove deposited paraffin wax from gathering and transportation pipelines It requires a careful operation in order to not break the pigs or not to make them stuck In addition, the production system has to shut down during a periodic pigging operation which may not be a cost-effective method if the operation cost is high Hence, the pigging is the third most common method of wax management, after pipe insulation and using chemical inhibitors such as PPD Flow assurance is an inclusive definition of operation, safeguard, optimization, and analysis that engineers will face Formation of gas hydrate, inorganic scales, internal corrosion, and severe slug are other challenges during the design of pipeline and transportation systems to assure a safety and economically transportation 66 NOMENCLATURE - Gas solubility, 𝑠𝑐𝑓/𝑆𝑇𝐵 Cross-section area, 𝑚2 or 𝑓𝑡 The American Petroleum Institute gravity Formation volume factor, 𝑏𝑏𝑙/𝑆𝐶𝐹 or bbl/STB Compressibility of oil, 𝑝𝑠𝑖 −1 - Froude number Enthalpy, 𝐽/𝑘𝑔 Liquid holdup Inner diameter, 𝑚 or 𝑓𝑡 or 𝑖𝑛 Length or a pipe segment, 𝑚 or 𝑓𝑡 Molecular weight, 𝑔/𝑚𝑜𝑙 or 𝑙𝑏𝑚/𝑚𝑜𝑙 Nusselt number Outer diameter, 𝑚 or 𝑓𝑡 or 𝑖𝑛 Pressure, 𝑝𝑠𝑖 or 𝑃𝑎 Prandtl number 𝑄 - 𝑅 - 𝑅𝑠 𝐴 𝐴𝑃𝐼 𝐵 𝑐𝑜 𝑐𝑝 𝐹𝑟 ℎ ℎ 𝐻 𝐼𝐷 𝐿 𝑀𝑊 𝑁𝑢, 𝑁𝑁𝑢 𝑂𝐷 𝑃 𝑃𝑟, 𝑁𝑃𝑟 𝐽 Heat capacity, 𝑘𝑔 𝐾 𝑊 𝐵𝑇𝑈 Heat transfer coefficient, 𝑚2 ℃ or ℎ𝑟 𝑓𝑡 ℉ Volumetric flow rate, 𝑚3 or 𝑓𝑡 𝑠 𝑠 𝑃𝑎 𝑚3 𝑝𝑠𝑖 𝑓𝑡 𝑔𝑚𝑜𝑙 𝐾 𝑙𝑏𝑚𝑜𝑙 °𝑅 The ideal gas constant, or - Thermal Resistance, 𝐾/𝑊 Reynolds number Perimeter, 𝑚 or 𝑓𝑡 Temperature, ℃ or ℉ Volume, 𝑚3 or 𝑓𝑡 Gas compressibility