CH.5. BALANCE PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - PowerPoint PPT Presentation

CH.5. BALANCE PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview  Balance Principles  Convective Flux or Flux by Mass Transport  Local and Material Derivative of a Volume Integral  Conservation of Mass  Spatial Form  Material Form  Reynolds Transport Theorem  Reynolds Lemma  General Balance Equation  Linear Momentum Balance  Global Form  Local Form 2

Overview (cont’d)  Angular Momentum Balance  Global Spatial  Local Form  Mechanical Energy Balance  External Mechanical Power  Mechanical Energy Balance  External Thermal Power  Energy Balance  Thermodynamic Concepts  First Law of Thermodynamics  Internal Energy Balance in Local and Global Forms  Reversible and Irreversible Processes  Second Law of Thermodynamics  Clausius-Planck Inequality 3

Overview (cont’d)  Governing Equations  Governing Equations  Constitutive Equations  The Uncoupled Thermo-mechanical Problem 4

5.1. Balance Principles Ch.5. Balance Principles 5

Balance Principles The following principles govern the way stress and deformation vary in the neighborhood of a point with time. REMARK  The conservation/balance principles : These principles are always valid, regardless of the type of  Conservation of mass material and the range of  Linear momentum balance principle displacements or deformations.  Angular momentum balance principle  Energy balance principle or first thermodynamic balance principle  The restriction principle :  Second thermodynamic law  The mathematical expressions of these principles will be given in,  Global (or integral) form  Local (or strong) form 6

5.2. Convective Flux Ch.5. Balance Principles 7

Convection  The term convection is associated to mass transport , i.e., particle movement.  Properties associated to mass will be transported with the mass when convective transport there is mass transport (particles motion)  Convective flux of an arbitrary property through a control A surface : S   amountof crossing A S S unitoftime 8

Convective Flux or Flux by Mass Transport  Consider:  An arbitrary property of a continuum medium (of any tensor order) A    x  The description of the amount of the property per unit of mass , , t ( specific content of the property ) . A  The volume of particles crossing a dV differential surface during the dS    interval is , t t dt     dV dS dh v n dt dS      dm dV v n dSdt  Then,  The amount of the property per unit of mass crossing the differential  surface per unit of time is: dm       d v n dS S dt 9

Convective Flux or Flux by Mass Transport  Consider:  An arbitrary property of a A inflow continuum medium (of any tensor order)   0 v n outflow  The specific content of ( the amount   A v n 0    x per unit of mass) . , t  Then,  The convective flux of through a spatial surface, , with unit S A normal is: n v   is velocity       v n Where: t dS  is density S s    If the surface is a closed surface, , the net convective flux is: S V         t v n dS = outflow - inflow  V  V 10 MMC - ETSECCPB - UPC 11/11/2015

Convective Flux REMARK 1 The convective flux through a material surface is always null. REMARK 2 Non-convective flux ( advection, diffusion, conduction ). Some properties can be transported without being associated to a certain mass of particles. Examples of non-convective transport are: heat transfer by conduction, electric current flow, etc. Non-convective transport of a certain property is characterized by the non-   convective flux vector ( or tensor) : , t q x        non-convectiveflu x convectivefl u x ; q n dS v n dS s s convective non-convective flux flux vector vector 11 MMC - ETSECCPB - UPC 11/11/2015

Example   Compute the magnitude and the convective flux which correspond to the S following properties: a) volume b) mass c) linear momentum d) kinetic energy 12

   Example - Solution      t v n dS S s a) If the arbitrary property is the volume of the particles:  A V The magnitude “property content per unit of mass” is volume per unit of mass, i.e., the inverse of density: 1 V     M The convective flux of the volume of the particles through the surface is: V S 1          VOLUME FLUX v n dS v n dS S s s 13

   Example - Solution      t v n dS S s b) If the arbitrary property is the mass of the particles:  A M The magnitude “property per unit of mass” is mass per unit of mass, i.e., the unit value: M    1 M The convective flux of the mass of the particles through the surface is: S M          1 MASS FLUX v n dS v n dS S s s 14

   Example - Solution      t v n dS S s c) If the arbitrary property is the linear momentum of the particles:  A v M The magnitude “property per unit of mass” is mass times velocity per unit of mass, i.e., velocity: M v    v M The convective flux of the linear momentum of the particles through the M v surface is: S        v v n dS MOMENTUM FLUX S s 15

   Example - Solution      t v n dS S s d) If the arbitrary property is the kinetic energy of the particles: 1  2 A M v 2 The magnitude “property per unit of mass” is kinetic energy per unit of mass, i.e.: 1 2 M v 1 2    2 v 2 M 1 The convective flux of the kinetic energy of the particles through the 2 2 M v surface is: S 1        2 KINETIC ENERGY FLUX v v n dS S 2 s 16

5.3. Local and Material Derivative of a Volume Integral Ch.5. Balance Principles 17

Derivative of a Volume Integral  Consider:  An arbitrary property of a continuum medium (of any tensor order) A  The description of the amount of the property per unit of volume    x (density of the property ) , , t A REMARK   and are related     through .  The total amount of the property in an arbitrary volume is: V   Q t        , Q t x t dV V     Q t t  The time derivative of this volume integral is:        Q t t Q t     lim Q t    t t 0 18 MMC - ETSECCPB - UPC 11/11/2015

Local Derivative of a Volume Integral    Consider: Q t         The volume integral , Q t x t dV     Q t t V Control Volume, V    The local derivative of is: Q t            , , x t t dV x t dV  local REMARK not       V V , lim x t dV   The volume is fixed in derivative   t t t 0 V space (control volume).  It can be computed as:            , , x t t dV x t dV         Q t t Q t        V V , lim lim x t dV        t t t t 0 t 0 V           [ , , ] x t t x t dV              , , , x x x t t t t      lim V lim dV dV        t t t 0 0 t t     V V     , x t      t 19

Material Derivative of a Volume Integral  Consider:         The volume integral , Q t x t dV V        Q t  The material derivative of is: Q t t Q t not d material       , x t dV derivative dt  V V t       REMARK      , , x t t dV x t dV   ( ) ( ) The volume is mobile in space V t t V t lim    0 t t and can move, rotate and deform (material volume).  It can be proven that:         d  d                        dV , x v  v   v  t dV dV dV dV       t dt t dt      V V V V V V       t convective material local derivative of derivative of derivative of the integral the integral the integral 20

5.4. Conservation of Mass Ch.5. Balance Principles 21