1. abj1 1.A Fluid Stream and Energy Transfer To/From A Fluid Stream in Incompressible Flow Machines 2.Convection Transport and N-Flux A Fluid Stream and Convection Transport 3.Convection Flux of Total Energy = Convection Flux of Thermal Energy + Convection Flux of Mechanical Energy 4.Conservation of (Mechanical) Energy and Mechanical Energy Loss (ME Loss) for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) Applications to 1) pump, 2) turbine, and 3) piping section 5.Hydraulic Power: MV and CV Interpretations 6.Hydraulic Head as Potential Height Equivalence for Hydraulic Power 7.Overall Efficiency of Hydraulic (Incompressible Flow) Machines 8.Conservation of (Mechanical) Energy for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) [Working Form for Piping System] Lecture 6.2.1: C-Energy for A Steady, Incompressible, Fluid Stream: Conservation of Mechanical Energy and Mechanical Energy Loss (ME Loss) CV includes the fluid stream only, no solid part. +

2. abj2 Very Brief Summary of Important Points and Equations [1] 1.A steady, incompressible fluid stream: 1.the energy of a fluid stream, N = Energy, 2.the convection of energy (through a cross section A, from one place to another): of a fluid stream, and 3.the addition of energy to (pump), and the extraction of energy from (turbine), a fluid stream. 2.The decomposition of energy into two forms: Thermal Energy (TE, U ) +Mechanical Energy (ME) 3.The convection of energy of a fluid stream at any one cross section can be decomposed into 1) the convection of thermal energy (u) + 2) the convection of mechanical energy (me): 1 2 N-Flux at 1 N-Flux at 2

3. abj3 Very Brief Summary of Important Points and Equations [2] + 4. C-(ME)-Energy and ME Loss for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) Physical Interpretation: The above C-Energy equation can be viewed as a conservation of mechanical energy (ME) of a fluid stream. The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2 = ME added to the fluid stream as work of all surface forces (excluding flow work) - ME Loss in CV (Irreversible conversion of ME to TE) CV includes the fluid stream only, no solid part. Work of all surface forces (excluding the flow work, pv, at inlet and exit), e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface.

4. abj4 Very Brief Summary of Important Points and Equations [3] 5.Hydraulic Power and Hydraulic Head Potential head/height equivalence (at the same mass flowrate ), H pump = Hydraulic Head where Hydraulic Power [CV Viewpoint] := Increase/Decrease in ME Flux of the fluid stream from CV inlet 1 to CV exit 2 [MV Viewpoint] := the time rate of change of mechanical energy of the fluid stream [ENERGY STORED] as it flows through CV [from CV inlet 1 to CV exit 2]. Recall the coincident MV(t) and CV(t)

5. abj5 Very Brief Summary of Important Points and Equations [4] 6. C-(ME)-Energy and ME Loss for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) Physical Interpretation: The above C-Energy equation can be viewed as a conservation of mechanical energy (ME) of a fluid stream. The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2 = ME added to the fluid stream at pump -ME extracted from the fluid stream at turbine - ME Loss in CV (Irreversible conversion of ME to TE)

6. abj6 A Fluid Stream A fluid stream here refers to  A stream of flowing fluid (we may choose a CV that excludes any solid part). Its side wall can be  real solid boundary, e.g.,  a stream of fluid flowing between two sections (1 and 2) of a pipe, or  imaginary wall, e.g.,  a stream of fluid flowing in a stream tube. 1 2 Pump Turbine abcd Mass flux: 1 2 No mass flow through side wall Mass flux: A stream tube as a fluid stream (stream surface as an imaginary side wall) A section of a piping system as a fluid stream

7. abj7 Example: Energy Transfer To/From A Fluid Stream Choices of Control Volumes for A Fluid Stream, 1) Forces and FBD, and 2) Energy Transfer as Work of Forces Example 2: CV includes the fluid stream only, no solid part. 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) 1 2 CV2 / MV2 Impeller Work Energy transfer as work at the rotating impeller surface Example 1: 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. 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) Shaft Work Energy transfer as work at the rotating shaft cross section 1 2 CV1 / MV1

8. abj8 Pump: A Cascade of Energy Transfers as Work Through 1) Rotating Shaft and 2) Rotating Impeller, and Finally To 3) the Fluid Stream Turbine: The Direction is Reversed Incompressible flow machines Pump Add (mechanical) energy to the fluid stream 1 2 CV1 / MV1 CV2 / MV2 ME 1 ME 2 ME 2 > ME 1 Turbine Extract (mechanical) energy from the fluid stream. 1 2 CV1 / MV1 CV2 / MV2 ME 1 ME 2 ME 2 < ME 1

9. abj9 Recall Relation Between 1) Forces and FBD and 2) Energy Transfer as Work of (All) Forces FBD and Energy Transfer as Work of All Forces Coincident MV(t) and CV(t) CV(t) MV(t) Pressure p Shear  Solid part 1. Moving solid surface 2. Stationary solid surface 3. Stationary imaginary surface 1.C-Energy: In the application of C-Energy, we must find energy transfer as work of all forces ( ) that act on our coincident MV/CV. 2.FBD: We should then recognize/draw a FBD for all the forces that act on our coincident MV/CV.

