上海交通大学：《材料加工原理 Principles of Materials Processing》教学资源（课件讲稿）凝固和铸造 Solidification and casting_Transport phenomena in solidification（1/2）

先进材料疑固实验室 Laboratory of Advanced Materials Solidification 1896 1920 1987 2006 Transport phenomena in solidification(1) Dr.Mingxu Xia nced Ma 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

1896 1920 1987 2006 Transport phenomena in solidification (1) Dr. Mingxu Xia

先进材料疑固实验室 Outline Laboratory of Advanced Materials Solidification Brief introduction Momentum transfer Energy transfer ⊕ Mass transfer Commonalities among phenomena 上游充通大￥ SHANGHAI JIAO TONG UNIVERSITY

Outline Brief introduction Momentum transfer Energy transfer Mass transfer Commonalities among phenomena

先进材料疑固实验室 Brief introduction Laboratory of Advanced Materials Solidification Transport phenomena concerns the exchange of mass,energy, and momentum between observed and studied systems.While it draws from fields as diverse as continuum mechanics and thermodynamics. Mass,momentum,and energy transport all share a very similar mathematical framework,and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others. 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Brief introduction Transport phenomena concerns the exchange of mass, energy, and momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics. Mass, momentum, and energy transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others

先进材料疑固实验室 Brief introduction Laboratory of Advanced Materials Solidification Theoretical framework: Three phenomena: Three experimental laws: Momentum transport Newton's law of viscosity Energy transport *Fourier law of heat conduction Mass transport Fick's first law Three conservation laws: Three analytic methods: Conservation law of mass The overall balance method Newton's first law Micro-element balance method The 1st law of Thermodynamics *Shell balance method 上游文通大学 SHANGHAI JIAO TONG UNIVERSITY

Brief introduction Three phenomena：  Momentum transport  Energy transport  Mass transport Three experimental laws:  Newton’s law of viscosity  Fourier law of heat conduction  Fick’s first law Three analytic methods：  The overall balance method  Micro-element balance method  Shell balance method Three conservation laws：  Conservation law of mass  Newton’s first law  The 1st law of Thermodynamics Theoretical framework:

先进材料疑固实验室 Brief introduction Laboratory of Advanced Materials Solidification Every aspect of transport phenomena is grounded in two primary concepts:the conservation laws and the constitutive equations. Almost all of these physical phenomena ultimately involve systems seeking their lowest energy state in keeping with the principle of minimum energy. ory a vanced Materi 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Brief introduction Every aspect of transport phenomena is grounded in two primary concepts : the conservation laws and the constitutive equations. Almost all of these physical phenomena ultimately involve systems seeking their lowest energy state in keeping with the principle of minimum energy

先进材料疑固实验室 Momentum transfer Laboratory of Advanced Materials Solidification The fluid is treated as a continuous distribution of matter.The study of momentum transfer,or fluid mechanics can be divided into two branches:fluid statics(fluids at rest),and fluid dynamics(fluids in motion). Wanced Materia 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Momentum transfer The fluid is treated as a continuous distribution of matter. The study of momentum transfer, or fluid mechanics can be divided into two branches: fluid statics (fluids at rest), and fluid dynamics (fluids in motion)

先进材料疑固实验室 Momentum transfer Laboratory of Advanced Materials Solidification When a fluid is flowing in the x direction parallel to a solid surface, the fluid has x-directed momentum,and its concentration is vp.By random diffusion of molecules there is an exchange of molecules in the z direction.Hence the x-directed momentum has been transferred in the =-direction from the faster-to the slower-moving layer. 1 V(y) V(y) 上游文通大学 SHANGHAI JIAO TONG UNIVERSITY

Momentum transfer When a fluid is flowing in the x direction parallel to a solid surface, the fluid has x-directed momentum, and its concentration is υxρ. By random diffusion of molecules there is an exchange of molecules in the z direction. Hence the x-directed momentum has been transferred in the z-direction from the faster- to the slower-moving layer. v V(y) v V(y)

先进材料疑固实验室 Momentum transfer Laboratory of Advanced Materials Solidification The equation for momentum transport is Newton's Law of Viscosity written as follows: dpvx 【=-V02 where is the flux of x-directed momentum in the z direction,v is kinematic viscosity,np,the momentum diffusivity,z is the distance of transport or diffusion,p is the density,and n is the viscosity. Newton's Law is the simplest relationship between the flux of momentum and the velocity gradient. 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Momentum transfer The equation for momentum transport is Newton's Law of Viscosity written as follows: where τzx is the flux of x-directed momentum in the z direction, ν is kinematic viscosity, η/ρ, the momentum diffusivity, z is the distance of transport or diffusion, ρ is the density, and η is the viscosity. Newton’s Law is the simplest relationship between the flux of momentum and the velocity gradient

先进材料疑固实验室 Momentum tansfer Laboratory of Advanced Materials Solidification Continuity equation in one dimension Pu dA=PudA P,P2---the average density at cross p2,u,,dA section A and cross section A2 pu,dA u,42---the average density at cross section A and cross section A2 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Momentum tansfer Continuity equation in one dimension ρ1, ρ2 --- the average density at cross section A1 and cross section A2 u1, u2 --- the average density at cross section A1 and cross section A2

先进材料疑固实验室 Momentum transfer Laboratory of Advanced Materials Solidification Euler equation for momentum transfer Assumptions:ideal fluid,no viscosity,no internal friction Only mass force as gravity,inertia force,electro- magnetic force et al.being considered. At point (x,y,z),a volumetric unit,with Volume V,surface area A and density p, The mass force here would be: d (.v)=d dF 1dF X dm p dv Then,the entire force on the volume is: F=∫pf(x,y,,0dV 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Momentum transfer Euler equation for momentum transfer • Assumptions: ideal fluid, no viscosity, no internal friction Only mass force as gravity, inertia force, electro￾magnetic force et al. being considered. At point (x, y, z), a volumetric unit, with Volume V, surface area A and density ρ, The mass force here would be: Then, the entire force on the volume is: