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momentum equation
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Book Chapter
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005450
EISBN: 978-1-62708-196-2
... Abstract This article is a comprehensive collection of fluid dynamic equations for properties of fluids, fluid statics, fluid motion, dimensional analysis, and boundary layer flow. It presents equations for analyzing problems in fluid mechanics, continuity equation, momentum equation...
Abstract
This article is a comprehensive collection of fluid dynamic equations for properties of fluids, fluid statics, fluid motion, dimensional analysis, and boundary layer flow. It presents equations for analyzing problems in fluid mechanics, continuity equation, momentum equation, and energy equation for solving various problems related to fluid dynamics.
Book Chapter
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... Abstract Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD...
Abstract
Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques. It introduces some common techniques for discretizing the fluid-flow equations and methods for solving the discrete equations. These include finite-difference methods, finite-element methods, spectral methods, and computational particle methods. The article describes the approaches for grid generation with complex geometries. It discusses the four-step procedures used in the CFD process for engineering design: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. The article also provides information on the engineering applications of the CFD. It concludes with a discussion on issues and directions for engineering CFD.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002444
EISBN: 978-1-62708-194-8
... Abstract Computational fluid dynamics (CFD) is reserved for computationally intensive three-dimensional simulations of thermal fluids systems where nonlinear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction...
Abstract
Computational fluid dynamics (CFD) is reserved for computationally intensive three-dimensional simulations of thermal fluids systems where nonlinear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques for their solution. It introduces discretization techniques that are used by finite-difference, finite-volume, finite-element, spectral, and some particle methods. Associated concepts of numerical stability and accuracy are also reviewed. The article describes two approaches for grid generation with complex geometries: the use of unstructured grids and the use of special differencing methods on structured grids. The article describes the four-step procedures of the CFD process: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. It provides information on the applications of the engineering CFD. Issues and directions for the engineering CFD are also described.
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005237
EISBN: 978-1-62708-187-0
... equations: continuity, momentum, energy, and solute conservation. The mushy zone is treated as a porous medium. More details pertaining to the transport equations, as well as modifications for numerical implementation, can be found in Ref 12 , 13 , 14 , 15 , 16 , 17 . Continuity Equation...
Abstract
In order to model macrosegregation, one must consider convection and the partitioning of segregating elements at the dendritic length scale. This article describes microsegregation with diffusion in the solid. It presents a continuum model of macrosegregation and illustrates the simulation of macrosegregation and microsegregation.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005525
EISBN: 978-1-62708-197-9
... Abstract This article presents conservation equations for heat, species, mass, and momentum to predict transport phenomena during solidification processing. It presents transport equations and several examples of their applications to illustrate the physics present in alloy solidification...
Abstract
This article presents conservation equations for heat, species, mass, and momentum to predict transport phenomena during solidification processing. It presents transport equations and several examples of their applications to illustrate the physics present in alloy solidification. The examples demonstrate the utility of scaling analysis to explain the fundamental physics in a process and to demonstrate the limitations of simplifying assumptions. The article concludes with information on the solidification behavior of alloys as predicted by full numerical solutions of the transport equations.
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005636
EISBN: 978-1-62708-174-0
... welding. The article presents the mathematical equations of mass, momentum, energy, and species conservation. It reviews the applications of heat transfer and fluid flow models for different welding processes. Finally, the article discusses the approaches to improve reliability of, and reduce uncertainty...
Abstract
This article provides a comprehensive review and critical assessment of numerical modeling of heat and mass transfer in fusion welding. The different fusion welding processes are gas tungsten arc welding, gas metal arc welding, laser welding, electron beam welding, and laser-arc hybrid welding. The article presents the mathematical equations of mass, momentum, energy, and species conservation. It reviews the applications of heat transfer and fluid flow models for different welding processes. Finally, the article discusses the approaches to improve reliability of, and reduce uncertainty in, numerical models.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005437
EISBN: 978-1-62708-196-2
... Abstract This article presents the governing equations for moving a solidification front, based on the balance of mass, momentum, energy, and solute. It reviews how material properties and geometry can be analyzed in the context of the governing equations. The article provides several example...
Abstract
This article presents the governing equations for moving a solidification front, based on the balance of mass, momentum, energy, and solute. It reviews how material properties and geometry can be analyzed in the context of the governing equations. The article provides several example problems that illustrate how the hierarchy of time and length scales associated with transport leads to the important features of cast microstructures. It includes equations for estimating microsegregation in cast alloys.