10. abj10 Example 1: FBD and Work Piping System: CV includes the fluid stream only, no solid part. 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) Stationary solid surface (no-slip) Pressure and shear  No work from both pressure and shear Moving solid surface (no-slip) Pressure and shear  There are works from both pressure and shear Inlet and Exit (stationary imaginary surface) Pressure  Work of pressure is included as flow work pv. Shear  If velocity is to the CS – hence to shear, no work of shear. pv Note: This shear work is often neglected in comparison to the flow work.

11. abj11 Example 2: FBD and Work Stream Tube: CV includes the fluid stream only, no solid part. 1 2 Streamlines Stationary imaginary surface (streamlines) Pressure  Pressure is velocity, no work of pressure Shear  There is work of shear. CS is streamline Inlet and Exit (stationary imaginary surface) Pressure  Work of pressure is included as flow work pv. Shear  If velocity is to the CS – hence to shear, no work of shear. pv Note: This shear work is often neglected in comparison to the flow work.

12. abj12 Convection Transport and N -Flux

13. abj13 The mass flux/flowrate carries with it (its own property) the flux/flowrate of N through a cross section A. This results in a convection transport of physical quantities (mass, momentum, energy, etc.) from one place to another. Convection (Transport) and N -Flux N flux: Mass flux: 1 2 A

14. abj14 1 2 Pump Turbine abcd 1 2 Mass flux: N-flux: Mass flux: N-flux: In this topic, we shall focus on the energy of the fluid stream, N = Energy, the convection of energy (from one place to another) of the fluid stream, and the addition of energy to (pump), and the extraction of energy from (turbine), the fluid stream.

15. abj15 Convection Flux of Total Energy = Convection Flux of Thermal Energy + Convection Flux of Mechanical Energy

16. abj16 Convection Flux of Energy = Convection Flux of Thermal Energy (TE) + Convection Flux of Mechanical Energy (ME) 1 2 Pump For an incompressible flow machines (hence, for turbine, it is often called hydraulic turbine): Pumpadds mechanical energy (me) to the fluid stream to increase the me of the stream via energy transfer as work through shaft and impeller Turbine extracts mechanical energy (me) from the fluid stream to convert it to useful shaft work via energy transfer as work through impeller and finally shaft Mechanical Energy Flux through a cross section A It is then convenient to decompose Convection flux of total energy (E) = Convection flux of thermal energy (TE) + Convection flux of mechanical energy (ME)

17. abj17 Conservation of (Mechanical) Energy and Mechanical Energy Loss (ME Loss) for A Steady, Incompressible, Fluid Stream (with no applied heat transfer)

18. abj18 Recall:C-Energy + 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

19. abj19 Conservation of (Mechanical) Energy for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) Net Convection Efflux TermNet Energy Transfer as Heat No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss. Net (ME) Energy Transfer as Work Work of all surface forces (excluding the flow work, pv, at inlet and exit), e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface. 1 2 Pump Turbine abcd 1 2 CV: CV includes the fluid stream only, no solid part. Assumptions: 1.Incompressible flow (  is steady and uniform) 2.All flow properties are steady (or steady-in-mean) 3.No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss. + Unsteady Term C-Energy

20. abj20 Work of all surface forces (excluding the flow work, pv, at inlet and exit), e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface. C-Energy Separate thermal energy from mechanical energy and group corresponding terms together: 1 2 1 2 Pump Turbine abcd TE 2 ME 2 ME 1 TE 1 +

21. abj21 The rise in mechanical energy of the fluid stream – from 1 to 2 – as measured by the difference in ME (me flux) is equal to + (mechanical) energy added to the fluid stream as work (of surface forces) - the thermal energy term 1 2 1 2 Pump Turbine abcd TE 2 ME 2 ME 1 TE 1 +

22. abj22 Mechanical Energy Loss: Irreversible Conversion of Mechanical Energy to Thermal Energy According to the Second Law of Thermodynamics 1 2 1 2 Pump Turbine abcd It can be proven from the second law of thermodynamics (see appendix) that, this thermal energy term – which we shall denote it by represents the time rate of irreversible conversion of mechanical energy to thermal energy according to the second law of themodynamics, or mechanical energy loss (ME Loss).