factor Diameter, 𝑚 or 𝑓𝑡 Energy, 𝐽𝑜𝑢𝑙𝑒 Moody’s friction factor Gravity acceleration, 𝑚/𝑠 or 𝑓𝑡/𝑠 𝑘 - 𝑚 𝑛 𝑞 𝑅𝑠 𝑟, 𝑅 𝑣 𝑤 𝑥 - Weight, 𝑘𝑔 or 𝑙𝑏𝑚 Mole Rate of heat transfer, 𝐽/𝑠 or 𝑊 or 𝐵𝑇𝑈/ℎ𝑟 Gas solubility, 𝑠𝑐𝑓/𝑆𝑇𝐵 Radius, 𝑚 or 𝑓𝑡 Velocity, 𝑚/𝑠 or 𝑓𝑡/𝑠 Rate of mass, 𝑘𝑔/𝑠 or 𝑙𝑏𝑚/𝑠 Mole fraction 𝑅 𝑅𝑒, 𝑁𝑅𝑒 𝑆 𝑇 𝑈 𝑉 𝑍, 𝑧 𝑑, 𝐷 𝑒 𝑓 𝑔 Overall heat transfer coefficient, 𝑊 𝑊 𝑚2 ℃ or 𝐵𝑇𝑈 ℎ𝑟 𝑓𝑡 ℉ 𝐵𝑇𝑈 Thermal conductivity, 𝑚 𝐾 or ℎ𝑟 𝑓𝑡 ℉ Greek Letters 𝛼 𝛼 𝛽 𝛾 𝛿 𝜀 - Thermal diffusivity, 𝑚2 /𝑠 or 𝑓𝑡 /ℎ𝑟 Gas fraction Ratio of length Specific gravity or Central angle, 𝑑𝑒𝑔𝑟𝑒𝑒 Thickness, 𝑚𝑚 Relative roughness, 𝑚𝑚 67 NOMENCLATURE 𝜂 𝜃 𝜆 𝜇 𝜌 𝜎 𝜏 - Joule-Thompson coefficient Angle of inclination, 𝑑𝑒𝑔𝑟𝑒𝑒 Volume fraction Viscosity, 𝑃𝑎 𝑠 or 𝑐𝑃 Density, 𝑘𝑔/𝑚3 or 𝑙𝑏𝑚/𝑓𝑡 Surface tension, 𝑁/𝑚 Shear stress, 𝑃𝑎 or 𝑝𝑠𝑖 Subscripts 𝑎 𝑏 𝑐 𝑐𝑜𝑛𝑑 𝑐𝑜𝑛𝑣 𝑒 𝑒 𝑓 𝐹 𝑔 𝑖 𝑖 𝐿 𝐿𝑆 𝑚 𝑛𝑠𝐿 𝑛𝑠𝑔 𝑂𝐷 𝑝𝑐 𝑝𝑟 𝑠 𝑠𝑐 𝑆𝐶 𝑆𝐿 𝑆𝑈 𝑆𝑔 𝑇𝐵 𝑤 - Actual condition At bubble point Core Conduction Convection Elevation Environment Friction Film At gas phase Interface Initial At liquid phase Liquid slug in flow Mixture No-slip liquid No-slip gas Dead oil or under saturated oil Pseudo-critical Pseudo-reduced Surface At standard condition Superficial gas core Superficial liquid Slug unit Superficial gas Taylor bubble Wall of pipe 68 REFERENCE [1] T.H Ahmed, Hydrocarbon Phase Behavior, Gulf Pub Co, 1989 [2] I.N Alves, F.J.S Alhanati, O Shoham, A Unified Model for Predicting Flowing Temperature Distribution in Wellbores and Pipelines, SPE Prod Eng (1992) 363–367 [3] A.M Ansari, N.D