Flow Rate Equation Research Articles

A MODIFIED NUMERICAL METHOD OF DETERMINING HYDRAULIC SYSTEM PARAMETERS FOR THE DEVELOPMENT OF MATHEMATICAL MODELS AND INFORMATION COMPLEXES OF COMPUTER SIMULATION MODELS FOR INDUSTRIAL CHEMICAL PRODUCTION

The paper introduces a modified numerical method for determining the parameters of hydraulic systems used in mathematical modeling and simulation-based computer systems for industrial chemical production. It explores the dynamics of fluids, gases, and their mixtures as they traverse pipelines equipped with valves, compressors, and other complex components. Traditional methods for solving nonlinear equation systems often face limitations, including sensitivity to initial approximations and the need for relaxation coefficient adjustments, which complicate simulation processes. The proposed method converts nonlinear equations into an optimization problem, us- ing the Nelder-Mead algorithm to minimize deviations between initial estimates and comput- ed parameters. This derivative-free approach employs a simplex geometric transformation, overcoming issues associated with iterative methods and delivering high accuracy, stable convergence, and computational efficiency. Its adaptability makes it suitable for handling in- tricate hydraulic systems. To validate the method, a practical example involving a pipeline system with multiple restricting devices is analyzed. The mathematical model includes equations for pressures and flow rates, supplemented by material balance constraints at branching points. Numerical ex- periments evaluate how input pressure influences system parameters, demonstrating the mod- el's robustness and precision. The results confirm that the method accurately simulates system parameters under steady-state conditions and identifies critical optimization points. This innovative approach offers significant advantages for designing and managing industrial chemical facilities. By simplifying model construction, enhancing reliability, and accelerating simulation processes, the method provides a powerful tool for optimizing the op- eration and control of complex technological systems.

Development of Model and Graphical User Interface for Leak Localization in Crude Oil Pipeline

Pipelines provide the most efficient and cost-effective means for fluid transport but the challenge posed by leaks has substantially increased the risks and hazards in pipeline fluid transportation. The development of efficient leak detection system pays off in quicker leak detection and localization, leading to quicker responses and remediation works by the pipeline emergency response team. This would ultimately bring about less severity of the pipeline leak in terms of financial losses, human and environmental consequences. Steady state modeling of crude oil pipeline flow with attempt to determine and localize leak has been achieved in this study. Mathematical models have been developed to detect and localize leaks in crude oil pipeline during leak occurrence. Leak detection was modeled using the conservation of mass equation while leak localization was modeled by modifying the Darcy-Weisbach pipeline liquid flow equations and utilizing the Swamee-Jain friction factor correlation. Equations for flowrate and pressure drop along the pipeline were developed for two cases: a case where there is no leak in the pipeline and a case where there is leak in the pipeline. Leak localization equation was determined by equating the fluid flow equation when there is no leak and when there is leak. The model was structured and simulated in Matlab software with the model tested for four field cases. The results from the simulation revealed swift leak detection, accurate localization of the leak, and determination of the pressure at the point of leak. Experimentally determined results from actual field measurements of leak incidences were used to validate the results. The result determined from the model developed proved accurate with only an average error of 0.216 miles.

A modified constitutive model for whole-life thermal-mechanical fatigue incorporating dynamic strain aging in 316LN stainless steel

A comprehensive analysis of experimental data is presented for 316LN stainless steel subjected to isothermal and thermal-mechanical fatigue loading conditions within the temperature range of 350–600 °C. The study aims to provide a thorough understanding of the cyclic behavior through meticulous data integration, supplementation, and the use of stress decomposition methods combined with a classical damage evolution definition. The modeling approach employed in this study is based on the classical AF-OW-Kang model, with the incorporation of the Arrhenius term in the plastic flow rate equation. Furthermore, to consider the effect of dynamic strain aging at varying temperatures, temperature-dependent terms are introduced into the equations that govern the evolution of isotropic stress and backstress, resulting in enhanced accuracy in describing cyclic hardening behavior. Additionally, a modified damage evolution equation is utilized, along with equations for isotropic stress and backstress evolution, to address cyclic softening. Simulation results confirm the effectiveness of the modified model in capturing the cyclic hardening/softening behavior of 316LN stainless steel throughout the whole-life time, under both isothermal and thermal-mechanical fatigue loading conditions.