23. abj23 C-Energy for A Steady, Incompressible Fluid Stream (with no applied heat transfer) Assumptions: 1.Incompressible flow (  is steady and uniform) 2.All flow properties are steady (or steady-in-mean) 3.No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss. 1 2 1 2 Pump Turbine abcd Work of all surface forces (excluding the flow work, pv, at inlet and exit), + ME 2 ME 1 CV includes the fluid stream only, no solid part.

24. abj24 1] Pump: Application of C-Energy for A Steady, Incompressible Fluid Stream 1 2 CV1 / MV1 CV2 / MV2 ME 1 ME 2 ME 2 > ME 1 Here, we use CV2/MV2 = Mechanical energy that the fluid stream actually receives from inlet 1 to exit 2. ME loss at bearings, etc. ME loss due to friction in the fluid stream itself in the pump (between 1 and 2). Recall the shear stress/Newton’s viscosity law in Chapter 2. Rubbing your hands together and they get warmer; friction converts mechanical energy (kinetic energy) to thermal energy. Impeller transfers (mechanical) energy as work at its surface to the fluid stream (+100 W). ME loss – ME being irreversibly converted to TE - in the stream as it flows from 1 to 2 (+20 W). [The loss occurs throughout the whole volume in CV.] The stream emerges at exit 2 with only (+80 W) more mechanical power than at inlet 1.

25. abj25 2] Turbine: Application of C-Energy for A Steady, Incompressible Fluid Stream 1 2 CV1 / MV1 CV2 / MV2 ME 1 ME 2 ME 2 < ME 1 Here, we use CV2/MV2 = Mechanical energy that the fluid stream actually gives up from inlet 1 to exit 2. ME loss at bearings, etc. ME loss due to friction in the fluid stream itself in the turbine (between 1 and 2). Recall the shear stress/Newton’s viscosity law in Chapter 2. Rubbing your hands together and they get warmer; friction converts mechanical energy (kinetic energy) to thermal energy. The fluid stream gives up its mechanical power as it flows from 1 to 2 (-100 W). ME loss – ME being irreversibly converted to TE - in the stream as it flows from 1 to 2 by (+20 W) [The loss occurs throughout the whole volume in CV.] The impeller actually receives the transfer of energy as work at its surface from the fluid stream only (- 80 W).

26. abj26 3] Pipe Section: Application of C-Energy for A Steady, Incompressible Fluid Stream Due to ME loss – ME being irreversibly converted to TE - in the stream as the fluid stream flows from 1 to 2, the stream emerges at 2 with (+20 W) less mechanical power than when it enters at 1. ME 2 < ME 1 ME 1 ME 2

27. abj27 Hydraulic Power: MV and CV Interpretations

28. abj28 Hydraulic Power: MV and CV Interpretations  Since we are interested in the change in mechanical energy of the fluid stream as it flows through CV, from CV inlet 1 to CV exit 2, we define = ME flux through cross section A 1 2 ME 1 ME 2 Hydraulic Power : = the time rate of change of mechanical energy of the fluid stream [ENERGY STORED] as it flows through CV [from CV inlet 1 to CV exit 2]. Hydraulic Power [MV Viewpoint]Hydraulic Power [CV Viewpoint] Recall the coincident MV(t) and CV(t) The two views are equivalent, see next slide.

29. abj29 MV and CV Aspect of Hydraulic Power: Another Application of RTT The coincident MV has the time rate of change of its mechanical energy = the net convection efflux of me = ME 2 – ME 1 =: Hydraulic Power 1 2 ME 1 ME 2 coincident MV RTT: MV View CV View

30. abj30 Recall The Three Regions I – II – III in the derivation of RTT The original MV that instantaneously coincides with the CV has the instantaneous time rate of increase of mechanical energy = coincident MV(t) and CV(t) at time t CV(t) at time t+dt Original MV at some later time t+dt MV (t+dt) Part of the original MV flows out to Region III New – but not yet of interest - MV flows into Region I II III I

31. abj31 Some Aspects of Hydraulic Power Hydraulic Power is the property of the fluid stream, not that of the solid shaft. must be evaluated from the properties of the fluid stream (i.e., pressure of fluid p, specific volume of fluid v, velocity of fluid V, etc.) - not that of the solid shaft. 1 2 ME 1 ME 2 = the time rate of change of mechanical energy of the fluid stream [ENERGY STORED] as it flows through CV [from CV inlet 1 to CV exit 2]. NOTE The term hydraulic “power” can be little misleading. It may cause confusion that this is the time rate of work of force (ENERGY TRANSFER). It is not. It is the time rate of change of mechanical energy of the fluid stream (ENERGY STORED).

