Transient Friction in Pressurized Pipes. II: Two-Coefficient Instantaneous Acceleration–Based Model

The goal in the field of modeling of hydraulic transients is a comprehensive model for pipe networks that is computationally fast and accurate. The fastest models are the one-dimensional (1D) models that use instantaneous acceleration–based (IAB) properties, but unfortunately these models are not as accurate as the more demanding 1D convolution-based (CB) models or quasi two-dimensional models. Focusing on a single pipe, this paper investigates the fundamental behavior of the much more accurate 1D CB model to find two coefficients for use with the two-coefficient formulation of the much-used modified IAB (MIAB) model for complete closing of a downstream valve. Two coefficients are found based on the weighting function used in the CB model, and these coefficients vary along the pipe length. Simulations are compared with two experimental results from tests performed at University of Adelaide in Australia in 1995. The experimental results are for different initial Reynolds numbers of approximately 2,000 and 5,800. The results show very good agreement between simulations and experiments. The improvement of the MIAB model is not general, and for the time being, only complete closure of a downstream valve in a single pipeline at low Reynolds numbers has been investigated.     

Range of Validity of the Transient Damping Leakage Detection Method

In a recent paper, an elegant, efficient, and easy to apply transient-based leakage detection method was proposed. The method exploits the fact that friction and leakage damp the modes of transient waves in a different manner. The method involves six major assumptions. These are: (1) the periodic motion in time of each mode is linearly independent of all other modes; (2) the amplitude of the induced transient is small; (3) the magnitude of the leak is small in comparison with the flow rate; (4) the wall friction can be represented by the Darcy–Weisbach equation; (5) the transient is initiated by an instantaneous small amplitude disturbance; and (6) the pipe system is a simple reservoir–pipe–valve type system or reservoir–pipe–reservoir type system. These six assumptions are relaxed and the validity of the transient damping method is assessed. The analysis shows that the first four assumptions do not pose any serious restriction to the applicability of the damping rate method provided that the mathematical model, used to generate the transient head trace in the leak-free pipe, accurately represents the frictional damping in the system. On the other hand, Assumptions (5) and (6) restrict the applicability of the method to systems that do not involve internal boundary conditions, such as junctions and pumps, and to transients triggered by impulses whose duration is smaller than the wave travel time. Extension of this method to complex pipe systems requires that the linearized waterhammer equations are solved under more general initial and boundary conditions. In addition, more investigation in relation to the frequency content of the input signal and its importance in leakage detection is warranted. The general framework used to derive the damping rate method has led to an efficient and direct algorithm for identifying leaks and future research should seek ways to adapt this framework to more complex pipe systems.     

Nonaqueous Phase Liquid Pool Dissolution as a Function of Average Pore Water Velocity

Bench-scale reactor experiments were performed to study the dissolution of a binary naphthalene-in-nonane mixture nonaqueous phase liquid (NAPL) pool over a wide range of average pore water velocities, vx (≈0.1–60 m/day). Experimental NAPL pool dissolution flux values were determined using a steady-state mass balance approach. The experimental flux data were compared to model predictions made assuming either local equilibrium or mass-transfer limited conditions. The local equilibrium model could describe the trends in the average effluent concentration and dissolution flux with 0.110?m/day. Data determined to be under mass-transfer limited conditions were fit to the nonequilibrium model to estimate values for an overall mass-transfer coefficient. The calculated overall mass-transfer coefficients had an average value of 0.407 m/day and showed no correlation with vx, probably due to mass-transfer resistance becoming dominated by the diffusional resistance in the NAPL. These results suggest that the nonequilibrium approach is better suited for describing high velocity (vx>10?m/day) dissolution of multicomponent NAPL pools, and that flushing of groundwater at very high velocities may not be an effective approach for enhancing NAPL-pool dissolution flux.     