Effectiveness of enhanced roof ventilation unit (ERU) based on venturi cap in different wind environments and new evaluation models: A numerical study

Reasonable building ventilation design can effectively improve indoor thermal comfort and reduce energy consumption. Therefore, an enhanced roof ventilation unit (ERU) was developed for this study, and its performance and influence law under various wind speeds and directions were explored through numerical simulation. Additionally, a new evaluation method for ERU was proposed through data fitting, and applicability was verified for different climate zones in China. The results showed that: (1) With a maximum difference of 3.11 °C, the ERU performed best in the N direction and poorest in the SW direction for indoor temperature. Furthermore, the impact of wind directions was more evident than wind speeds. (2) The wind speed affected the volume flow rate significantly in the NE direction (7.77 m3/s) and insignificantly not in the SE direction (0.32 m3/s). Nevertheless, the performance was the opposite in the NE and SE directions when the wind speed increased from 0.5 m/s (3) The air age performed best in the NE and worst in the SE directions. As wind speeds surpassed 2.0 m/s, the air age variation steadily stabilized in various wind directions. (4) The fitted equation for indoor temperature has applicability in typical cities in five climate zones, and the fitted equation for volume flow rate is applicable in four climate zones except the hot summer and warm winter zone. The study results can provide references for the ERU to improve indoor thermal ventilation, contributing to sustainable building energy efficiency.

Regression equations for peak expiratory flow rate in children aged 5-10 years of Western Maharashtra, India: a cross-sectional study

Background: Incidence of pulmonary diseases in urban children is on the rise. Peak expiratory flow rate, a component of pulmonary function tests, is a useful measure for initial pulmonary assessment. It is easily implementable on large population due to its simplicity. Studies presenting such data for Indian children are severely lacking, as PEFR is anthropometric, population and region specific. The present study aimed to establish an equation for predicting PEFR in urban children in Pune city in western Maharashtra, India. Methods: The cross-sectional study was done in Pune, Maharashtra, India. Children from different schools were selected by cluster sampling. The number of participants screened was 2100 of which 1760 were selected. Best of three readings of PEFR was recorded for each child using an EU Scale Peak Flow meter (Breath-O meter, Cipla). Anthropometric data like height, weight, chest expansion, waist/hip ratio were measured and BMI was calculated along with PEFR of each child. Results: Out of the 1760 children included in the study 933 (53%) were boys and 827 (47%) were girls. The stepwise regression analyses were carried out using age, height, weight and waist/hip ratio as predictor variables. Conclusions: Final regression equation was derived using height. Approximately 80% of the data was used for prediction of regression equation and remaining 20% data was used as a control group to validate the derived equation. The regression equation formulated shall offer a predicted PEFR value as guideline for healthcare workers in Pune city, Maharashtra, India.

The effect of flow rate on the ammonia oxidation rate in the riverway biofilm in-situ remediation and modeling

The biofilm in-situ remediation is one of the most important and practical techniques for river remediation. The biochemical function of biofilms and the hydraulics of the river are both significant aspects of river remediation study, however, they belong to distinctive research points. To date, little research has combined the biochemical process of biofilm and the hydraulics of the river. In this study, ammonia oxidation rate (AOR) was used to characterize the activity of biofilms and measured at different flow rates. It was increased with the flow rate increasing before the peak (26.13 g-N/m3/h) at 0.3 m/s and then decreased to 0 at 0.414 m/s directly. Meanwhile, the skewness equation of flow rate (Fr)−AOR was established based on previous hydraulics and biology research. The kinetic parameters of the equations were determined by fitting the measured values of AOR (R2 > 0.95), which could be used to simulate the AOR of the river simulators and open channels. Besides, the distribution characteristics of the AOR at various flow rates in different sections were obtained. Overall, this paper provides a novel perspective by studying the internal biochemical model of biofilm and the hydraulic model with multiple influencing factors further to improve the biofilm-hydraulics coupling model.

