Energy as A Conserved Quantity Conservation of Energy for An Isolated System

1. + 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) • Modes of Energy Transfer (Heat/TE + Work/ME + Others) • Forms of Energy Stored (TE + ME + Others) • LHS: [Modes of] Energy Transfer • Decomposition of Energy Transfer: Heat + Work + [Others, if any] • Energy Transfer As Work (of A Force) • 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 • 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.

2. + C-Energy for A MV C-Energy (Working Forms) for A CV e-pv - form u-me - form h - form ho - form = stagnation enthalpy Very Brief Summary of Important Points and Equations

3. Energy as A Conserved Quantity/Scalar • Conservation of Energy for An Isolated System • Conservation of Energy for A MV (Closed System)

4. “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.] • According to classical mechanics, energy – like mass – is a conserved scalar/quantity.

5. Energy as A Conserved Quantity/Scalar Conservation of Energy for An Isolated System Universe (Isolated System) EU = Constant (Conserved)  dEU = 0

6. Universe (Isolated System) EU = Constant (Conserved)  dEU = 0 Surroundings, ESur EMV MV (Closed System) EU = EMV + ESur = Constant Total Amount - dESur Universe (Isolated System) EU = Constant (Conserved)  dEU = 0 Surroundings dEMV Change/Increase in Energy Stored dEU = dEMV + dESur = 0 - dESur = dEMV The amount of energy transferred to a system must come from its surroundings. Relation between changes of various parts Relation Between Changes of Various Parts U = MV + Surroundings An Isolated System

7. Universe (Isolated System) EU = Constant (Conserved)  dEU = 0 Energy Transfer to MV from its surroundings in various modes - dESur dEMV Change/Increase in Energy Stored in MV in various forms dEU = dEMV + dESur = 0 Surroundings Let’s denote the LHS instead by dET.(= - dESur) Energy as A Conserved Quantity/Scalar Conservation of Energy for Focus on a MV (closed system) as a part of the Universe A MV (Closed System)

8. Conservation of Energy for A MV (Closed System) • Modes of Energy Transfer • Energy Transfer As Heat ( , Thermal Energy Transfer) • Energy Transfer As Work ( , Mechanical Energy Transfer) • Other Modes of Energy Transfer ( ) • Forms of Energy Stored • Thermal Energy (TE) • Mechanical Energy (ME) • Other Forms of Energy Stored

9. dET Energy Transfer to MV from its surroundings in various modes dEMV Change/Increase in Energy Stored in MV in various forms MV (Closed System) Surroundings • Modes of Energy Transfer • Energy Transfer as Heat dQ = Thermal energy transfer • Energy Transfer as WorkdW = Mechanical energy transfer • Other modes of energy transferdET (e.g., electromagnetic radiation, etc.) • Forms of Energy Stored • Thermal energy TE (= U) • Mechanical energyME (= KE) • Other forms of energy stored( e.g., electrical, chemical, etc.) Conservation of Energy for Modes of Energy Transfer and Forms of Energy Stored A MV (Closed System) KEY: 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 thatEUis conserved or- dESur = dEMV(a simple balance law)

10. + Key: If some other forms of energy are also excited/changed, they must be taken into accounted according to the conservation of energy. Energy Transfer to MV from its surroundings in various modes Change/Increase in Energy Stored in MV in various forms Surroundings MV (Closed System) dET = dQ + dW + [dET ] dQ = Heat = Thermal energy transfer dW = Work = Mechanical energy transfer dET= Other modes of energy transfer EMV = TE + ME [+ Other forms] TE = Thermal energy ME = Mechanical energy …... = Other forms of energy stored A MV (Closed System) Conservation of Energy for In most of our problems of interest, only 1) Thermal Energy (TE) and 2) Mechanical Energy (ME) are excited/changed

11. + Time Rate of Energy Transfer to MV from its surroundings in various modes Time Rate of Change/Increase in Energy Stored in MV in various forms Surroundings MV (Closed System) C-Energy for A MV (Closed System)

12. z y x Work of Body Force mg and Potential Energy [1] Scratch Note: Proof of Work of mg

13. Form 1 Form 2 The Two Forms of C-Energy for A MV (Closed System)(according to where we put the work of mg / potential energy)

14. + + + Physics: - input causes E-increase Physics: - input causes E-increase Physics: - input causes E-increase + Physics: - output causes E-decrease Similar can be said for 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.

15. LHS: [Modes of] Energy Transfer • Energy Transfer as Heat [Thermal Energy Transfer] • Energy Transfer as Work [Mechanical Energy Transfer] C-Energy for A MV (Closed System)

16. LHS = Energy Transfer to MV Mechanical Energy Transfer (as Work ofForces) Thermal Energy Transfer (as Heat ) Energy Transfer in Other Modes Like 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. • Recall in C-Mom • Keys • Recognize varioustypes of forces. • Know how to findthe resultantof various types of forces (e.g., pressure, etc.). • Sum all the external forces. • Keys: Energy Transfer to MV • Recognize varioustypes/modes of energy transfers. • Know how to findthe energy transferof various types/modes (e.g., heat (TE), work (ME), electrical (EE), etc.). • Sum all the energy transfers to MV. Modes of Energy Transfer on The LHS

17. + Heat ( ) Work of Forces (input-positive) Work of Surface Force/Stress Work of Body Force/mg Normal (Pressure) Stress vector Tangential (Shear) (input-positive) e.g. electrical, electromagnetic, etc. Through a finite surface S : (input-positive) (input-positive) Work of mg is later accounted for as potential energy (input-positive) Energy Transfer Modes (between a system and its surroundings) Work If any other Other Modes of Energy Transfer If there are other body forces besides mg, all must be accounted for.

18. Energy Transfer As Work of A Force [Mechanical Energy Transfer]

19. Coincident CV(t) and MV(t) Pressure p CV(t) MV(t) Shear t FBD Volume/Body Force • Work is the mode of (mechanical) energy transfer. • Work is work of a force, • In order to apply C-Energy, 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 Forces in Fluids and FBD : Energy Transfer as Work (Mechanical Energy Transfer)

20. Recall 1: Recall all and various types of forces. must be the sum of the works of all the forces on MV(t). Pressure p Coincident CV(t) and MV(t) CV(t) 2. Distributive Surface Force (in fluid part) MV(t) Shear t FBD • Concentrated/Point Surface Force Volume/Body Force Net Surface Force Net Volume/Body Force 1. Concentrated/Pointed Surface Force 2. Distributive Surface Force in Fluid [Pressure p + Friction t ] and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)

21. Work of A Force ( ) Concept Work = Force x Displacement in the direction of the force (per unit time) Particle Recall 2: Energy Transfer as Work of A Force (Mechanical Energy Transfer)

22. Work of A Force , Same Concept Work = Force x Displacement in the direction of the force Continuum Body Particle • Same concept, just that • there are more types of forces to be accounted for: Surface force and Body force (and…) • Each type is described differently • 3) As before, how to sum them all. Energy Transfer as Work of A Force (Mechanical Energy Transfer)Particle VS Continuum Body

23. Pressure p Coincident CV(t) and MV(t) CV(t) 2. Distributive Surface Force (in fluid part) MV(t) Shear t FBD • Concentrated/Point Surface Force Volume/Body Force Note = Shaft work is work due to shear stress (surface force) at the cross section of a shaft. Work of All Forces

24. + CV(t) MV(t) Surroundings S • Work of pressure force on CS/MS: • Infinitesimal work of pressure force: • Rate of work (power) done on a finite closed surface S: • Work of shear force on CS/MS: • Infinitesimal work of shear stress: • Rate of work (power) done on a finite closed surface S: Work of Surface Forces: 1) Pressure Force (Flow Work), 2) Shear Force Recall the coincident CV(t) and MV(t)

25. Finite Control Volume Formulation of Physical Laws C-Energy

26. + Energy transfer as heat Surroundings dEMV/dt Physical Laws Energy transfer as work of forces Material Volume(MV) p, t CV(t), MV(t) C-Energy: RTT Finite CV Formulation of Physical Laws: C- Energy Recall the coincident CV(t) and MV(t)

27. Apply RTT to dEMV/dt To save some symbols, here we redefine at various steps. Finite CV Formulation of Physical Laws: C- Energy

28. + Energy transfer as heat Surroundings dEMV/dt Energy transfer as work of forces Material Volume(MV) p, t CV(t), MV(t) e-pv - form u-me - form h - form ho - form = stagnation enthalpy C-Energy (Working Forms) Recall the coincident CV(t) and MV(t)

29. Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force]

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., • 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.

31. 3. Stationary Imaginary surface (where there is mass flow in/out.) Pressurep Sheart Solid part • Moving solid surface • (e.g., pump impeller surface, cross section of a rotating solid shaft) 2. Stationary solid surface (e.g., pump casing) Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force]

32. 3. Stationary Imaginary surface (where there is mass flow in/out.) In general, Work due to pressure force here is later moved to the RHS and included as flow work, pv, in the convection flux term: Pressurep Sheart Solid part 2. Stationary solid surface (e.g., pump casing) 1. Moving solid surface (e.g., pump impeller surface, cross section of a rotating solid shaft) In general, Note: For moving imaginary surface, we may use the decomposition Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force]

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

34. 1 Turbine Pump 1 1 2 2 2 a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) Surface Force: Normal and shear stress over the moving/rotating cross section of a solid shaft Surface Force: Pressure and shear on moving/rotating impeller surface Surface Force Pressure and shear Surface Force Pressure and shear MV MV Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces • CV includes the fluid stream only, no solid part. • 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. • FBD • Surface force: pressure/normal and shear stresses, over all surfaces. [Body force is not shown.]

35. Surroundings Surroundings MV MV 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). MV Pressure and shear stresses on the rotating impeller surfaces act on the moving fluid  Energy transfer as work to MV (fluid stream)

36. Surroundings Surroundings MV • [Pump] • Pressure force pushes fluid, • Shear force drags fluid, • such that the fluid at the material surface has velocity . MV Energy transfer as work of force at the rotating impeller surface Energy transfer as work of forces at the surface of the moving/rotating impeller

37. Energy transfer as work of force at the rotating cross section of a solid shaft. Surroundings MV MV • 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). = External force and torque due to surroundings on our MV (Recall the concept of FBD and Newton’s Second Law) Energy transfer as work of forces at the cross section of a solid shaft

38. MV Surroundings Direction of mechanical energy transfer as work Motor Turbine [Motor, Turbine] MV gives up its own mechanical energy to the surroundings. Motor/Turbine drives its Pump/Load [Pump, Load] MV receives mechanical energy from the surroundings. Pump Load Surroundings MV

39. CV1 / MV1 • CV1 / MV1 [See , but do not see .] • [FBD] sees the shear stress at the rotating shaft cross section, • [Work] sees the energy transfer as work at the rotating shaft cross section. CV1 / MV1 2 1 • CV2 / MV2 [See , but do not see .] • [FBD] sees the pressure and shear stresses on the rotating impeller surface. • [Work] sees the energy transfer as work at the rotating impeller surface. CV2 / MV2 CV2 / MV2 Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces