Lecture 6.3: C-Angular Momentum and Turbomachines

Lecture 6.3: C-Angular Momentum and Turbomachines
abj Lecture 6.3: C-Angular Momentum and Turbomachines Introduction to Fluid Machinery and Turbomachine Classification of Fluid Machinery Turbomachines and Their Energy Transfer Aspects Examples of Axial-Flow Machine Radial-Flow Machine Mixed-Flow Machine Dominant Velocity Components in Axial- and Radial-Flow Machines Performance Parameters Hydraulic / Fluid Stream VS Impeller VS Mechanical/Shaft Angular Momentum Angular Momentum About A (Fixed) Point C of A Particle VS of A Continuum Body Angular Momentum and RTT Moment of Forces: Point Force VS Distributive Force Moment of Surface Forces: Pressure and Shear Forces abj

C-Angular Momentum (MV) C-Angular Momentum (CV)
abj C-Angular Momentum C-Angular Momentum (MV) C-Angular Momentum (CV) Force, Torque, and Energy Transfer as Work in Turbomachines Relation between 1) Surface Force (Pressure + Shear) on the Moving/Rotating Impeller Surface 2) Energy Transfer as Work at the Impeller Surface, and 3) Impeller Torque and Shaft Torque Impeller Torque VS Shaft Torque Euler Turbomachine Equation, Hydraulic Torque, and The Associated Power Equation Various CV’s and The Corresponding Net/Resultant Moment for Turbomachine Analysis Euler Turbomachine Equation Hydraulic Torque Hydraulic Torque VS Impeller Torque VS Mechanical Torque at Shaft The Associated Power Equation abj

Analysis for The Performance of Idealized Turbomachines Problem
abj Analysis for The Performance of Idealized Turbomachines Problem Blade b (and Flow a) Angles Convention Sketch the Blade Shape Shockless-Entry/Exit Condition Relative Velocity Relation and Velocity Diagram Is it a pump or a turbine? Hydraulic Power and Hydraulic Head In Circular Cylindrical Coordinates r-q-z: Axial-Flow Machine Basic of Velocity Diagram Appendix and Review Recall: The Terminologies: Hydraulic / Fluid Stream VS Mechanical / Shaft Recall: Free-Body-Diagram (FBD) Concept abj

Very Brief Summary of Important Points and Equations [1]
1. Euler Turbomachine Equation: 2. The Associated Power Equation: 3. Relative Velocity Relation and Velocity Diagram: abj

Fluid Machinery and Turbomachine
Introduction to Fluid Machinery and Turbomachine abj

Classification of Fluid Machinery
abj Classification of Fluid Machinery Fluid Machinery Positive Displacement (Confined volume) Dynamic/Turbomachines (Dynamic effect between fluid stream and solid component/rotor) Classification by Direction of Energy Input energy into fluid stream Pump, fan, compressor Extract energy from fluid stream Hydraulic/Wind/Gas/Steam turbine Classification by Direction/Path of Flow (as it passes through the blade passage.) Radial-Flow Mixed-Flow Axial-Flow abj

Turbomachines and Their Energy Transfer Aspect
Energy extracted from a high-energy fluid stream to be converted to useful shaft work Energy added to a low-energy fluid stream to raise the energy level of the stream Hydraulic turbine installation: PE of a fluid stream is extracted and converted to shaft work OUT 1 2 A fluid stream Energy is added to a fluid stream as shaft work IN A fluid stream 1 2 Centrifugal fan (Radial-flow) Low E fluid stream High E fluid stream Streamtube 1 2 A fluid stream High E fluid stream Low E fluid stream KE of a fluid stream is extracted and converted to shaft work OUT abj

Axial-Flow Machine c q Reference axis abj
Axial Flow: Blade Rows (From Fluid Dynamics and Heat Transfer of Turbomachinery, Lakshminarayana, B., John Wiley & Son., 1996, p. 9. Photograph courtesy of FIAT.) q c Reference axis The dominant velocity components are z (axial) – for carrying the flow through the blade passage/machine, and q (tangential) – for change in angular momentum  torque Gas Turbine (Axial Flow) PW Commercial High Bypass Turbofan Engines (From abj