Transient Friction in Pressurized Pipes. I: Investigation of Zielke’s Model

This paper investigates the well-known model for unsteady friction developed by Zielke in 1968. The model is based on weights of past local bulk accelerations and is analytically correct for laminar flow, but computationally demanding. Different models have been proposed using dynamic properties, typically based on instantaneous accelerations (IAB) that are more rapid in computational schemes. Unfortunately, they are not as accurate as Zielke’s model and fail to model certain types of transients. This paper points out that the water hammer transient is dominated by a periodicity varying along the pipe. Because of this, the unsteady friction calculated by the Zielke model is distributed nonuniformly along the pipe, and changes in the pipe length change the local unsteady friction. This phenomenon may explain why IAB models using calibrated coefficients to match experimental results have a large span in value for the reported coefficients. This paper will hopefully contribute to further work to find highly accurate and rapid models. The subject deserves to be brought up for discussion as a part of a total understanding of the problem.     

Head- and Flow-Based Formulations for Frequency Domain Analysis of Fluid Transients in Arbitrary Pipe Networks

Applications of frequency-domain analysis in pipelines and pipe networks include resonance analysis, time-domain simulation, and fault detection. Current frequency-domain analysis methods are restricted to series pipelines, single-branching pipelines, and single-loop networks and are not suited to complex networks. This paper presents a number of formulations for the frequency-domain solution in pipe networks of arbitrary topology and size. The formulations focus on the topology of arbitrary networks and do not consider any complex network devices or boundary conditions other than head and flow boundaries. The frequency-domain equations are presented for node elements and pipe elements, which correspond to the continuity of flow at a node and the unsteady flow in a pipe, respectively. Additionally, a pipe-node-pipe and reservoir-pipe pair set of equations are derived. A matrix-based approach is used to display the solution to entire networks in a systematic and powerful way. Three different formulations are derived based on the unknown variables of interest that are to be solved: head-formulation, flow-formulation, and head-flow-formulation. These hold significant analogies to different steady-state network solutions. The frequency-domain models are tested against the method of characteristics (a commonly used time-domain model) with good result. The computational efficiency of each formulation is discussed with the most efficient formulation being the head-formulation.     

钢管张力减径过程传热模型

为了提高钢管张力减径过程的轧制质量、降低能耗以及控制终轧温度的准确性,从而为钢管出炉温度提供科学设定依据,通过对传热机理分析,建立了钢管张力减径过程传热模型,给出了除鳞、轧制及空冷阶段钢管边界热流的计算式.基于塑性材料的变分原理,建立了轧制变形区的变形热计算模型.结果表明:变形热对钢管温度分布影响不可忽略;该模型能真实反映钢管在张力减径过程中的温度变化,与实测结果吻合较好,可用于钢管再加热和张力减径过程中的参数分析及工艺优化.     

Modeling surface effects on hydrogen permeation in metals

A new mathematical model for analyzing hydrogen permeation in solids, in which surface effects and traps influence hydrogen transport, is presented and solved. The new model combines the McNabb and Foster equations for diffusion with concomitant trapping^[1] and a surface-limited mass-transfer boundary condition. An important result of the new model is the introduction of a new variable,h _m, which is defined as the surface-limited mass-transfer coefficient. Theh _m coefficient can account for all possible surface effects and may be experimentally evaluated.     

Numerical Sensitivity Study of Unsteady Friction in Simple Systems with External Flows

This paper investigates the importance of unsteady friction effects when performing water hammer analyses for pipe systems with external fluxes due to demands, leaks, and other system elements. The transient energy equation for a system containing an orifice-type external flow is derived from the two-dimensional, axial momentum equation. A quasi-two-dimensional flow model is used to evaluate the relative energy contribution of total friction, unsteady friction, and the external flow, in a 1,500?m pipeline, with orifice flows ranging from steady-state flows of 2–70% of the mean pipe flow, and a Reynolds number of 600,000. It is found that for initial lateral flows larger than around 30% of the mean flow, unsteady friction effects can probably be neglected, whereas for external flows smaller than this, unsteady friction should generally be considered. Overall, the relative role of unsteady friction is found to diminish as the external flux increases, implying that unsteady friction is not critical for systems with large external flows. These results imply that unsteady friction may have a significant impact on the validity of transient leak detection techniques that have been derived assuming quasi-steady friction. To demonstrate this point, an existing transient leak detection method, originally derived under quasi-steady conditions, is tested with unsteady friction included.     