Sylvester, O Shoham, J.P Brill, A Comprehensive Mechanistic Model for Upward TwoPhase Flow in Wellbores, in: All Days, SPE, 1990 [4] D Barnea, Transition from annular flow and from dispersed bubble flow—unified models for the whole range of pipe 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gasliquid flow, AIChE J 22 (1976) [27] J.Ø Tengesdal, A.S Kaya, C Sarica, Flow-Pattern Transition and Hydrodynamic Modeling of Churn Flow, SPE J (1999) [28] J.O Tengesdal, C Sarica, Z Schmidt, D Doty, A Mechanistic Model for Predicting Pressure Drop in Vertical Upward Two-Phase Flow, J Energy Resour Technol 121 (1999) [29] M.T Thap, Tính tốn đảm bảo dịng chảy q trình vận chuyển sản phẩm khai thác từ giàn X2 - WHP X1-CPP, Ho Chi Minh University of Technology, 2014 [30] R Valinejad, A.R Solaimany Nazar, An experimental design approach for investigating the effects of operating factors on the wax deposition in pipelines, Fuel 106 (2013) 843–850 70 APPENDIX APPENDIX A NEWTON-RAPHSON ITERATIVE PROCEDURE AND THE EXISTENCE OF THE SOLUTION FOR SLUG FLOW MODEL (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 (𝑣𝑔𝐿𝑆 − 𝑣𝐿𝐿𝑆 ) − 𝑣𝐿𝐿𝑆 = ⇔ (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 + 𝐻𝑔𝐿𝑆 𝑣𝐿𝐿𝑆 − 𝑣𝐿𝐿𝑆 = ⇔ (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 − 𝑣𝐿𝐿𝑆 (1 − 𝐻𝑔𝐿𝑆 ) = ⇔ (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 − 𝑣𝑇𝐵 (𝐻𝑔𝑇𝐵 − 𝐻𝑔𝐿𝑆 ) − (1 − 𝐻𝑔𝑇𝐵 )𝑣𝐿𝑇𝐵 (1 − 𝐻𝑔𝐿𝑆 ) (1 − 𝐻𝑔𝐿𝑆 ) = ⇔ (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 − 𝑣𝑇𝐵 (𝐻𝑔𝑇𝐵 − 𝐻𝑔𝐿𝑆 ) + (1 − 𝐻𝑔𝑇𝐵 )𝑣𝐿𝑇𝐵 = ⇔ (𝑣𝑆𝑔 + 𝑣𝑆𝐿 ) − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 + 𝑣𝑇𝐵 𝐻𝑔𝐿𝑆 − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 + (1 − 𝐻𝑔𝑇𝐵 )𝑣𝐿𝑇𝐵 = Assume that 𝐴 = 𝑣𝑚 − 𝐻𝑔𝐿𝑆 𝑣𝑔𝐿𝑆 + 𝑣𝑇𝐵 𝐻𝑔𝐿𝑆 Then the equation is simplified to ⇔ 𝑣𝐿𝑇𝐵 (1 − 𝐻𝑔𝑇𝐵 ) − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 + 𝐴 = ⇔ 9.916√𝑔𝐷 (1 − √𝐻𝑔𝑇𝐵 ) (1 − 𝐻𝑔𝑇𝐵 ) − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 + 𝐴 = The left-hand side of equation is denoted as ⇔ 𝐹(𝐻𝑔𝑇𝐵 ) = 9.916√𝑔𝐷 (1 − √𝐻𝑔𝑇𝐵 ) (1 − 𝐻𝑔𝑇𝐵 ) − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 + 𝐴 Taking derivative of equation with respect to 𝐻𝑔𝑇𝐵 , ′ 2 ′ ′ 𝐹 ′ (𝐻𝑔𝑇𝐵 ) = 9.916√𝑔𝐷 {[(1 − √𝐻𝑔𝑇𝐵 ) ] (1 − 𝐻𝑔𝑇𝐵 ) + (1 − √𝐻𝑔𝑇𝐵 ) (1 − 𝐻𝑔𝑇𝐵 ) } − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 1 − ′ 2 ′ ⇔ 𝐹 (𝐻𝑔𝑇𝐵 ) = 9.916√𝑔𝐷 { (1 − √𝐻𝑔𝑇𝐵 ) (1 − √𝐻𝑔𝑇𝐵 ) (1 − 𝐻𝑔𝑇𝐵 ) + (1 − √𝐻𝑔𝑇𝐵 ) (1 − 𝐻𝑔𝑇𝐵 ) } ′ ′ − 𝑣𝑇𝐵 𝐻𝑔𝑇𝐵 1 − 2 1 ⇔ 𝐹 (𝐻𝑔𝑇𝐵 ) = 9.916√𝑔𝐷 { (1 − √𝐻𝑔𝑇𝐵 ) (− ) (1 − 𝐻𝑔𝑇𝐵 ) + (1 − √𝐻𝑔𝑇𝐵 ) (−1)} − 𝑣𝑇𝐵 2√𝐻𝑔𝑇𝐵 ′ ⇔ 𝐹 ′ (𝐻𝑔𝑇𝐵 ) = 9.916√𝑔𝐷 [ 0.5 (1 − 𝐻𝑔𝑇𝐵 ) + (1 − 𝐻 ) ] + 𝑣𝑇𝐵 √ 𝑔𝑇𝐵 [(1 − √𝐻 )𝐻 ]0.5 𝑔𝑇𝐵 𝑔𝑇𝐵 71 APPENDIX APPENDIX B Stratified Flow Model The dimensionless liquid height, ℎ𝐿 ℎ𝐿 = 𝑅 − 𝑅 cos ⇔ 𝛾 ℎ𝐿 𝛾 = − cos 𝑅 ⇔ ℎ𝐿 = ℎ𝐿 𝛾 = (1 − cos ) 𝐷 2 (B-1) The dimensionless perimeter of liquid phase, 𝑆𝐿 𝛾 𝑆𝐿 𝜋𝐷 − 𝐷 𝛾 𝛾 𝛾 𝑆𝐿 = = = 𝜋 − = 𝜋 − cos−1 (cos ( )) = 𝜋 − cos −1 (2 (1 − cos ) − 1) 𝐷 𝐷 2 2 𝑆𝐿 = 𝜋 − cos−1 (2ℎ𝐿 − 1) (B-2) The dimensionless perimeter of gas phase, 𝑆𝑔 𝛾 𝑆𝑔 𝐷 𝛾 𝛾 𝑆𝑔 = = = cos−1 (cos ( )) = cos −1 (2 (1 − cos ) − 1) 𝐷 𝐷 2 𝑆𝑔 = cos−1 (2ℎ𝐿 − 1) (B-3) The dimensionless