A Closer View of Turbine Cascade in Axial-Flow Machine
abj A Closer View of Turbine Cascade in Axial-Flow Machine Turbine Cascade (From Flow through a turbine cascade, outlet Mach number = 0.68 (~ 200 m/s). (From Visualized Flow, The Japan Society of Mechanical Engineers, Pergamon Press, 1988, p.101.) abj

(From http://www.windpower.org/en/pictures/multimeg.htm)
Wind turbine (From Duct fan (Axial-flow) (From abj

Radial-Flow Machine c q Reference axis abj
The dominant velocity components are r (radial) – required for carrying the flow through the blade passage/machine, and q (tangential) – required for change in angular momentum  torque q c Reference axis Centrifugal fan (Radial-flow) (From abj

(From http://www.peerlessxnet.com/documents/8175_Prod.bmp)
abj Centrifugal pump (From abj

Radial/Mixed compressor impeller
abj Mixed-Flow Machine Radial/Mixed compressor impeller (From abj

abj Turbocharger (From abj

Dominant Velocity Components in Axial- and Radial-Flow Machines
Axial Flow: Blade Rows (From Fluid Dynamics and Heat Transfer of Turbomachinery, Lakshminarayana, B., John Wiley & Son., 1996, p. 9. Photograph courtesy of FIAT.) Radial Flow (Centrifugal fan) (From q (tangential) required for change in angular momentum  torque Axial-Flow Machines Radial-Flow Machines n (normal) required for carrying the flow through the blade passage/ machine. abj

Performance Parameters
Hydraulic / Fluid Stream VS Impeller VS Mechanical/Shaft abj

Recall: Hydraulic Power VS Impeller Power VS Mechanical Power at Shaft [See also Appendix and Review: Recall The Terminologies: Hydraulic / Fluid Stream VS Mechanical / Shaft ] Hydraulic Power The actual amount of mechanical energy me that the fluid stream receives/gives up from inlet 1 to exit 2. Properties of fluid stream, not those of solid shaft, e.g., pressure p of fluid, velocity V of fluid, etc. 1 2 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 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 abj

Performance Parameters
With the intended uses of turbomachines, the following performance parameters are of interest. [See summary next slide.] Torque input (pump, fan, etc.) or torque output (turbine, etc.) Hydraulic Torque VS Impeller Torque VS Mechanical / Shaft Torque Power input/output. Hydraulic Power VS Impeller Power VS Mechanical/Shaft Power Other parameters of interest are, e.g., flowrate, hydraulic head, total and static pressure rise, etc. Recall Terminologies: Hydraulic/Fluid Stream Quantities Properties of a fluid stream, Evaluated from fluid stream properties Mechanical / Shaft Quantities Properties of a solid shaft Evaluated from shaft See also Appendix and Review: Recall: The Terminologies: Hydraulic / Fluid Stream VS Mechanical / Shaft abj

Hydraulic / Fluid Stream Side
Summary of Important Quantities: Hydraulic / Fluid Stream VS Mechanical/Shaft Hydraulic / Fluid Stream Side (subscripted h) Mechanical / Shaft Side (subscripted s) Hydraulic Torque Shaft Torque Torque Hydraulic Power Shaft Power Power Hydraulic Head Head Associated Power Equation: Euler Turbomachine Equation: (for idealized machines) = Hydraulic torque x shaft angular velocity (mixed quantity) abj

Angular Momentum abj

Angular Momentum About A (Fixed) Point C. of A Particle. VS
Angular Momentum About A (Fixed) Point C of A Particle VS of A Continuum Body Particle Continuum body x y Observer A C m x y Observer A C Angular momentum of a body about a point c is defined as the moment of linear momentum of the body about the point c. abj

Angular Momentum and [Recall] The RTT
x y Observer A C MV(t) and CV(t) Coincident MV and CV Reynolds Transport Theorem (RTT): abj

Example 1: Finding The Time Rate of Change of The Angular Momentum of
Example 1: Finding The Time Rate of Change of The Angular Momentum of an MV By The Use of A Coincident CV and The RTT Problem: Given that the velocity field is steady-in-mean and the flow is incompressible, and we evaluate the mean properties. 1. State whether or not the time rate of change of the angular momentum about the z-axis passing through point c (Hc,z) of the material volume MV(t) that instantaneously coincides with the control volume CV shown below vanishes. 2. if not, state also - whether they are positive or negative, and - whether there should be the corresponding net moment (Mz) acting on the MV/CV, and - whether the corresponding net moment is positive or negative. 3. Find it. abj

c c c y x r r + + z z c c + + + tangential exit radial exit
axial inlet + c z r tangential exit axial inlet + c z r abj

Point Force VS Distributive Force
Moment of Forces Point Force VS Distributive Force abj

Moment of Forces about A Point C: Point Force VS Distributive Force
Surface Force: results in Surface Integral Volume Force: Volume Integral abj

Because the evaluation of the moment of distributive forces results in
It is to be understood that when we simply write we refer to the corresponding surface or volume integral. Surface Integral Surface Force: Volume Integral Volume Force: abj

Moment of Surface Forces: Pressure and Shear Forces
Closed surface C A To find the net moment of a surface force (pressure or shear) on an MV, we just simply sum (integrate) it over the whole MS. c A Open surface Moment about c due to pressure force: Moment about c due to shear/frictional force: Note: The shear force – and in fact all the surface forces - can be written in terms of the area vector as where is the stress tensor corresponding to that force. Because it is a little more complicated at this point, we shall leave it at that. abj

C-Angular Momentum [ MV ]
Here, we shall limit ourselves to an observer in IFR only. abj

C-Angular Momentum [MV]
x y Observer A, IFR FBD (again) Pressure p Shear t c Concentrated force Solid part Distributive force Physical Law (for an observer in an IFR and a fixed point C) = Net/Resultant moment of all external forces/moments (hence, FBD again) on the MV(t) about a fixed point c. abj

Net/Resultant moment of all external forces/moments
Net/Resultant moment of all external forces/moments (hence, FBD again) on the MV(t) about a fixed point c It is emphasized that in the physical law is the net/resultant moment of all external forces/moments (hence, FBD again) on the MV(t) about a fixed point c. Recall the FBD Concept In C-Angular Momentum, we need to sum all the external moments of all the external forces/moments about C on the MV, FBD Concept abj

FBD Concept Define the system of interest clearly.
Know and Recognize various types of forces (/moments). With 2, recognize all forces (/moments) that act on the current system of interest. Know how to find their resultant force , and resultant moment Recall in case of fluid flow: Forces in fluids (or solid – i.e., continuum, - for that matters) Forces in Fluid: Line Force Surface Force Body Force Surface forces = Pressure/Normal Friction/Tangential Need to sum the contributions due to surface forces on all surfaces of the MS. abj

Examples c c Solid part Solid part
Pressure p Shear t c Concentrated force Solid part Distributive force in fluid Cross section of a solid shaft Normal p in solid surface force  surface integral Pressure p Shear t c Solid part Normal p For surface force, we need to sum the contributions of surface forces over all surfaces of MV, and both solid and fluid parts. For solid part, p refer to normal stress and it must also be included in Recognize that is in fact the resultant moment/couple of shear stress (surface force) distribution over a cross section of a solid shaft. abj

Note on Notations Thus, depending on the forces and moments that act on the current system of interest, the specific form for depends on that specific force/moment system. According to the physical law of C-Angular Momentum (hence, FBD) = sum of all external moments due to all external forces/moments on MV Generically, though, we may simply write or even to emphasize the distributive nature of forces in fluid. However, for example, if there are concentrated forces on the MV, it is to be understood that all these forces must be taken into account in , e.g., abj