Performance evaluation and model of spacesuit cooling by hydrophobic hollow fiber-membrane based water evaporation through pores

A comprehensive mathematical model is presented that accurately estimates and predicts failure modes through the computations of heat rejection, temperature drop and lumen side pressure drop of the hollow fiber (HF) membrane-based NASA Spacesuit Water Membrane Evaporator (SWME). The model is based on mass and energy balances in terms of the physical properties of water and membrane transport properties. The mass flux of water vapor through the pores is calculated based on Knudsen diffusion with a membrane structure parameter that accounts for effective mean pore diameter, porosity, thickness, and tortuosity. Lumen-side convective heat transfer coefficients are calculated from laminar flow boundary layer theory using the Nusselt correlation. Lumen side pressure drop is estimated using the Hagen-Poiseuille equation. The coupled ordinary differential equations for mass flow rate, water temperature and lumen side pressure are solved simultaneously with the equations for mass flux and convective heat transfer to determine overall heat rejection, water temperature and lumen side pressure drop. A sensitivity analysis is performed to quantify the effect of input variability on SWME response and identify critical failure modes. The analysis includes the potential effect of organic and/or inorganic contaminants and foulants, partial pore entry due to hydrophilization, and other unexpected operational failures such as bursting or fiber damage. The model can be applied to other hollow fiber membrane-based applications such as low temperature separation and concentration of valuable biomolecules from solution.

Study of the Lubricity of Oil Film at Constant Cutting Speed Under Different Loads of a Hydrostatic Turntable

This article studies the tribological behavior of the lubricant film when the Q1-224 hydrostatic turntable is operated at constant cutting speed. First, the rotational speed equation of the worktable under the constraint of constant cutting speed is derived. Based on the lubrication mechanism of hydrostatic turntables, the flow-rate equation of the oil film, the temperature-rise equation of the oil film, and the bearing-capacity equation of the double rectangular cavity are deduced. Furthermore, the prediction model of the lubrication performance of hydrostatic turntable operating at constant cutting speed is established. Meanwhile the quasi-steady-state analysis is carried out by numerical simulation to obtain the change law for the lubricating oil film temperature, pressure, and other performance indicators with time under the different loads. Finally, these laws are verified by designed experiments. It is found that the load has a large effect on the change of oil film performance; it affects the temperature rise of the oil film by influencing the dissipation capacity, and there is a nonlinear relationship between the increase of the load and the average temperature rise of the oil film.

Research and calculation of controlled damping elastic kinematic joint of manipulation robots

Based on the analysis of an open shell, the design of a damping elastic kinematic joint of a lever mechanism used in controlled dampers, precision robots and manipulators is considered. The control problem is being solved according to the maximum principle using the connections of the obtained controls with the equations of the gas flow rate entering the spring cavity. A relationship is established between the kinematics of the spring, the shape of its cross section and filling with gas. The problem of optimal control of this connection in terms of speed is solved. Keywords: manipulation robot, pneumatic drive, spring opening, Euler—Lagrange equations, Lagrange multipliers, speed, gas flow, throttling. vichizhikov@gmail.com, kurnasov@mirea.ru

Characteristics of Gas–Solid Flow in an Intermittent Countercurrent Moving Bed

The calculation equations of pressure drop and solid flow rate are given in an intermittent countercurrent moving bed with multiple optimized structures. The flow fields are investigated by experimental methods under different conditions, e.g., ratio of position of the gas distributor to the bed width (rp = 0.5~1.5), gas superficial velocity (ug = 0~0.1591 m/s), solid outlet diameter (Do = 5~25 mm) and cone angle (α = 0~60°). It is found that when rp ≥ 1.0, the flow field in the bed is little affected by rp. Flow patterns are divided into three modes: continuous discharging, intermittent discharging (synchronous, asynchronous) and particle bridging. During continuous discharging, the calculation formula of the solid flow rate, which is closely related to Do and gas–solid slip velocity vslip, is established by referring to the modified De Jong and Beverloo formulas, and its error ≤ ±12.5%. The pressure drop of the total bed consists of the pressure drops of the granular bed and the solid outlets, which are affected by Do, vslip and α; it is built by referring to the Ergun formula, and its error ≤ ±17.0%. As the solid flow rate and pressure drop influence each other through vslip, an iterative algorithm is proposed to enhance the computation accuracy.

Stepwise mathematical derivation of the Herschel–Bulkley laminar fluid flow equations—in pipes

Stepwise derivation of flow equations of the Herschel–Bulkley (HB) model is not available in the literature. These equations are crucial for mechanical, chemical and petroleum engineering academia and industries where fundamental works on non-Newtonian fluids may be done to reach future models and estimation methods. Therefore, this work focuses on derivation of laminar flow equations and estimation methods of HB fluids through pipes. In this work, first, stepwise derivation of the HB fluid flow parameters consisting of fluid velocity, flow rate, average velocity and relative velocity equations is presented, followed by a straightforward mathematical model for use in numerical solution. Next, stepwise mathematical derivation of the laminar pressure drop equations by Merlo et al. (An innovative model for drilling fluid hydraulics. Paper presented at the SPE Asia Pacific oil and gas conference, Kuala Lumpur, Malaysia, 1995) and Gjerstad and Time (SPE J 20:1–18, 2014) is presented, and finally practical and user-friendly calculation procedures for different estimation methods are presented. The step-by-step derivation procedures presented in this work contribute to effective learning for engineering students and practitioners in addition to providing a clear example derivation guideline for future researchers to reach other more accurate non-Newtonian hydraulics models and estimation methods.

Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections

Slope siphon drainage is a convenient and efficient above-ground drainage method that is free of manual power and can effectively maintain the stability of potential landslides and prevent the loss of life and property. The complex engineering topography inevitably requires the use of siphon drains with a total length of more than 150 m and a horizontal section length of more than 80 m, which significantly increases the difficulty of calculating the drainage capacity and thus affects the actual utilization of the project. The traditional siphon flow rate equation does not apply to long-pipe siphon conditions, especially when the lift is close to the limit, and there are significant errors in the calculation results, for which we propose a new calculation method. The proposed method considers both air release and flow-pattern classification. Thirty-six sets of experiments were conducted to validate our proposed calculation method. The results showed that our method not only calculated the siphon flow velocity well but also predicted the main flow pattern in the siphon in the experiment well. Furthermore, the equation for calculating the siphon flow velocity was extended to the siphon operation mode with long horizontal sections.

Soft computing and statistical approach for sensitivity analysis of heat transfer through the hybrid nanoliquid film in rotating heat pipe

In this paper, the numerical solution for heat transfer through a rotating heat pipe is studied and a sensitivity analysis is presented by using statistical experimental design technique. Graphene oxide-molybdenum disulfide (GO-MoS2) hybrid nanofluid is taken as working fluid inside the pipe. The impact of the heat pipe parameters (rotation speed, initial mass, temperature difference) on the heat transfer and liquid film thickness is investigated. The mathematical model coupling the fluid mass flow rate and liquid film evolution equations in evaporator, adiabatic, and condenser zones of the heat pipe is constructed. The mathematical model is solved by implementation of “Particle Swarm Optimization” along with the finite difference method. The outcomes demonstrate that hybrid nanoparticles help to improve the heat transfer through the heat pipe and reduce liquid film thickness. The heat transfer rises with increasing temperature difference and reducing inlet mass, and it reduces slightly with rising rotation speed. The difference in liquid film thickness between the evaporator and condenser zones increases with increasing temperature difference and decreasing rotation speed. The impact of increasing the volume fraction of GO on the liquid film thickness is higher than that in the case of the MoS2 nanoparticles. However, an increase of the heat transfer is noticed in case of increasing the volume fraction of GO relative to increasing MoS2 concentration. Statistical analysis of the computed numerical data and the identification of significant parameters for total heat transfer are found using the response surface method. At 95% level of significance, the GO concentration in the hybrid nanofluid, inlet mass of the hybrid nanofluid and the temperature difference inside the evaporator zone of the pipe are found to be significant linear parameters for increasing heat transfer.

A novel model for oil-water two-phase relative permeability of shale formation during flowback based on fractal method

As an abundant type of unconventional hydrocarbon resources on the earth, shale oil has been widely concerned by scholars across the world. During flowback period, flow-back resistance of fracturing fluids increases because of the existence of shale oil. Therefore, it is of practical significance to study the flow-back capability of fracturing fluid for high well performance. In this paper, fractal method of porous media has been used for calculation of relative permeability of oil-water two phase flow in shale pores. First, we built up flow equation of single nanopore with four layers: boundary layer, bulk wetting phase layer, oil-water two phase layer, and nonwetting phase layer. Furthermore, boundary slip, boundary, viscosity change in boundary layer and threshold pressure gradient is considered for correction flow rate equation. The pores of shale matrix can be simplified into capillary bundle model, and fractal theory is suitable for this model. Aperture fractal dimension and tortuosity fractal dimension is used to expand its scale to porosity media with fractal method. Then the model is verified by experimential results and classic models. Furthermore, serval main factors affecting the relative permeability of water are primarily analyzed, and the more essential factors out of the aforementioned ones are extracted and further studied. The results show that with the increase in the thickness of oil-water two phase layer, the relative permeability of water is affected more seriously; The relative permeability of water is positively correlated with slip length and thickness of boundary layer. This work provides a more convenient approach to calculate relative permeability with no experiments, and provide permeability for numerical simulation in flow back period.

Influence of surface roughness on methane flow in shale kerogen nano-slits

Understanding the mechanism of methane transport in real kerogen nano-pores is critical to evaluate gas productivity in shale formations accurately. This work presents a novel numerical method to generate nano-slits with different surface roughness for Type II-D kerogen in molecular dynamics (MD) simulations. The methane adsorption and transport phenomenon in the kerogen nano-slits are then studied. A modified velocity model is proposed by considering the effect of surface roughness on the slip velocity and the effect of gas rarefaction on the methane viscosity. The ratio of the mass flow rate in the adsorption region to the total flow rate is between 57% and 78% at a channel width of 2.5 nm depending on the surface roughness and pressure, but it drops to between 6% and 16% at a channel width of 10 nm. Hence, two mathematical equations are derived to estimate the mass flow rate in the adoption region from either (1) the adsorption per apparent surface area, or (2) the product of the bulk density and the width of the adsorption region, respectively. Finally, a new flow enhancement model is derived by combining the modified velocity model and the equation for mass flow rate in the adsorption region. The flow enhancement increases with the Knudsen number but decreases with the increase of surface roughness. The proposed flow enhancement model can match the MD simulation well. The proposed model for methane flow enhancement covers the slip flow and transition flow regimes and can potentially be implemented to study the large-scale flow properties, such as permeability, of the shale matrix at various conditions to better evaluate the shale gas production performance.

Manual Hydraulic Calculation Method of the Fire Sprinkler System Design Considering Velocity Pressure

Theoretical calculation methods for hydraulic calculation of fire sprinkler systems considering the velocity pressure have been proposed; however, there has been no specific application method to the design. Understanding the design considering the velocity pressure is important because it is the actual situation that occurs in the pipe when calculating the friction loss. Therefore, the procedure for the design considering the velocity pressure is explained in detail from beginning to end of the design. Two simple points can be summarized for the hydraulic calculation procedure considering the velocity pressure. 1) the sprinklers discharge flow rate on the branch is calculated by the discharge flow rate equation due to normal pressure. 2) Flow is adjusted at the intersection point of the pipe where the branch pipe is connected and two different pressures are present. The calculated value according to the manual hydraulic calculation procedure considering the velocity pressure presented in this paper is 12.9642 bar for pump demand pressure, and it is consistent with the value (12.96 bar) calculated under the same condition by the PIPENET computer program. As for the required flow rate of the pump, the value directly calculated in this study is 3137 lpm, which is 99.81% of the value (3143 lpm) calculated by PIPENET computer program, indicating that the calculation procedure is accurate.

Study of the Fluid Passing through the Screen in the Three Products Hydrocyclone Screen (TPHS): A Theoretical Analysis and Numerical Simulation

The three products hydrocyclone screen (TPHS), a branch of the hydrocyclone, effectively removes the fish-hook effect, which has been used in the industrial field. The current cylindrical screen in the TPHS generates the characteristic flow known as the screen underflow, which has a significant impact on device performance. To investigate the flow behaviour of the fluid passing through the screen, a combination of a dynamic analysis and a numerical simulation was used. The permeating process in the TPHS was abstracted by a simple fan mode in this work to generate the flow-rate equations and the driving-force models. The pressure difference was the driving force for the screen penetration in the ideal fluid, but it also included a viscous force in the viscous fluid. Furthermore, at the same inlet velocity, the viscous fluid had a higher flow rate than the ideal, indicating that the viscosity promoted the fluid penetration. Meanwhile, as the inlet velocity increased, the mass flow of the screen backflow increased, while the corresponding proportion first rose to a peak then dropped and then gradually stabilised. Furthermore, a flow equation for the screen underflow in the TPHS was developed, which is related to the structural parameters (the rotation radius, the length of the cylindrical screen, the aperture size, and the open-area percentage) and the process parameters (the dynamic viscosity of the fluid and the pressure difference between the feed inlet and the screen outlet).

The effect of the altitude on the performance of a solar chimney

This study aimed to show the relationship between the altitude and the performance of a solar chimney based on simulation analysis and numerical calculation. Steady-state heat transfer from isothermal vertical walls of a chimney is modelled analytically. The paper introduces the basic concepts and governing equations for modelling heat transfer and air flow rates within the solar chimney at various altitudes. The relationship between altitude, temperature, and mass flow rates for the buoyancy-driven chimney flow is complex. A multidimensional equation comprising the inverse tangent and inverse hyperbolic tangent functions is developed. A Matlab program was developed to overcome the problem of the non-linearity of the equation. The results obtained showed that the altitude has modest influence but not substantial on the performance of the solar chimney if we assumed a constant ambient air temperature for all altitudes. But in real climatic conditions, the ambient temperature is altitude dependent. In these circumstances, the performance of the solar chimney is significantly influenced by the altitude. Therefore, microclimatic conditions should be used rather than large-scale area climatic data for solar chimney design.

Predictive regression equation and nomogram of peak expiratory flow rate in healthy school going children of Kolhapur, Maharashtra, India

The Journal is the primary organ of Continuing Paediatric Medical Education in Sri Lanka. The journal also has a website. Free full text access is available for all readers.The Sri Lanka Journal of Child Health is now indexed in SciVerse Scopus (Source Record ID 19900193609), Index Medicus for South-East Asia Region (IMSEAR), CABI (Centre for Agriculture and Bioscience International Global Health Database), DOAJ and is available in Google, as well as Google Scholar.The policies of the journal are modelled on the Committee on Publication Ethics (COPE) Guidelines on Principles of Transparency and Best Practice in Scholarly Publishing. Sri Lanka Journal of Child Health is recognised by the International Committee of Medical Journal Editors (ICMJE) as a publication following the ICMJE Recommendations.

Experimental and numerical analysis on vortex cavitation morphological characteristics in u-shape notch spool valve and the vortex cavitation coupled choked flow conditions

The u-shape notch of spool valve is usually combined with the other typical throttling notches to achieve proportional flowrate adjustments in practical hydraulic control applications. Due to the vacuum-suction nozzle-structural effect, u-shape notch is more likely for cavitation arising. Traditionally, notch throttling is based on the continuity and Bernoulli equation to explain the Harvey nuclei bubble flow from development to collapse, which is not entirely agreed with present proliferated large vapor vortex cavitation researches. In this paper, experimental and numerical analysis discuss the morphological characteristics of u-shape notch vortex cavitation and the choked flow issues caused by vortex cavitation. The large vapor vortex cavitation morphological reproduced by the Large Eddy Simulation (LES) model associated with the multi-phase cavitation model is very close to experimental observations. Further, based on the clear cavity entity obtained by simulation, it could be found that, as notch opening increases, the cavity starting position moves back and the cavity moves upward. And as pressure difference increases, the cavity length increases smoothly until critical pressure value, after which the length increases sharply and maintains in long length level. In addition, it seems that the normalized measurement manner of cavitation length could be proposed, if using local cavitation number dividing incipient cavitation as abscissa to measure the cavity length. Moreover, the cavitating flowrate equation is proposed considering the cavitation-induced serious choked flow problem. The vortex cavitation, especially its length, is assumed to represent intense enough notch vortex flow friction loss. Based on the added flow loss term, the classical flowrate equation is revised, according to which the cavitating flowrate could be predicted within the accuracy of 0.5–2.0%.