abj1  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 (Heat/TE + Work/ME + Others) 2.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. Lecture 6.2: Conservation of Energy (C-Energy), and Energy Transfer as Work of (Surface) Forces +

abj2 Very Brief Summary of Important Points and Equations + C-Energy for A MV C-Energy (Working Forms) for A CV = stagnation enthalpy u-me - form h - form h o - form e-pv - form

abj3 Energy as A Conserved Quantity/Scalar Conservation of Energy for An Isolated System Conservation of Energy for A MV (Closed System)

abj4 “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 kg of mass in the universe. Also Conservation of Energy  According to classical mechanics, there is a total of 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.

abj5 Energy as A Conserved Quantity/Scalar Conservation of Energy for Universe (Isolated System) E U = Constant (Conserved)  dE U = 0 An Isolated System

abj6 Relation Between Changes of Various Parts U = MV + Surroundings Universe (Isolated System) E U = Constant (Conserved)  dE U = 0 E U = E MV + E Sur = Constant E MV MV (Closed System) Surroundings, E Sur Total Amount dE MV Change/Increase in Energy Stored - dE Sur Universe (Isolated System) E U = Constant (Conserved)  dE U = 0 dE U = dE MV + dE Sur = 0 - dE Sur = dE MV Surroundings Relation between changes of various parts The amount of energy transferred to a system must come from its surroundings. An Isolated System

abj7 Energy as A Conserved Quantity/Scalar Conservation of Energy for Focus on a MV (closed system) as a part of the Universe Universe (Isolated System) E U = Constant (Conserved)  dE U = 0 dE U = dE MV + dE Sur = 0 dE MV Change/Increase in Energy Stored in MV in various forms - dE Sur Energy Transfer to MV from its surroundings in various modes Surroundings Let’s denote the LHS instead by  E T.(= - dE Sur ) A MV (Closed System)

abj8 Conservation of Energy for A MV (Closed System) 1. Modes of Energy Transfer 1. Energy Transfer As Heat (, Thermal Energy Transfer) 2. Energy Transfer As Work (, Mechanical Energy Transfer) 3. Other Modes of Energy Transfer ( ) 2. Forms of Energy Stored 1.Thermal Energy (TE) 2. Mechanical Energy (ME) 3. Other Forms of Energy Stored

abj9 Conservation of Energy for Modes of Energy Transfer and Forms of Energy Stored dE MV Change/Increase in Energy Stored in MV in various forms MV (Closed System) ETET Energy Transfer to MV from its surroundings in various modes Surroundings 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 that E U is conserved or - dE Sur = dE MV (a simple balance law) Forms of Energy Stored  Thermal energy TE (= U)  Mechanical energy ME (= KE )  Other forms of energy stored ( e.g., electrical, chemical, etc.) Modes of Energy Transfer  Energy Transfer as Heat  Q = Thermal energy transfer  Energy Transfer as Work  W = Mechanical energy transfer  Other modes of energy transfer  E T (e.g., electromagnetic radiation, etc.) A MV (Closed System)

abj10 Conservation of Energy for In most of our problems of interest, only 1) Thermal Energy (TE) and 2) Mechanical Energy (ME) are excited/changed  E T =  Q +  W + [  E T ]  Q = Heat = Thermal energy transfer  W = Work = Mechanical energy transfer  E T = Other modes of energy transfer E MV  TE + ME [+ Other forms]  = Thermal energy ME = Mechanical energy …... = Other forms of energy stored Energy Transfer to MV from its surroundings in various modes Surroundings Change/Increase in Energy Stored in MV in various forms MV (Closed System) Key: If some other forms of energy are also excited/changed, they must be taken into accounted according to the conservation of energy. + A MV (Closed System)

abj11 C-Energy for A MV (Closed System) Surroundings 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 MV (Closed System) +

abj12 Scratch Note: Proof of Work of mg Work of Body Force mg and Potential Energy [1] x y z

abj13 The Two Forms of C-Energy for A MV (Closed System) (according to where we put the work of mg / potential energy) Form 1 Form 2

abj14 Energy input into a system causes increase in energy of the system. Energy extracted from a system causes decrease in energy of the system. Sign Conventions for The Energy Equation Physics: - output causes E- decrease Physics: - input causes E- increase Similar can be said for

abj15 C-Energy for A MV (Closed System) LHS: [Modes of] Energy Transfer 1.Energy Transfer as Heat [Thermal Energy Transfer] 2.Energy Transfer as Work [Mechanical Energy Transfer]

abj16 Recall in C-Mom Keys 1.Recognize various types of forces. 2.Know how to find the resultant of various types of forces (e.g., pressure, etc.). 3.Sum all the external forces. Keys: Energy Transfer to MV 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. 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. LHS = Energy Transfer to MV Mechanical Energy Transfer (as Work of Forces ) Energy Transfer in Other Modes Thermal Energy Transfer (as Heat ) Modes of Energy Transfer on The LHS

abj17 + Through a finite surface S : (input-positive) Heat ( ) If any other Other Modes of Energy Transfer (input-positive) e.g. electrical, electromagnetic, etc. Work Energy Transfer Modes ( between a system and its surroundings ) Work of Forces (input-positive) Stress vector Tangential (Shear) (input-positive) Normal (Pressure) (input-positive) Work of Surface Force/Stress Work of mg is later accounted for as potential energy Work of Body Force/mg If there are other body forces besides mg, all must be accounted for.

abj18 Energy Transfer As Work of A Force [Mechanical Energy Transfer]

abj19 : Energy Transfer as Work (Mechanical Energy Transfer) CV(t) MV(t) Pressure p Shear  Coincident CV(t) and MV(t) Volume/Body Force FBD  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

abj20 and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t) 1. Concentrated/Pointed Surface Force 2. Distributive Surface Force in Fluid [Pressure p + Friction  ] Net Surface ForceNet Volume/Body Force CV(t) MV(t) Pressure p Shear  2. Distributive Surface Force (in fluid part) 1.Concentrated/Point Surface Force Coincident CV(t) and MV(t) Volume/Body Force FBD Recall 1: Recall all and various types of forces. must be the sum of the works of all the forces on MV(t).

abj21 Recall 2: Energy Transfer as Work of A Force (Mechanical Energy Transfer) Particle Concept Work = Force x Displacement in the direction of the force (per unit time) Work of A Force ( )

abj22 Energy Transfer as Work of A Force (Mechanical Energy Transfer) Particle VS Continuum Body Work of A Force, Same Concept Work = Force x Displacement in the direction of the force Particle Continuum Body Same concept, just that 1)there are more types of forces to be accounted for: Surface force and Body force (and…) 2)Each type is described differently 3) As before, how to sum them all.

abj23 Work of All Forces CV(t) MV(t) Pressure p Shear  2. Distributive Surface Force (in fluid part) 1.Concentrated/Point Surface Force Coincident CV(t) and MV(t) Volume/Body Force FBD Note = Shaft work is work due to shear stress (surface force) at the cross section of a shaft.

abj24 Work of shear force on CS/MS: Infinitesimal work of shear stress: 1.Rate of work (power) done on a finite closed surface S : Work of Surface Forces: 1) Pressure Force (Flow Work), 2) Shear Force Work of pressure force on CS/MS: Infinitesimal work of pressure force: 1.Rate of work (power) done on a finite closed surface S : + Recall the coincident CV(t) and MV(t) S Surroundings MV(t) CV(t)

abj25 Scratch Note: Proof of Work of mg Work of Body Force mg and Potential Energy [2] [Another Approach for A Continuum Body] x y z

abj26 Finite Control Volume Formulation of Physical Laws C-Energy

abj27 Finite CV Formulation of Physical Laws: C- Energy C-Energy: Physical Laws RTT Recall the coincident CV(t) and MV(t) + Material Volume (MV) dE MV /dt Surroundings CV(t), MV(t) Energy transfer as heat Energy transfer as work of forces p, 

abj28 Finite CV Formulation of Physical Laws: C- Energy To save some symbols, here we redefine at various steps. Apply RTT to dE MV /dt

abj29 C-Energy (Working Forms) + Recall the coincident CV(t) and MV(t) = stagnation enthalpy u-me - form h - form h o - form e-pv - form Material Volume (MV) dE MV /dt Surroundings CV(t), MV(t) Energy transfer as heat Energy transfer as work of forces p, 

abj30 Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force]

abj31 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.

abj32 Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] Pressure p Shear  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) 3. Stationary Imaginary surface (where there is mass flow in/out.)

abj33 Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] Pressure p Shear  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, 3. Stationary Imaginary surface (where there is mass flow in/out.) In general, Note: For moving imaginary surface, we may use the decomposition Work due to pressure force here is later moved to the RHS and included as flow work, p v, in the convection flux term:

abj34  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

abj35 Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) 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.] 1 2 Surface Force: Pressure and shear on moving/rotating impeller surface Surface Force Pressure and shear MV 1 2 Surface Force Pressure and shear Surface Force: Normal and shear stress over the moving/rotating cross section of a solid shaft MV

abj36 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 Surroundings Pressure and shear stresses on the rotating impeller surfaces act on the moving fluid  Energy transfer as work to MV (fluid stream) MV

abj37 Energy transfer as work of forces at the surface of the moving/rotating impeller [Pump] Pressure force pushes fluid, Shear force drags fluid, such that the fluid at the material surface has velocity. Surroundings MV Surroundings Energy transfer as work of force at the rotating impeller surface

abj38 Energy transfer as work of forces at the cross section of a solid shaft 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). MV Surroundings = External force and torque due to surroundings on our MV (Recall the concept of FBD and Newton’s Second Law) MV Energy transfer as work of force at the rotating cross section of a solid shaft.

abj39 MV Surroundings MV Surroundings Motor/Turbine drives its Pump/Load [Pump, Load] MV receives mechanical energy from the surroundings. [Motor, Turbine] MV gives up its own mechanical energy to the surroundings. Motor Pump Turbine Load Direction of mechanical energy transfer as work

abj40 Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces 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. 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. CV1 / MV1 CV2 / MV2 1 2 CV1 / MV1 CV2 / MV2