C-Angular Momentum [ CV ]
abj

C-Angular Momentum [CV]
Pressure p Shear t c Concentrated force Solid part Distributive force MV(t) CV(t) Coincident MV(t) and CV(t) Physical Law (for an observer in IFR and a fixed point C) RTT Physical Laws RTT C-Angular Momentum abj

Force, Torque, and Energy Transfer as Work in Turbomachines
abj

Impeller and shaft as a system
Relation between 1) Surface Force (Pressure + Shear) on the Moving/Rotating Impeller Surface 2) Energy Transfer as Work at the Impeller Surface, and 3) Impeller Torque and Shaft Torque Impeller and shaft as a system According to the Energy-Work Principle: A moving solid body in a fluid stream (in this case, a moving/rotating solid impeller) is required in order to have energy transfer as work between the solid body and the fluid stream. Force results in Energy Transfer as work Force results in Torque abj Centrifugal fan (Radial-flow, from

Impeller Torque VS Shaft Torque
z c Impeller Torque defined as the net moment due to surface force on the impeller Note that shaft torque, and 2. impeller torque, are related through the FBD of the section of the solid impeller as shown. However, they are generally not equal due to, e.g., frictional torque, at the bearings abj

Euler Turbomachine Equation, Hydraulic Torque,
and The Associated Power Equation abj

Various MV/CV’s and The Corresponding Net/Resultant Moment for Turbomachine Analysis
CV1/MV1: CV includes fluid stream, impeller, and part of solid shaft CV2/MV2: CV includes fluid stream only, no solid part. Recall the concept of FBD c z 1 2 c z 1 2 Moving solid impeller surface (Recall the outward normal to the system of interest) = Shaft torque Shaft torque for CV1/MV1 No shaft torque for CV2/MV2 abj

Example 1: Radial-Flow Machines [1]
Centrifugal fan (Radial-flow) (From z c View this way at solid shaft cross section 1 2 r1 1 2 Inlet 1 = Inner cylindrical surface Exit 2 = Outer cylindrical surface CV/MV: CV includes fluid stream, impeller (and its back plate), and cuts across the solid shaft at the back plate. abj

Note on Notations To avoid messy diagram, it is to be understood that
to evaluate , all moments due to all forces must be taken into account. For surface forces, need to sum the surface forces on all surfaces of the MS/CS Surface forces = Pressure/Normal Friction/Tangential Hence, to remind us once in a while or to focus on some surfaces, we shall simply use the figure of the area vector or simply in place of the more complete one and only on some surfaces instead of on all surfaces. abj

Example 2: Radial-Flow Machines [2]
We can also include the fluid stream only. This time we will not see the shaft torque But we see instead additional forces (pressure and shear) on the impeller surface (fluid as a system) whose moment must be taken into account in and r1 1 2 Example 1 CV/MV CV/MV: CV includes fluid stream only, no solid part. r1 1 2 For clarity, forces on other surfaces are omitted from the diagram. abj

Example 3: Radial-Flow Machines [3] [Compare to Example 1]
z z c View this way 2 1 View this way r1 Inlet 1 = Inlet duct circular cross section Exit 2 = Outer cylindrical surface CV/MV: CV includes fluid stream, impeller (and its back plate), and cuts across the solid shaft at the back plate. It also includes part of the axial inlet duct. abj

Example 4: Axial-Flow Machines
Annular exit plane Annular inlet plane z c A single blade CV covers the whole rotor and cuts through the cross section of a solid shaft. Note, however that the inlet and exit planes are annular. Turbine Cascade (From abj

Euler Turbomachine Equation
1 2 Coincident MV(t) and CV(t) c z 1 2 z CV/MV includes fluid stream, impeller, part of solid shaft, and inlet and exit ducts. It cuts across a cross section of the solid shaft. This CV illustrates the global nature of the Euler Turbomachine equation - without having to deal with the blade geometry, i.e., torque = net angular momentum efflux. CV/MV [CV in Example 1] includes fluid stream, impeller (and its back plate), and cuts across a cross section of the solid shaft (and no inlet and exit ducts). Later, this CV is used in developing the associated power equation. abj