32. abj32 Hydraulic Power VS Impeller Power VS Mechanical Power at Shaft Shaft Work [Mechanical] Energy transfer as work (between one part of the solid shaft to another part of the solid shaft) at the solid cross section of a shaft. 1 2 Shaft Power Impeller work [Mechanical] Energy transfer as work (between the moving solid impeller and the fluid stream) at the moving solid impeller surface. (Solid-Fluid Interaction] 1 2 Impeller Power Hydraulic Power The actual amount of mechanical energy me that the fluid stream receives/gives up from inlet 1 to exit 2. Hydraulic Power Properties of fluid stream, not those of solid shaft, e.g., pressure p of fluid, velocity V of fluid, etc. 1 2

33. abj33 Hydraulic Head as Potential Height Equivalence for Hydraulic Power

34. abj34 Hydraulic Power and Hydraulic Head H as the Potential Power Equivalence Potential head/height equivalence (at the same mass flowrate ), H pump For ease of grasping the physical sense of the magnitude of the hydraulic power, we equate the hydraulic power to the amount of power that is required to lift the fluid at the same mass flowrate to the height H (hydraulic head), i.e., = Hydraulic Head where

35. abj35 In order to determine the hydraulic power, the hydraulic head H must be stated together with the mass flowrate ( ) since, according to the definitions Stating only the head H is not enough. Hydraulic Power and Hydraulic Head H m Hydraulic Head H at m

36. abj36 Overall Efficiency of Hydraulic (Incompressible Flow) Machines Pump and (Hydraulic) Turbine Note: Not Gas or Steam Turbine where the flow inside is compressible.

37. abj37 Overall Efficiency of Pump and (Hydraulic) Turbine PUMP Overall Pump Efficiency Mechanical power that the fluid stream receives from inlet 1 to exit 2 Mechanical power input at shaft = Hydraulic Power Turbine Overall Turbine Efficiency Mechanical power that the fluid stream gives up from inlet 1 to exit 2 Mechanical power output at shaft = Hydraulic Power

38. abj38 Shaft Power Impeller Power 1 2 1 2 1 2 Hydraulic Power 1 2 1 2 1 2 ME loss at bearings, etc. ME loss due to friction in the fluid stream itself in the pump (between 1 and 2). ME loss at bearings, etc. ME loss due to friction in the fluid stream itself in the pump (between 1 and 2). PUMP Overall Pump Efficiency Turbine Overall Turbine Efficiency For simplicity in notation, later on we shall drop the ‘s’ for ‘shaft.’

39. abj39 Conservation of (Mechanical) Energy and Mechanical Energy Loss (ME Loss) for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) [Working Form for Piping System]

40. abj40 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) Her, we consider many CV’s and sum their equations together. CV’s include fluid stream only, no solid part. CV: 1-a (pipe) CV: a-b (pump) CV: b-c (pipe) CV: c-d (turbine) CV: d-2 (pipe) + + + + LHS = ME change from inlet to exit CV: 1  2

41. abj41 Conservation of (Mechanical) Energy for A Steady, Incompressible, Fluid Stream (with no applied heat transfer) 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) 1 2 Assumptions: 1.Incompressible flow (  is steady and uniform) 2.All flow properties are steady (or steady- in-mean) 3.No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss. Physical Interpretation: The above C-Energy equation can be viewed as a conservation of mechanical energy (ME) of a fluid stream. The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2 = ME added to the fluid stream at pump -ME extracted from the fluid stream at turbine - ME Loss in CV (Irreversible conversion of ME to TE)

42. abj42 “Walking” (from 1 to 2) Interpretation 1 2 Pump Turbine a 1(pump) b 2(pump) c 1(turbine) d 2(turbine) 1 2 Physical Interpretation: The above C-Energy equation can be viewed as a conservation of mechanical energy (ME) of a fluid stream. ME at 2 = ME at 1 (starting ME) + Increase in ME at pump -Decrease in ME at turbine -ME Loss in CV (Irreversible conversion of ME to TE, piping sections)

43. abj43 What’s coming up? Chapter 10: Turbomachine We will learn how to find the hydraulic power and hydraulic head for idealized machines in terms of the machine’s geometrical and kinematical parameters. Chapter 8: Viscous Internal Flow We will learn how to correlate ME Loss in piping components in terms of mechanical quantities (not thermal quantities). Chapter 6: Bernoulli’s Equation For a steady incompressible flow in a stream tube, in the absence of energy transfer as work and in the absence of friction (ME Loss), mechanical energy of the fluid stream is conserved. For an infinitesimal stream tube, i.e., a streamline, this becomes the conservation of mechanical energy along the streamline.

44. abj44 Special Case 1: No energy transfer as heat or work except heat transfer that is associated with ME Loss. 1 2 1 2 Pump Turbine abcd ME Loss is simply the difference between ME at 1 and 2. Assumptions: 1.Incompressible flow (  is steady and uniform) 2.All flow properties are steady (or steady-in-mean) 3.No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss. 4.No energy transfer as work of surface forces, except the flow work pv. No energy transfer as heat or work except -heat transfer that is associated with ME Loss, and - the flow work pv.

45. abj45 Special Case 2: No energy transfer as heat or work. NO ME Loss, 1 2 1 2 Pump Turbine abcd In the absence of 1) energy transfer as heat or work (except the flow work pv), and 2) ME Loss, the mechanical energy of a steady, incompressible fluid stream is conserved. This leads to the Bernoulli’s equation as the Conservation of Mechanical Energy Equation for such a fluid stream In chapter 6: Incompressible, Inviscid flow, we will derive this Bernoulli’s equation from another principle, the C-Mom Equation. Assumptions: 1.Incompressible flow (  is steady and uniform) 2.All flow properties are steady (or steady-in-mean) 3.No energy transfer as heat. 4.No energy transfer as work (except the flow work pv). 5.No ME Loss,. [Essentially, we assume no friction/viscous force, i.e., inviscid flow.] No energy transfer as heat or work, except the flow work pv.

46. abj46 NOTE:Conversion Between ME and TE Two mechanisms that can cause transformation between ME and TE METE 1. Compressibility (Compression/Expansion work) Reversible process, ME and TE can be converted back and forth If the flow is incompressible, ME cannot be converted to TE via this mode, and vice versa. 1. ME transfer as work of force cannot be converted to TE. ME transfer as work of force results in increase in ME of the MV only. 2. TE transfer as heat cannot be converted to ME. TE transfer as heat results in increase in TE of the MV only. METE 2. Friction / Viscous Dissipation Irreversible process, convert ME to TE only If the flow is inviscid, ME cannot be converted to TE via this mode.

47. abj47 Special Case 2: C-ME Energy in Relation to the The Bernoulli’s Equation As a result, for a 1. steady, 2. incompressible (No reversible ME  TE conversion via compressibility), 3. inviscid flow (No irreversible ME  TE conversion via viscous dissipation), [MV view] the mechanical energy of the coincident MV in a fluid stream, or [CV view] the mechanical energy flux through any cross section of the coincident CV in the fluid stream, is conserved. This is true even though there is energy transfer as heat so long as the flow remains incompressible, since the incompressibility condition suppresses TE  ME via mechanism 1. There can be heat transfer, but it only results in the increase in TE ( u ) in incompressible flow. Certainly, in order for the ME to be conserved there can be no (mechanical) energy transfer as work since that will affect the ME of the fluid stream directly, Finally, in order for [MV view] the mechanical energy of the coincident MV in a fluid stream, or [CV view] the mechanical energy flux through any cross section of the coincident CV in the fluid stream, to be conserved (in case of CV, conserved = ME flux through any cross sections remain the same), the assumptions in Special Case 2 can be replaced by the above 3 assumptions + 4. no energy transfer as work. [ 1) Drop the no energy transfer as heat assumption, and 2) replace no ME Loss by inviscid/no friction. ] NOTE: If the fluid is a compressible fluid such as air, energy transfer as heat generally couples with the change in density, hence not an incompressible flow and the above result does not apply.

48. abj48 APPENDIX (Related Proof for) ME Loss = Irreversible Conversion of ME to TE According to The Second Law of Thermodynamics + 1 MV, CV 2 A simple way to think about this: But, if ds and  q/T are to have some definite values, their difference must equal something, [ let that something be called  P s : ] and that something has to be non-negative,. In order to consider the ME Loss, we consider the coincident MV: 1.Second law of thermodynamics: Rewrite it in a more convenient form: (1) where  P s is the production in entropy and is never negative:. 2.Gibb’s equation for a simple compressible substance: (2) 3.(1) = (2): 4.For incompressible flow, dv = 0. Thus, With the above assumptions, this equation basically states that the sum of the net increase in TE of an incompressible system and the net (TE) energy transfer out as heat (-  q ) is never negative. Recall the RTT and the relation between the coincident MV and CV.