Deoxidatlon rate of copper droplet levitated in Ar-H2 gas stream

Levitated copper droplets, 5 mm in diameter with initial oxygen contents of 0.036 to 1.9 wt pct, were deoxidized at about 1970 K in an Ar-H₂ gas stream. The Ar-H₂ gas mixture having hydrogen partial pressure less than 4 kPa was introduced into a silica reaction tube of 11-mm ID at gas flow rates up to 2 x 10^-4 Nm³s^-1. The effects of initial oxygen content of the droplets, hydrogen partial pressure, and gas flow rate on the deoxidation process were examined. A mixed control model for the deoxidation rate involving both gas and liquid film mass-transfer resistances was combined with a thermodynamic relationship for the dissolved species in molten copper. The value of 2 × 10^-4 m 73x00D7; s^-1 was assigned to the liquid film mass-transfer coefficient of dissolved oxygen throughout all experimental conditions. Under the experimental conditions of low initial oxygen content and high hydrogen partial pressure, the liquid film mass-transfer resistance was significant. When a droplet of high initial oxygen content was deoxidized, transition phenomena from gas to liquid film mass-transfer control were noticed in the later stage of reaction. It was deduced from the present model that the accumulation of dissolved hydrogen was indispensable to these phenomena. Formerly Student     

Timescale Behavior of the Wall Shear Stress in Unsteady Laminar Pipe Flows

Based on two-dimensional (2D) flow model simulations, the effects of the radial structure of the flow (e.g., the nonuniformity of the velocity profile) on the pipe wall shear stress, τw, are determined in terms of bulk parameters such as to allow improved 1D modeling of unsteady contribution of τw. An unsteady generalization, for both laminar and turbulent flows, of the quasi-stationary relationship between τw and the friction slope, J, decomposes the additional unsteady contribution into an instantaneous energy dissipation term and an inertial term (that is, based on the local average acceleration-deceleration effects). The relative importance of these two effects is investigated in a transient laminar flow and an analysis of the range of applicability of this kind of approach of representing unsteady friction is presented. Finally, the relation between the additional inertial term and Boussinesq momentum coefficient, is clarified. Although laminar pipe flows are a special case in engineering practice, solutions in this flow regime can provide some insight into the behavior of the transient wall shear stress, and serve as a preliminary step to the solutions of unsteady turbulent pipe flows.     

Wave Propagation in a Pipe Pile for Low-Strain Integrity Testing

This paper presents an analytical solution methodology for a tubular structure subjected to a transient point loading in low-strain integrity testing. The three-dimensional effects on the pile head and the applicability of plane-section assumption are the main problems in low-strain integrity testing on a large-diameter tubular structure, such as a pipe pile. The propagation of stress waves in a tubular structure cannot be expressed by one-dimensional wave theory on the basis of plane-section assumption. This paper establishes the computational model of a large-diameter tubular structure with a variable wave impedance section, where the soil resistance is simulated by the Winkler model, and the exciting force is simulated with semisinusoidal impulse. The defects are classified into the change in the wall thickness and Young’s modulus. Combining the boundary and initial conditions, a frequency-domain analytical solution of a three-dimensional wave equation is deduced from the Fourier transform method and the separation of variables methods. On the basis of the frequency-domain analytic solution, the time-domain response is obtained from the inverse Fourier transform method. The three-dimensional finite-element models are used to verify the validity of analytical solutions for both an intact and a defective pipe pile. The analytical solutions obtained from frequency domain are compared with the finite-element method (FEM) results on both pipe piles in this paper, including the velocity time history, peak value, incident time arrival, and reflected wave crests. A case study is shown and the characteristics of velocity response time history on the top of an intact and a defective pile are investigated. The comparisons show that the analytical solution derived in this paper is reliable for application in the integrity testing on a tubular structure.     

Improved Control of Pressure Reducing Valves in Water Distribution Networks

The behavior of transients in water pipe networks is well understood but the influence of modulating control valves on this behavior is less well known. Experimental work on networks supplied through pressure reducing valves (PRVs) has demonstrated that, in certain conditions, undesirable phenomena such as sustained or slowly decaying oscillation and large pressure overshoot can occur. This paper presents results from modeling studies to investigate interaction between PRVs and water network transients. Transient pipe network models incorporating random demand are combined with a behavioral PRV model to demonstrate how the response of the system to changes in demand can produce large or persistent pressure variations, similar to those seen in practical experiments. A proportional-integral-derivative (PID) control mechanism, to replace the existing PRV hydraulic controller, is proposed and this alternative controller is shown to significantly improve the network response. PID controllers are commonly used in industrial settings and the methods described are easy to implement in practice.     

Efficient Meshfree Computation with Fast Treatment of Essential Boundary Conditions for Industrial Applications

This paper presents a fast treatment of essential boundary conditions in three-dimensional (3D) meshfree computation for computational efficiency. Due to the loss of Kronecker delta properties in the meshfree shape functions, the imposition of essential boundary conditions is tedious, especially in 3D applications. The proposed boundary singular kernel (BSK) method introduces singularities to the kernel functions associated with the essential and kinematically constrained boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values, and consequently constraints can be imposed directly. In this work, the recovery of nodal value properties on essential boundary nodes is proved for general n-dimensional geometries. The extension of previously proposed two-dimensional BSK method to 3D formulation thus becomes straightforward, and essential boundary treatment consumes almost no additional cost to meshfree computation and makes the method affordable for industrial applications. The effectiveness of the proposed method is demonstrated in 3D metal forming examples.     

Solute Mixing Models for Water-Distribution Pipe Networks

The spreading of solutes or contaminants through water-distribution pipe networks is controlled largely by mixing at pipe junctions where varying flow rates and concentrations can enter the junction. Alternative models of solute mixing within these pipe junctions are presented in this paper. Simple complete-mixing models are discussed along with rigorous computational-fluid-dynamics models based on turbulent Navier–Stokes equations. In addition, a new model that describes the bulk-mixing behavior resulting from different flow rates entering and leaving the junction is developed in this paper. Comparisons with experimental data have confirmed that this bulk-mixing model provides a lower bound to the amount of mixing that can occur within a pipe junction, while the complete-mixing model yields an upper bound. In addition, a simple scaling parameter is used to estimate the actual (intermediate) mixing behavior based on the bounding predictions of the complete-mixing and bulk-mixing models. These simple analytical models can be readily implemented into network-scale models to develop predictions and bounding scenarios of solute transport and water quality in water-distribution systems.     

Second-Order Radiation Boundary Condition for Water Wave Simulation with Large Angle Incidence

A finite-element method (FEM) is used to simulate water wave propagation with large angle incidence at exterior boundaries. In this paper, the radiation boundary condition is expanded to a second-order approximation and a quadratic shape function is used in the FEM wave model. Cases used for verifications include wave scattering around a vertical cylinder and wave propagation over a submerged circular shoal with concentric contours. Numerical calculations based on this second-order radiation boundary condition are found to be in good agreement with theoretical and experimental results available. The numerical predictions show that this model has made a very good improvement over the first-order radiation boundary conditions for oblique wave incidence in coastal engineering.     

Pilot-Scale Verification and Analysis of Iron Release Flux Model

An original model by Mutoti in 2003 was developed mathematically, and empirically, to predict the increase in total iron concentration in distribution systems. This model, referred to as a flux model, relates the increase in iron concentration in a reach of unlined or galvanized iron pipe to the surface area of the pipe in contact with the water. A flux term, defined with a dimension of mass per area per time was used. The effects of water chemistry, pipe material and hydraulic conditions were incorporated into the flux term. This paper describes the verification of the flux model using independent pilot data obtained with variable water quality under worst case, laminar flow conditions. The original model accurately predicted iron release for this independent verification data, with an overall R2 of 0.80. For laminar flow conditions, the increase in iron concentration is proportional to the flux and the hydraulic residence time, and is inversely proportional to the pipe diameter.     

Interaction of Nonconservative 1D Continuum and Moving MDOF Oscillator

This paper presents a method for computing the response of a 1D elastic continuum induced by a multi-degree-of-freedom (MDOF) oscillator traveling over it. The continuum and the oscillator are nonconservative systems with proportional damping. Unlike most studies in the field, the solution method does not address a particular type of continuous structure and oscillator. Instead, a rigorous mathematical formulation is presented that can be applied to a broad class of proportionally damped 1D continua and MDOF oscillators, regardless of boundary conditions. The problem is reduced to the integration of a system of linear differential equations with time-dependent coefficients. These coefficients are found to depend on natural frequencies, damping ratios, and eigenfunctions and eigenvectors of the continuum and the oscillator. The method is tested on numerical examples and results are compared to those available in the literature. As a practical application, the method can be used to analyze vehicle-bridge interaction problems.     

Modified Form of the Extended Kostiakov Equation Including Various Initial and Boundary Conditions

The extended Kostiakov equation is intensively used in surface irrigation applications. Traditionally, the extended Kostiakov infiltration formula is calibrated for specific field conditions. However, there is a dependence of the extended Kostiakov coefficients on both initial and boundary conditions. In this paper, a new simplified methodology is developed to account extended Kostiakov κ variation for these effects. The purely empirical extended Kostiakov equation is transformed to a form of a modified version of the classical Philip two-term equation. This modification relates a physical parameter, the soil sorptivity S, with the purely empirical coefficient κ of the extended Kostiakov formula. Then, the variation of the sorptivity S for various water levels and initial water contents is given theoretically by a simple algebraic equation. The proposed correction was compared with numerical infiltration data with varying initial (water content) and boundary conditions (ponding depth) for two contrasting soils. Results indicate that the corrected infiltration curves converge well with the simulated ones.     

Approximation of Turbulent Wall Shear Stresses in Highly Transient Pipe Flows

Theoretical predictions of wall shear stresses in unsteady turbulent flows in pipes are developed for all flow conditions from fully smooth to fully rough and for Reynolds numbers from 103 to 108. A weighting function approach is used, based on a two-region viscosity distribution in the pipe cross section that is consistent with the Colebrook–White expression for steady-state wall friction. The basic model is developed in an analytical form and the resulting weighting function is then approximated as a sum of exponentials using a modified form of an approximation due to Trikha. A straightforward method is presented for the determination of appropriate values of coefficients for any particular Reynolds number and pipe roughness ratio. The end result is a method that can be used relatively easily by analysts seeking to model unsteady flows in pipes and ducts.     

Semianalytical Model for Solid Concentration in Turbulent Pipe Flows with Dispersed Particles

This paper presents a semianalytical model for the radial distribution of the solid concentration in a fully developed vertical turbulent pipe two-phase flow. A simplified momentum equation in the radial direction for solid phase in a two-phase flow with dilute suspended particles was first obtained. A linear empirical closure relation for the mean gas and solid velocities along the pipe direction was constructed using published experimental data. By incorporating the closure relation, an analytical solution to the simplified solid momentum equation with the appropriate boundary conditions at the pipe center and wall was obtained. The results from this semianalytical model are able to describe the core-annulus phenomenon commonly occurring in two-phase turbulent pipe flows. Very good agreements were found between the model predictions and published experimental data.