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005208
EISBN: 978-1-62708-187-0
... Abstract This article begins with balance equations for mass, momentum, energy, and solute and the necessary boundary conditions for solving problems of interest in casting and solidification. The transport phenomena cover a vast range of length and time scales, from atomic dimensions up...
Abstract
This article begins with balance equations for mass, momentum, energy, and solute and the necessary boundary conditions for solving problems of interest in casting and solidification. The transport phenomena cover a vast range of length and time scales, from atomic dimensions up to macroscopic casting size and from nanoseconds for interface attachment kinetics to hours for casting solidification. The article describes how to determine which phenomena are most important at the particular length and time scale for the problem. It concludes with several examples of the application of transport phenomena in solidification, focusing on microstructure formation.
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005234
EISBN: 978-1-62708-187-0
... A COMPLETE MODEL of the solidification of a metal alloy involves coupling of energy, species, mass, and momentum conservation equations ( Ref 1 , 2 , 3 , 4 , 5 , 6 , 7 ). This article examines critical features of four key areas of modeling transport phenomena associated with casting processes...
Abstract
This article examines the critical features of four key areas of modeling transport phenomena associated with casting processes. These include heat and species transport in a metal alloy, flow of the liquid metal, tracking of the free metal-gas surface, and inducement of metal flow via electromagnetic fields. Conservation equations that represent important physical phenomena during casting processes are presented. The article provides a discussion on how the physical phenomena can be solved. It provides information on a well-established array of general and specific computational tools that can be readily applied to modeling casting processes. The article also summarizes the key features of the conservation equations in these tools.
Series: ASM Handbook
Volume: 4F
Publisher: ASM International
Published: 01 February 2024
DOI: 10.31399/asm.hb.v4F.a0007008
EISBN: 978-1-62708-450-5
... differential equations for the fluid flow and heat-transfer phenomena during quenching. No solid-state transformations or stress-strain relationships are considered. The models consist of a set of differential equations based on the conservation of mass (continuity equation), momentum (motion equation...
Abstract
Computational fluid dynamics (CFD) provides an efficient, alternate, virtual approach for simulating and analyzing quenching processes with an impact on component design, manufacturing process, and quality. This article provides domain insights for quenching researchers and CFD practitioners for the modeling of the industrial quenching process and for supporting the diverse multifunctional needs in an industry, ranging from primary metallurgical companies (steel, aluminum, and other alloys), original equipment manufacturers, engineering companies, captive and commercial heat treating facilities, quench system manufacturers, and quench fluid suppliers. It describes the governing differential equations for the fluid flow and heat-transfer phenomena during quenching. The article also discusses different modeling categories to determine a CFD methodology for quenching.
Book: Composites
Series: ASM Handbook
Volume: 21
Publisher: ASM International
Published: 01 January 2001
DOI: 10.31399/asm.hb.v21.a0003402
EISBN: 978-1-62708-195-5
... respectively. V barrel = π DN , where D is the diameter of the screw and N is the speed in revolutions per second. Physical Laws Once the system is identified, one would express conservation of mass, momentum, and energy within the system boundaries ( Ref 21 , 22 ). The conservation equations...
Abstract
This article provides information on the classification of various composites manufacturing processes based on similar transport processes. The composites manufacturing processes can be grouped into three categories: short-fiber suspension methods, squeeze flow methods, and porous media methods. The article presents an overview of the modeling philosophy and approach that is useful in describing composite manufacturing processes.
Image
Published: 01 February 2024
Fig. 85 Schematic representation of a quenching system showing boundary conditions for momentum, mass, and energy equations. 1, inlet flow; 2, outlet flow; 3, nonslip wall conditions; 4, free surface; 5, internal mass and momentum source
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Image
in Advanced Industrial Quench System Design—Fluid Dynamics Analysis
> Quenchants and Quenching Technology
Published: 01 February 2024
Fig. 4 Schematic representation of a quenching system showing boundary conditions for momentum, mass, and energy equations. 1, inlet flow; 2, outlet flow; 3, nonslip wall condition; 4, free surface; 5, internal mass and momentum source
More
Book Chapter
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... that total energy, in all its forms, must be conserved These three principles lead to the basic three equations of motion: the mass, or continuity, equation; the momentum equation; and the total energy equation. Each of the articles that follow in this Section develops these fundamental balance...
Abstract
Computational fluid dynamics (CFD) is one of the tools available for understanding and predicting the performance of thermal-fluids systems. This article qualitatively describes the basic principles of CFD. The numerical methods, such as geometry description and discretization, used to solve the CFD equations are discussed. The article also demonstrates the application of CFD to a few casting problems.