interfacial width between gas and liquid phases, 𝑆𝑖 𝑆𝑖 = 𝑆𝑖 𝐷 𝛾 2√𝑅2 (1 − cos2 ) 2𝑅 𝛾 √(1 − cos2 ) ⇔ 𝑆𝑖 = = 𝐷 𝐷 𝛾 𝛾 ⇔ 𝑆𝑖 = 2√ (1 − cos ) (1 + cos ) 2 𝛾 𝛾 ⇔ 𝑆𝑖 = 2√ (1 − cos ) (2 − + cos ) 2 𝛾 𝛾 ⇔ 𝑆𝑖 = 2√ (1 − cos ) − (1 − cos ) 2 2 ⇔ 𝑆𝑖 = 2√ℎ𝐿 − ℎ𝐿 = √4ℎ𝐿 − 4ℎ𝐿 = √1 − (2ℎ𝐿 − 1) 𝑆𝑖 = √1 − (2ℎ𝐿 − 1) The dimensionless hydraulic diameter of liquid, 𝐷𝐿 4𝐴𝐿 𝐴𝐿 𝐴𝐿 𝐷𝐿 𝐸𝐿 𝑆 𝑆𝐿 𝑆𝐿 𝑆 𝐷𝐿 = = = = 𝐿 = 𝐴 𝐷 𝐷 𝑆𝐿 𝜋𝐷 𝑆 𝜋𝐷 72 (B-4) APPENDIX 𝐷𝐿 = 𝐸𝐿 𝑆 𝑆𝐿 (B-5) The dimensionless hydraulic diameter of liquid, 𝐷𝑔 𝜋𝐷 (𝐴 − 𝐴𝐿 ) 4𝐴𝑔 𝐴 𝑆 (𝐴 − 𝐴𝐿 ) 𝑆 (1 − 𝐿 ) 𝑆 ∗ 𝐻𝑔 𝐷𝑔 𝑆𝑔 + 𝑆𝑖 𝐷 (𝐴 − 𝐴𝐿 ) 𝜋𝐷 𝐴 = 𝐴 𝐷𝑔 = = = = = = 𝐷 𝐷 𝑆𝑔 + 𝑆𝑖 𝑆𝑔 + 𝑆𝑖 𝑆𝑔 + 𝑆𝑖 𝑆𝑔 + 𝑆𝑖 𝑆𝑔 + 𝑆𝑖 𝐷𝑔 = 𝑆 ∗ 𝐻𝑔 𝑆𝑔 + 𝑆𝑖 (B-6) The dimensionless cross-section area of liquid phase, 𝐴𝐿 𝐴𝐿 = 𝐴𝐿 = 0.25 [𝜋 − cos−1 (2ℎ𝐿 − 1) + (2ℎ𝐿 − 1) √1 − (2ℎ𝐿 − 1) ] 𝐷 𝐴𝐿 = 0.25 [𝑆𝐿 + (2ℎ𝐿 − 1) 𝑆𝑖 ] (B-7) The dimensionless cross-section area of gas phase, 𝐴𝑔 𝐴𝑔 = 𝐴𝑔 −1 √1 − (2ℎ𝐿 − 1) ] = 0.25 [cos − 1) − − 1) (2ℎ (2ℎ 𝐿 𝐿 𝐷2 𝐴𝑔 = 0.25 [𝑆𝑔 − (2ℎ𝐿 − 1) 𝑆𝑖 ] (B-8) The dimensionless actual velocity of liquid phase, 𝑣𝑔 𝑣𝐿 = 𝑣𝐿 𝑣𝑆𝐿 (B-9) 𝑣𝐿 = 𝐴 𝐴𝐿 (B-10) 𝑣𝐿 = 𝐴 𝐴𝐿 (B-11) 𝑣𝑔 = 𝑣𝑔 𝑣𝑆𝑔 (B-12) 𝑣𝑔 = 𝐴 𝐴𝑔 (B-13) 𝑣𝑔 = 𝐴 𝐴𝑔 (B-14) The dimensionless actual velocity of gas phase, 𝑣𝑔 The liquid volume fraction, 𝐻𝐿 𝑅2 (𝛾 − sin 𝛾) (𝛾 − sin 𝛾) 𝐴𝐿 𝐻𝐿 = = = 𝐴 𝜋𝑅 2𝜋 𝐻𝐿 = (𝛾 − sin 𝛾) 2𝜋 The interfacial width, 𝑆𝑖 73 (B-15) APPENDIX 𝑆𝑖 = 2𝑅 sin 𝛾 (B-16) 𝑆𝑖 = 𝑅√2(1 − 𝛾) (B-17) The central angle, 𝛾 𝛾 𝑆𝑖 sin = 2𝑅 𝛾 = sin−1 ( 𝑆𝑖 ) 2𝑅 (B-18) The dimensionless liquid height ℎ𝐿 is calculated from Equation (B-19) by substituting Equations (2-120) and (2-121) into Equation (2-119) which is −𝜏𝑤𝐿 𝑆𝑔 𝑆𝐿 1 + 𝜏𝑤𝑔 + 𝜏𝑖 𝑆𝑖 ( + ) + 𝑔 sin 𝜃 (𝜌𝐿 − 𝜌𝑔 ) = 𝐴𝐿 𝐴𝑔 𝐴𝐿 𝐴𝑔 74 (B-19) APPENDIX APPENDIX C THE DERIVATION OF EQUATIONS FOR TEMPERATURE PREDICTION The mass balance for the mixture is: 𝑑 (𝜌𝑣) = 𝑑𝐿 (C-1) the momentum balance for the mixture can be express as 𝑑 𝑑𝑃 𝜏𝜋𝑑 (𝜌𝑣 2) = − − 𝜌𝑔 sin 𝜃 − 𝑑𝐿 𝑑𝐿 𝐴𝑝 (C-2) while the energy balance for the mixture is given by 𝑑 𝑑𝑃 𝑞𝜋𝑑 (𝑃𝑣) − 𝜌𝑣𝑔 sin 𝜃 − [𝜌𝑣 (𝑒 + 𝑣 2)] = − 𝑑𝐿 𝑑𝐿 𝐴𝑝 (C-3) Applying the mass balance, the Equations (C-2) and (C-3) are reduced to 𝜌𝑣 ⟺ 𝑑𝑣 𝑑𝑃 𝜏𝜋𝑑 =− − 𝜌𝑔 sin 𝜃 − 𝑑𝐿 𝑑𝐿 𝐴𝑝 𝑑𝑃 𝑑𝑣 𝜏𝜋𝑑 = −𝜌𝑔 sin 𝜃 − 𝜌𝑣 − 𝑑𝐿 𝑑𝐿 𝐴𝑝 (C-4) In the absence of external work, the energy equation of steady-state flow can be expressed as 𝜌𝑣 𝑑 𝑃 𝑑𝑣 𝑞𝜋𝑑 (𝑒 + ) = −𝜌𝑣𝑣 − 𝜌𝑣𝑔 sin 𝜃 − 𝑑𝐿 𝜌 𝑑𝐿 𝐴𝑝 ⇔ 𝑑 𝑃 𝑑𝑣 𝑞𝜋𝑑 (𝑒 + ) = −𝑣 − 𝑔 sin 𝜃 − 𝑑𝐿 𝜌 𝑑𝐿 𝑤 ⟹ 𝑑ℎ 𝑑𝑣 𝑞𝜋𝑑 = −𝑣 − 𝑔 sin 𝜃 − 𝑑𝐿 𝑑𝐿 𝑤 (C-5) where the subscript 𝑝 is the pipeline segment The heat transfer to the surroundings 𝑞 can be determined from the overall heat-transfer coefficient, 𝑈 𝑞 = 𝑈(𝑇 − 𝑇𝑒 ) (C-6) where the overall heat-transfer is related to the hydrocarbon mixture, pipeline configuration and the environment (𝑟𝑛 𝑈)−1 𝑛 ln (𝑟𝑖+1 ) 𝑓(𝑡) 𝑟𝑖 = +∑ + 𝑟1 ℎ𝑓 𝑘𝑖 𝑘𝑒 𝑖=1 −1 𝑛 ln (𝑟𝑖+1 ) 1 𝑓(𝑡) 𝑟𝑖 𝑈= ( +∑ + ) 𝑟𝑛 𝑟1 ℎ𝑓 𝑘𝑖 𝑘𝑒 (C-7) 𝑖=1 The subscript 𝑛 describes the outer most layer and 𝑒 is the environment condition Combining Equations (C-5) and (C-6) yields 𝑑ℎ 𝑑𝑣 𝑈𝜋𝑑 (𝑇 − 𝑇𝑒 ) = −𝑣 − 𝑔 sin 𝜃 − 𝑑𝐿 𝑑𝐿 𝑤 (C-8) or using thermodynamic principles, the specific enthalpy can be expanded as 𝑑ℎ 𝜕ℎ 𝑑𝑃 