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Lecture 6.2: Conservation of Energy (C-Energy), and Energy Transfer as Work of (Surface) Forces Energy as A Conserved Quantity Conservation of Energy for An Isolated System • Conservation of Energy for A MV (Closed System) 1. Modes of Energy Transfer LHS: (TE + ME + Others) • Decomposition of Energy Transfer: • Energy Transfer As Work (of A Force) Heat + Work + [Others, if any] Decomposition of Work of Surface Force: Pressure + Shear Finite Control Volume Formulation of Physical Laws: C-Energy Conservation of Energy (Working Forms) Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] Example of Energy Transfer as Work of (Surface) Forces: Pump and Turbine • abj • + [Modes of] Energy Transfer • Q (Heat/TE + Work/ME + Others) 2. Forms of Energy Stored W • Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces Here we limit ourselves to an observer in an inertial frame of reference (IFR) only. Note that kinetic energy (KE) – being defined from velocity - is frame of reference dependence, i.e., observers moving relative to each other observe different amount of KE for the same mass. 1 Very Brief Summary of Important Points and Equations Q W + Q W ...... C-Energy for A MV Time Rate of Energy Transfer to MV from its surroundings in various modes (as heat and work,etc.) dEMV dt Time Rate of Change/Increase in Energy Stored in MV in various forms (TE and ME) C-Energy (Working Forms) for A CV d Q W e( dV ) (e pv)( V f / s dA); dt CV CS : e-pv - form e pv V2 eu gz 2 V2 u pv gz, 2 me : : : abj : u-me - form h - form ho - form V2 gz pv ke pe 2 u me , me : pv V2 h gz, 2 h : u pv ho gz, V2 ho : h = stagnation enthalpy 2 W : W shaft W shear W others 2 Energy as A Conserved Quantity/Scalar • Conservation of Energy for An Isolated System • Conservation of Energy for A MV (Closed System) abj 3 “According to Classical Mechanics” Let’s say, the universe – that we are a part of - is an isolated system. Conservation of Mass According to classical mechanics, there are 249 689 127 954 677 702 907 942 097 982 129 076 250 067 682 009 482 730 602 701 620 707 616 740 576 190 705 687 196 070 561 076 076 104 051 876 549 701 707 617 048 651 671 076 017 057 901 710 461 765 379 480 547 610 707 617 019 641 127 kg of mass in the universe. Also Conservation of Energy According to classical mechanics, there is a total of 580 140 804 219 884 603 733 864 586 354 599 887 940 543 537 431 687 943 187 603 734 360 687 465 465 075 940 408 562 545 546 454 651 326 406 306 302 135 543 067 654 987 651 861 684 616 846 516 516 576 516 546 165 131 986 543 074 921 975 970 297 249 027 290 579 540 410 434 573 805 706 076 J of energy in the universe. Of course, the numbers are not real (I made them up, obviously), but you get the idea of the concept of conservations of mass and energy. [Both are conserved scalar/quantity.] abj According to classical mechanics, energy – like mass – is a conserved scalar/quantity. 4 Energy as A Conserved Quantity/Scalar Conservation of Energy for An Isolated System Universe (Isolated System) EU = Constant (Conserved) abj dEU = 0 5 Relation Between Changes of Various Parts U = MV + Surroundings Universe (Isolated System) EU = Constant (Conserved) EU = EMV + ESur An Isolated System Surroundings, ESur dEU = 0 EMV = Constant MV (Closed System) Total Amount - dESur Universe (Isolated System) EU = Constant (Conserved) - dESur = abj changes dEU = 0 dEMV dEU = dEMV + dESur = 0 Relation between various parts Surroundings dEMV of Change/Increase in Energy Stored The amount of energy transferred to a system must come from its surroundings. 6 Energy as A Conserved Quantity/Scalar Conservation of Energy for A MV (Closed System) Focus on a MV (closed system) as a part of the Universe Universe (Isolated System) EU = Constant (Conserved) - dESur dEU = 0 Energy Transfer to MV from its surroundings in various modes dEMV dEU = dEMV + dESur = 0 Change/Increase in Energy Stored in MV in various forms Surroundings dESur Let’s denote the LHS instead by ET.(= - dESur) E T Energy Transfer to MV from its surroundings in various modes abj dEMV dEMV Change/Increase in Energy Stored in MV in various forms 7 Conservation of Energy for A MV (Closed System) 1. Modes of Energy Transfer 1. 2. 3. 2. Q , Thermal Energy Transfer) W , Mechanical Energy Transfer) Other Modes of Energy Transfer ( E ) T Q, Energy Transfer As Work (W , Energy Transfer As Heat ( Forms of Energy Stored 1. Thermal Energy (TE) 2. Mechanical Energy (ME) 3. Other Forms of Energy Stored abj 8 Conservation of Energy for Modes of Energy Transfer Energy Transfer to MV from its surroundings in A MV (Closed System) and Forms of Energy Stored ET various modes dEMV Change/Increase in Energy Surroundings Stored in MV in various forms MV (Closed System) E T Energy Transfer to MV from its surroundings in various modes dEMV Change/Increase in Energy Stored in MV in various forms Modes of Energy Transfer Forms of Energy Stored Energy Transfer as Heat Q = Thermal energy transfer Thermal energy Energy Transfer as Work W = Mechanical energy transfer Mechanical energy ME Other modes of energy transfer ET (e.g., electromagnetic radiation, Other forms of energy stored ( e.g., etc.) electrical, chemical, etc.) KEY: abj TE (= U) (= KE) Regardless of the number of modes of energy transfer and forms of energy stored, the basic idea of the conservation of energy is that All must be accounted for so that EU is conserved or - dESur = dEMV (a simple balance law) 9 Conservation of Energy for In most of our problems of interest, only are excited/changed A MV (Closed System) 1) Thermal Energy (TE) and 2) Mechanical Energy (ME) Energy Transfer ET Q W ET ,others Key: If some other forms of energy are also excited/changed, they must be taken into accounted according to the conservation of energy. to MV from its surroundings in dEMV d (TE ME ) MV various modes Change/Increase in Energy Surroundings Stored in MV in various forms MV (Closed System) E T Energy Transfer to MV from its surroundings in various modes Q W + dEMV Change/Increase in Energy Stored in MV in various forms Q W ET dEMV Energy Q ET dEMV dt Energy Time W ET = Q + W + [ET ] Q = Heat W = Work abj EMV TE + ME [+ Other forms] = Thermal energy transfer TE = Thermal energy = Mechanical energy transfer ME = Mechanical energy ET = Other modes of energy transfer …... = Other forms of energy stored 10 C-Energy for A MV (Closed System) Time Rate of Energy Transfer Q W Wothers (if any) to MV from its surroundings in dEMV dt various modes W Q Time Rate of Change/Increase in Energy + Surroundings Stored in MV in various forms MV (Closed System) Q W ..... Time Rate of Energy Transfer to MV from its surroundings in various modes (as heat and work, etc.) abj dEMV dt Time Rate of Change/Increase in Energy Stored in MV in various forms (TE and ME) 11 Work Body mgof mg and Potential Energy [1] Scratch of Note: ProofForce of Work z W mg mg V ˆ m( gk ) V mgV z dz dt d (mgz ) dt mg d ( PE ) MV W mg , dt abj g g kˆ y x mg V PE : mgz 12 The Two Forms of C-Energy for A MV (Closed System) (according to where we put the work of mg / potential energy) Q W Form 1 dEMV , dt : 1 E TE ME U KE U mV 2 2 TE Form 2 ME : W must include work of mg , : 1 E TE ME U ( KE PE ) U mV 2 mgz 2 TE : abj ME W must not include work of mg , Q W Q W W ... W mg ... W mg d ( PE ) MV Q W dt Q W d (U KE ) MV dt d (U KE ) MV , dt d (U KE ) MV , dt d (U KE PE ) MV dt W ... W mg ... W W W mg d ( PE ) MV W mg dt 13 Sign Conventions for The Energy Equation Energy input into a system causes increase in energy of the system. Energy extracted from a system causes decrease in energy of the system. W Q + W Q + dEMV Equation : Q W dt dEMV Equation : Q W dt Equation : positive Q Physics: Q - input causes positive dEMV / dt Equation : positive W Physics: W - input causes positive dEMV / dt Equation : positive Q Physics: Q - input causes positive dEMV / dt Similar can be said for abj Q + causes causes E-increase E-increase E-increase Equation : positive W causes negative dEMV / dt W - output Physics: W causes W causes E-decrease Q + 14 C-Energy for A MV (Closed System) Q W ...... Time Rate of Change/Increase in Energy Stored in MV in various forms (TE and ME) Time Rate of Energy Transfer to MV from its surroundings in various modes (as heat and work) LHS: abj dEMV dt [Modes of] Energy Transfer 1. Energy Transfer as Heat Q [Thermal Energy Transfer] 2. Energy Transfer as Work W [Mechanical Energy Transfer] 15 Modes of Energy Transfer on The LHS LHS = Energy Transfer to MV dEMV (t ) dt Q W Time rate of energy transfer to MV (t ) as heat and work Time rate of change of energy of MV (t ) Thermal Energy Transfer Mechanical Energy Transfer ) (as Heat Q ) (as Work of Forces W Like Recall in C-Mom Keys F 1. Recognize various types of forces. 2. Know how to find the resultant of various types of forces (e.g., pressure, etc.). 3. abj Energy Time Energy Transfer in Other Modes F W others in C-Mom, regardless of how it is written or notations used, the key idea is to sum all (the modes of) the energy transfers to MV. Keys: Energy Transfer to MV Q W ... 1. Recognize various types/modes of energy transfers. 2. Know how to find the energy transfer of various types/modes (e.g., heat (TE), work (ME), electrical (EE), etc.). 3. Sum all the energy transfers to MV. Sum all the external forces. 16 W Energy Transfer Modes (between a system and its surroundings) Heat Q + Work ( Q q dA ) If any other Work of Forces Other Modes W F V dF of Energy Transfer (input-positive) Work of Surface Force/Stress Work of Body Force/mg W S V dFS V T dA W B V dFB V BdV Stress vector T Tnormal Ttangential Normal (Pressure) Tnormal peˆn Tangential (Shear) Ttangential Tshear Through a finite surface S : Q q dA (input-positive) S abj W p pv V dA S (input-positive) W shear Tshear V dA S (input-positive) Work of mg is later accounted for as potential energy W others (input-positive) e.g. electrical, electromagnetic, etc. 17 If there are other body forces besides mg, all must be accounted for. Energy Transfer As Work of A Force W F [Mechanical Energy Transfer] abj 18 W : Energy Transfer as Work (Mechanical Energy Transfer) Q W ..... Time rate of energy transfer to MV (t ) as heat and work Pressure p Coincident CV(t) and MV(t) CV(t) MV(t) Shear t Fi FBD Volume/Body Force gdm g ( dV ) Work is the mode of (mechanical) energy transfer. Work is work of a force, W F In order to apply C-Energy, W on the LHS must be the sum of all the energy transfers as work, i.e., the sum of works of all the forces. Recall then abj Forces in Fluids and FBD 19 Recall 1: Recall all and various types of forces. W must be the sum of the works of all the forces on MV(t). F and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t) Net external force Coincident CV(t) and MV(t) CV(t) MV(t) Pressure p 2. Distributive Surface Force (in fluid part) 1. Concentrated/Point Surface Force Shear t FBD Fi Net Surface Force FS 1. Concentrated/Pointed Surface Force Volume/Body Force gdm g ( dV ) F Fi 2. Distributive Surface Force in Fluid [Pressure p + Friction t ] abj FS FB Net Volume/Body Force FB mg g ( dV ) CV MV 20 Recall 2: Energy Transfer as Work of A Force (Mechanical Energy Transfer) Work of A Force F ( WF , W F ) F Concept V WF F V dS Vdt Work = Force x Energy Time Displacement in the direction of the force (per unit time) Particle WF F dS WF F V abj Energy Energy Time 21 Energy Transfer as Work of A Force (Mechanical Energy Transfer) Particle VS Continuum Body Work of A Force F , WF , W F F V WF V dF Energy Time Particle WF F dS WF F V W FB V dFB V BdV V dF V T dA abj W S FS V dFB Energy Time Continuum Body Energy Same concept, just that Energy Time W F (V B)dV B V V Work = Force x Displacement in the direction of the force WF F V dFS Same Concept dS Vdt dA W F V T dA S S 1) there are more types of forces to be accounted for: Surface force and Body force (and…) 2) Each type is described differently dFS T dA , dFB BdV , Force T Area Force B Volume 3) As before, how to sum them all. 22 Work of All Forces W Pressure p 2. Distributive Surface Force Shear t (in fluid part) 1. W Note abj Concentrated/Point Surface Force W S ( Surface force) W p ( pressure) W s ( shear) W s (shaft) = Fi Coincident CV(t) and MV(t) CV(t) MV(t) FBD Volume/Body Force gdm g ( dV ) W B ( Body force) W mg Energy Time Wconcentrated surface force Wothers (if Shaft work is work due to shear stress (surface force) at the cross section of a shaft. 23 any) Work of Surface Forces: 1) Pressure Force (Flow Work), 2) Shear Force dA Recall the coincident CV(t) and MV(t) dFS dFpressure dFshear CV(t) W Q V MV(t) Surroundings + S dFS dF pressure dFshear ( pdA) dFshear Work of pressure force on CS/MS: Work of shear force on CS/MS: • • Infinitesimal work of pressure force: W p V dFp V pdA pV dA pv(V dA) Infinitesimal work of shear stress: W shear V dFshear input into MV positive input into MV positive 1. Rate of work (power) done on a finite closed surface S: W p pv V dA abj 1. Rate of work (power) done on a finite closed surface S: W p W p S V Tshear dA W shear W shear S input into MV positive S W shear V Tshear dA S input into MV positive 24 Finite Control Volume Formulation of Physical Laws C-Energy abj 26 Finite CV Formulation of Physical Laws: C- Energy Recall the coincident CV(t) and MV(t) W Q Q Surroundings dEMV/dt + W Material Volume (MV) Energy transfer as heat Energy transfer as work of forces p, t CV(t), MV(t) C-Energy: N E, e Physical Laws Q W Time rate of energy transfer to MV (t ) as heat and work dEMV (t ) dt Time rate of change of energy of MV (t ) dECV (t ) dt Time rate of change of energy of CV (t ) e( V f / s dA), CS (t ) dm dQ Energy Time Net convectionefflux of energy through CS (t ) RTT abj 27 Finite CV Formulation of Physical Laws: C- Energy ~ d E MV (t ) Q W , dt : ~ E U KE , 1 e~ u V 2 , 2 W W W : W p W shear shaft mg Energy Time , W others ~ dE MV (t ) Q W Wmg , W : W p W shear W shaft W others dt d (U KE ) MV (t ) d ( PE ) MV (t ) d ( PE ) MV (t ) , W mg dt dt dt dEMV (t ) 1 Q W , N E U KE PE , e u V 2 gz dt 2 : W : W p W shear W shaft W others dEMV (t ) Q W W p , W : dt Apply RTT to dEMV/dt dECV (t ) e( V f / s dA) dt CS dm dQ W shear W shaft W others , CS pv( V f / s dA), dm dQ W p pv( V f / s dA) CS dm dQ dECV (t ) (e pv)( V f / s dA) dt CS dm dQ dE (t ) Q W CV (e pv)( V f / s dA) dt CS dm dQ d e( dV ) (e pv)( V f / s dA) dt CV CS dm dQ abj : W : W shaft W shear W others at various steps. To save some symbols, here we redefine W 28 C-Energy (Working Forms) Recall the coincident CV(t) and MV(t) Q Surroundings W dEMV/dt Q W Material Volume (MV) + Energy transfer as heat Energy transfer as work of forces p, t CV(t), MV(t) d (e pv)( V dA); Q W e ( dV ) f /s dt CV CS : e-pv - form e pv V2 eu gz 2 V2 u pv gz, 2 me : : : abj : u-me - form u me , h - form h ho - form V2 gz, 2 ho gz, W : W shaft W shear W others V2 me : pv gz pv ke pe 2 h : u pv V2 ho : h = stagnation enthalpy 2 29 Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] abj 30 Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] Later on, we will be writing the C-Energy in various specialized forms, e.g., d Q W e( dV ) (e pv)( V f / s dA), dt CV CS : W W shaft W shear W others 1 e : u V 2 gz 2 Here, we will first focus and emphasize the basic idea of energy transfer as work of (surface) forces first. So, let us step back one step by moving the flow work term (pv) back to the LHS. d Q W e( dV ) (e)( V f / s dA), dt CV CS abj 1 e : u V 2 gz 2 31 Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] 3. Stationary Imaginary surface (where there is mass flow in/out.) Pressure p Shear t V Solid part V 0 1. Moving solid surface (e.g., pump impeller surface, cross section of a rotating solid shaft) abj V 2. Stationary solid surface (e.g., pump casing) 32 Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] 3. Stationary Imaginary surface Work due to pressure force here is later moved to the RHS and included as flow work, pv, in the convection flux term: (where there is mass flow in/out.) In general, W p,t V dFp,t , 0 Pressure p Shear t V 0 ( e p v) ( V d A ) f /s CS (except dF V ) V Note: For moving imaginary surface, we may use the decomposition W p ,t V dF p ,t , V Vs V f / s (Vs V f / s ) dF p ,t Solid part V 0 V 2. Stationary solid surface (e.g., pump casing) W p ,t V dF p,t , V 0 (no slip) 0 1. Moving solid surface (e.g., pump impeller surface, cross section of a rotating solid shaft) In general, W p,t V dFp,t , abj 0 V 0 (no slip) (except dF V ) 33 Example of Energy Transfer as Work of (Surface) Forces: Pump and Turbine Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces abj 34 Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces m 1 1 Turbine Pump a 1(pump) b c d 2(pump) 1(turbine) 2(turbine) • CV includes the fluid stream only, no solid part. 1 2 2 m 2 • CV includes the fluid stream, the solid impeller, and a section of the solid shaft. • It cuts through the cross section of a solid shaft. Surface Force: Pressure and shear on moving/rotating impeller surface 1 2 Surface Force: Normal and shear stress over the moving/rotating cross section of a solid shaft MV MV Surface Force Pressure and shear abj Surface Force Pressure and shear FBD • Surface force: pressure/normal and shear stresses, over all surfaces. [Body force is not shown.] 35 • Energy transfer as work of (surface) forces occurs at moving material surfaces where there are surface forces act. There can be no energy transfer as work of forces at a stationary material surface. In order to have energy transfer as work of forces (in this case, surface forces), • the point of application of the force must have displacement (in the direction of the force). dF V r Surroundings V r Q W F MV V W F Pressure and shear stresses on the rotating impeller surfaces act on the moving fluid Energy transfer as work to MV (fluid stream) abj Surroundings dEMV , dt V dF Energy Time dF t dA eˆ dF MV V MV 36 W f Energy transfer as work of forces at the surface of the moving/rotating impeller Surroundings Surroundings MV W F V dF [Pump] • Pressure force pushes fluid, V W f Energy transfer as work of force at the rotating impeller surface abj MV • Shear force drags fluid, such that the fluid at the material surface has velocity V . 37 Energy transfer as work of forces at the cross section of a solid shaft W s Energy transfer as work of force at the rotating cross section of a solid shaft. W F V dF V W s W F V dF MV dF ( r ) dF Surroundings MV T Vector triple product identity : ( r ) dF (r dF ) ( r dF ) dT : r dF dT , dT W dF dF (dAzt z )eˆ Shear stress at a cross section of a solid shaft. • It is due to the other section of the shaft (surroundings) acting on our section of the shaft (MV). shaft shaft cross sec tion dT shaft cross sec tion W shaft T dF , T = External force and torque due to surroundings on our MV abj (Recall the concept of FBD and Newton’s Second Law) 38 dF dF V Surroundings Motor Motor/Turbine drives its Pump/Load dF T Direction of mechanical energy transfer as work [Motor, Turbine] Turbine 0 Wshaft (T ) MV 0 Pump Load Surroundings MV abj W shaft T T V MV (T ) MV 0 MV gives up its own mechanical energy to the surroundings. [Pump, Load] MV receives mechanical energy from the surroundings. W shaft (T ) MV 0 39 Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces W s CV2 / MV2 CV1 / MV1 W s W f CV1 / MV1 CV1 / MV1 [See W s , but do not see W f .] • [FBD] sees the shear stress at the rotating shaft cross section, 1 W f CV2 / MV2 • [Work] sees the energy transfer as work at the rotating shaft cross section. 2 CV2 / MV2 [See W f , but do not see W s .] • [FBD] sees the pressure and shear stresses on the rotating impeller surface. • [Work] sees the energy transfer as work at the rotating impeller surface. abj 40