Book Chapter
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005510
EISBN: 978-1-62708-197-9
... interface. Analogous to the magnetic diffusion equation, this interface is internal to the domain because the continuity of the interface shear force at the slag-metal interface is inherent in the momentum equation. The special aspects of the boundary conditions that arise due to the presence of the slag...
Abstract
This article provides an overview of the studies on computational modeling of the vacuum arc remelting (VAR) and electroslag remelting (ESR) processes. These models involve the axisymmetric analysis of the electromagnetic, flow, heat-transfer, and phase-change phenomena to predict the pool shape and thermal history of an ingot using two-dimensional axisymmetric models for VAR and ESR. Analysis of segregation of alloying elements during solidification that gives rise to macrolevel compositional nonuniformity in titanium alloy ingots is also described. The article discusses the important features of the control-volume-based computational method to review the unique aspects of the processes. Measurement of the properties of alloys and slags is explained and an analysis of the process variants for improving the predictive accuracy of the models is presented.
Series: ASM Handbook
Volume: 6
Publisher: ASM International
Published: 01 January 1993
DOI: 10.31399/asm.hb.v06.a0001480
EISBN: 978-1-62708-173-3
.... From another viewpoint, the equilibrium of forces is expressed by the conservation of momentum and the definition of the stress tensor, σ, the traction vector, τ, and the body force, b , in the differential equation: (Eq 17) ∇ ⋅ σ + b = m x ¨ In welding, it is typical to assume...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005587
EISBN: 978-1-62708-174-0
... the momentum and continuity equation. The thermal flux from the arc is prescribed on the weld pool and arc interface. With these data, the energy equation determines the position of the liquid-solid interface. With a new temperature distribution, the momentum equation is solved. If this iteration procedure...
Abstract
This article is a comprehensive collection of formulas and numerical solutions, addressing many heat-transfer scenarios encountered in welds. It provides detailed explanations and dimensioned drawings in order to discuss the geometry of weld models, transfer of energy and heat in welds, microstructure evaluation, thermal stress analysis, and fluid flow in the weld pool.
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005238
EISBN: 978-1-62708-187-0
... to solve them, and a few examples of recent applications in shape castings and continuous casting. Governing Equations The modeling of mechanical behavior requires solution of 1) the equilibrium or momentum equations relating force and stress, 2) the constitutive equations relating stress and strain...
Abstract
This article summarizes some issues and approaches in performing computational analyses of mechanical behavior, distortion, and hot tearing during solidification. It presents the governing equations and describes the methods used to solve them. The article reviews the finite element formulation, multidomain approaches, and arbitrary Lagrangian Eulerian method in solidification modeling. It illustrates the sand casting of braking disks and continuous casting of steel slabs.
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005898
EISBN: 978-1-62708-167-2
... Abstract This article focuses on the basic turbulent flow, and the thermal, mass-transfer, and hydrodynamic phenomena for use in modeling physical processes during induction melting. It provides a discussion on transport phenomena equations that includes the approximation of convective terms...
Abstract
This article focuses on the basic turbulent flow, and the thermal, mass-transfer, and hydrodynamic phenomena for use in modeling physical processes during induction melting. It provides a discussion on transport phenomena equations that includes the approximation of convective terms in the transport equation and computational schemes for the fluid dynamics equation. The aspects of computational algorithms for specific magnetohydrodynamic problems with mutual influence of the magnetic field and melt flow due to the changing shape of the free surface are also considered. The article illustrates the application of the basic equations and approaches formulated for electromagnetic field and melt turbulent flow for the numerical study of an induction crucible furnace.
Series: ASM Handbook
Volume: 8
Publisher: ASM International
Published: 01 January 2000
DOI: 10.31399/asm.hb.v08.a0003302
EISBN: 978-1-62708-176-4
... properties of materials. Several inelastic properties such as hardness, yield strength, and fracture toughness appear in a majority of these equations. However, the exponent of the velocity term in these models has been observed to range between 2 and 5. Based on the momentum considerations, these authors...
Abstract
This article describes a method for determining the dynamic indentation response of metals and ceramics. This method, based on split Hopkinson pressure bar testing, can determine rate-dependent characteristics of metals and ceramics at moderate strain rates. For example, dynamic indentation testing reveals a significant effect of loading rates on the hardness and the induced plastic zone size in metals and on the hardness and induced crack sizes of brittle materials. The article also explains the rebound and pendulum methods for dynamic hardness testing.
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