𝜕ℎ 𝑑𝑇𝑓 =( ) +( ) 𝑑𝐿 𝜕𝑃 𝑇𝑓 𝑑𝐿 𝜕𝑇𝑓 𝑃 𝑑𝐿 where 75 (C-9) APPENDIX ( 𝜕ℎ ) = −𝜂𝑐𝑃 , 𝜕𝑃 𝑇𝑓 (C-10) 𝜕ℎ ) = 𝑐𝑃 𝜕𝑇𝑓 𝑃 (C-11) ( Therefore, the overall enthalpy change in a flowing fluid is 𝑑ℎ 𝑑𝑇 𝑑𝑃 = 𝑐𝑃 − 𝜂𝑐𝑃 𝑑𝐿 𝑑𝐿 𝑑𝐿 (C-12) where 𝑐𝑃 is the specific capacity at a specific pressure and 𝜂 is the Joule-Thompson coefficient From Equation (C-9), the third term of the right-hand side describes the heat loss per unit length It can be expressed as 𝑑𝑞 𝑈𝜋𝑑(𝑇 − 𝑇𝑒 ) = 𝑑𝐿 𝑤 (C-13) Equations (C-9) and (C-12) can be combined to yields 𝑐𝑃 ⇔ 𝑑𝑇 𝑑𝑃 𝑑𝑣 𝑈𝜋𝑑 (𝑇 − 𝑇𝑒 ) − 𝜂𝑐𝑃 = −𝑣 − 𝑔 sin 𝜃 − 𝑑𝐿 𝑑𝐿 𝑑𝐿 𝑤 𝑑𝑇 𝑈𝜋𝑑 𝑈𝜋𝑑 𝑑𝑃 𝑑𝑣 + 𝑇= 𝑇 + (𝜂𝑐𝑃 −𝑣 − 𝑔 sin 𝜃) 𝑑𝐿 𝑤𝑐𝑃 𝑤𝑐𝑃 𝑒 𝑐𝑃 𝑑𝐿 𝑑𝐿 (C-14) Defining 𝐴 as a relaxation distance 𝑎= 𝑤𝑐𝑃 𝑈𝜋𝑑 (C-15) and Φ as a dimensionless parameter Φ= (𝜌𝜂𝑐𝑃 𝑑𝑃 𝑑𝑣 − 𝜌𝑣 − 𝜌𝑔 sin 𝜃) 𝑑𝐿 𝑑𝐿 , 𝑑𝑃 𝑑𝐿 (C-16) If the surrounding temperature varies linearly with depth, 𝑇𝑒 = 𝑇𝑒𝑖 − 𝑔𝑒 𝐿 sin 𝜃 (C-17) 𝑑𝑇 𝑇 𝑇𝑒𝑖 𝑔𝑒 𝐿 sin 𝜃 Φ 𝑑𝑃 + = − + 𝑑𝐿 𝑎 𝑎 𝑎 𝜌𝑐𝑃 𝑑𝐿 (C-18) the Equation (C-14) is reduced to For a certain pipe segment, Equation (C-18) can be integrated by assuming constant values for 𝑈, 𝑐𝑃 , 𝜂, 𝑔𝑒 , 𝜃, 𝑣, and 𝑑𝑃 𝑑𝐿 𝑑𝑣 𝑑𝐿 According to Alves et al [2], the resulting solution is 𝐿 𝐿 𝑇 = (𝑇𝑒𝑖 − 𝑔𝑒 𝐿 sin 𝜃) + (𝑇𝑖 − 𝑇𝑒𝑖 )𝑒 −𝑎 + 𝑎𝑔𝑒 sin 𝜃 [1 − 𝑒 −𝑎 ] + 𝐿 Φ𝑎 𝑑𝑃 [1 − 𝑒 −𝑎 ] 𝜌𝑐𝑃 𝑑𝐿 (C-19) For a multiphase flow system, the enthalpy gradient can be written as the sum of the enthalpy gradient of the individual phases 𝑑ℎ𝑔 𝑑ℎ 𝑑ℎ𝐿 = (𝑤𝑔 + 𝑤𝐿 ) 𝑑𝐿 𝑤 𝑑𝐿 𝑑𝐿 (C-20) From Equation (C-12), the enthalpy gradients of each phase are 𝑑ℎ𝑔 𝑑𝑇 𝑑𝑃 = 𝑐𝑃,𝑔 − 𝜂𝑔 𝑐𝑃,𝑔 𝑑𝐿 𝑑𝐿 𝑑𝐿 76 (C-21) APPENDIX and 𝑑ℎ𝐿 𝑑𝑇 𝑑𝑃 = 𝑐𝑃,𝐿 − 𝜂𝐿 𝑐𝑃,𝐿 𝑑𝐿 𝑑𝐿 𝑑𝐿 (C-22) where Joule-Thompson coefficient of gas is 𝜂𝑔 = 𝜕 1 ( )] − } 𝜕𝑇 𝜌𝑔 𝜌𝑔 (C-23) 𝜕 1 {𝑇 [ ( )] − } 𝑐𝑃,𝐿 𝜕𝑇 𝜌𝐿 𝑃 𝜌𝐿 (C-24) 𝑐𝑃,𝑔 {𝑇 [ 𝑃 and that of oil phase is 𝜂𝐿 = For the real gas and incompressible liquid, the Equations (C-23) and (C-24) are become 𝜂𝑔 = − 𝑇 𝜕𝑧 {− ( ) } 𝑐𝑃,𝑔 𝜌𝑔 𝑧 𝜕𝑇 𝑃 (C-25) 𝑐𝑃,𝐿 𝜌𝐿 (C-26) and 𝜂𝐿 = − The flow in pipes is under the influence of heat transfer to environment, hence the Joule-Thompson coefficient 𝜂𝑔 > 𝜕𝑧 and (𝜕𝑇) < 𝑃 Substituting Equations (C-25) and (C-26) into (C-21) and (C-22) accordingly yields 𝑑ℎ𝑔 𝑑𝑇 𝑇 𝜕𝑧 𝑑𝑃 = 𝑐𝑃,𝑔 + [− ( ) ] 𝑑𝐿 𝑑𝐿 𝜌𝑔 𝑧 𝜕𝑇 𝑃 𝑑𝐿 (C-27) 𝑑ℎ𝐿 𝑑𝑇 𝑑𝑃 = 𝑐𝑃,𝐿 + 𝑑𝐿 𝑑𝐿 𝜌𝐿 𝑑𝐿 (C-28) and Substituting Equations (C-27) and (C-28) into (C-20) yields 𝑑ℎ 𝑑𝑇 𝑇 𝜕𝑧 𝑑𝑃 𝑑𝑇 𝑑𝑃 = (𝑤𝑔 {𝑐𝑃,𝑔 + [− ( ) ] } + 𝑤𝐿 [𝑐𝑃,𝐿 + ]) 𝑑𝐿 𝑤 𝑑𝐿 𝜌𝑔 𝑧 𝜕𝑇 𝑃 𝑑𝐿 𝑑𝐿 𝜌𝐿 𝑑𝐿 (C-29) Rearranging Equation (C-29) gives 𝑑ℎ 𝑤𝑔 𝑐𝑃,𝑔 + 𝑤𝐿 𝑐𝑃,𝐿 𝑑𝑇 𝑤𝑔 𝑇 𝜕𝑧 𝑤𝐿 𝑑𝑃 = + { [− ( ) ] + } 𝑑𝐿 𝑤 𝑑𝐿 𝑤 𝜌𝑔 𝑧 𝜕𝑇 𝑃 𝜌𝐿 𝑑𝐿 (C-30) According to Equation (C-30), for the two-phase mixture, the average heat capacity is 𝑐𝑃 = 𝑤𝑔 𝑐𝑃,𝑔 + 𝑤𝐿 𝑐𝑃,𝐿 𝑤 (C-31) and the average Joule-Thompson coefficient is expressed as 𝜂=− 𝑤𝑔 𝑇 𝜕𝑧 𝑤𝐿 { [− ( ) ] + } 𝑐𝑃 𝑤 𝜌𝑔 𝑧 𝜕𝑇 𝑃 𝜌𝐿 (C-32) 𝜂=− 𝑇 𝜕𝑧 {𝑦𝑔 [− ( ) ] + 𝑦𝐿 } 𝑐𝑃 𝜌𝑚 𝑧 𝜕𝑇 𝑃 (C-33) or 77 APPENDIX where 𝑦𝑔 and 𝑦𝐿 are the no-slip holdup of gas and liquid, respectively Hence, the dimensionless parameter Φ for the two-phase flow will be expressed as the following equation Φ=− 𝜌 𝑇 𝜕𝑧 𝑑𝑣 𝑑𝑃 − 𝜌𝑔 sin 𝜃) {𝑦𝑔 [− ( ) ] + 𝑦𝐿 } + (−𝜌𝑣 𝜌𝑚 𝑧 𝜕𝑇 𝑃 𝑑𝐿 𝑑𝐿 78 (C-34)