1 z c 2 Assumptions All flow properties are steady in mean.
Evaluate mean properties. Incompressible flow Neglect all other torques, except shaft torque. [Neglect torques due to surface force (pressure + shear), body force (mg), frictional torque at bearings, etc.]. c z 1 2 C-Angular Momentum abj

1 z c 2 Euler Turbomachine Equation: 2 1 r1 Unsteady Term:
Net Convection Efflux Term: Net Moment about c: C-Angular Momentum becomes Euler Turbomachine Equation: abj

Hydraulic Torque Euler’s Turbomachine Equation:
In a similar manner as hydraulic power, we define the RHS (and the RHS only) as the hydraulic torque. Hydraulic Torque [CV Viewpoint] = Net convection efflux of angular momentum through CS. Hydraulic Torque : = the time rate of change of angular momentum of the fluid stream as it flows through CV [from CV inlet 1 to CV exit 2]. Hydraulic Torque [MV Viewpoint] Like hydraulic power, through the RTT, we can see that the two views are equivalent. abj

Properties of fluid stream
Hydraulic Torque is the property of fluid stream, not shaft evaluated from the properties of fluid stream, not shaft. abj

Euler Turbomachine Equation
Ideal Case In an ideal case (idealized machine) where the flow has steady-in-mean flow properties no other torque except shaft torque, the Euler Turbomachine Equation states that Euler’s Turbomachine Equation: Shaft Torque = Hydraulic Torque abj

In a real case, however, since there are other torques
z 1 2 Shaft Torque Hydraulic Torque In a real case, however, since there are other torques moments due to pressure and friction on MS/CS, frictional torque at bearings, etc., acting on the MV/CV, the two torques are not equal. Real Case abj

Assumption 5: Further Assume Uniform
c z 1 2 Assumption 5: Uniform at each cross section. Or, evaluate at some reference radius. Euler Turbomachine Equation: = Velocity of fluid at inlet/exit C-Mass also gives abj

Recall: Hydraulic Power VS Impeller Power VS Mechanical Power at Shaft [See also Appendix and Review: Recall The Terminologies: Hydraulic / Fluid Stream VS Mechanical / Shaft ] Hydraulic Power The actual amount of mechanical energy me that the fluid stream receives/gives up from inlet 1 to exit 2. Properties of fluid stream, not those of solid shaft, e.g., pressure p of fluid, velocity V of fluid, etc. 1 2 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 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 abj

In the Same Manner to Power Hydraulic Torque VS Impeller Torque VS Mechanical Torque at Shaft
the time rate of change of angular momentum of the fluid stream as it flows through CV [from CV inlet 1 to CV exit 2]. 1 2 Impeller Torque defined as the net moment due to surface force on the impeller Shaft Torque Net moment due to surface stress (shear stress) distribution over a cross section of a solid shaft z c With the concept of resultant moment, we can see that can be written in terms of stress in the same manner as The area integral in this case is the solid shaft cross section. For this generic definitions, we leave the reference point and axis unspecified first. abj

The Associated Power Equation
1 2 If we focus on the impeller and look at the CV below: Euler’s Turbomachine Equation: With the Euler’s Turbomahine Equation, which gives an ideal shaft torque, we can find the mechanical power of the shaft as is the tangential velocity (of the blade). The Associated Power Equation: Essentially, we define the associated power from the hydraulic torque not shaft torque abj

In Summary 1 z c 2 Euler Turbomachine Equation:
Assumptions All flow properties are steady in mean. Evaluate mean properties. Incompressible flow Neglect all other torques, except shaft torque. [Neglect torques due to surface force (pressure + shear), body force (mg), frictional torque at bearings, etc.]. In Summary c z 1 2 r1 1 2 = Velocity of fluid at inlet/exit = The tangential velocity (of the blade). Euler Turbomachine Equation: Assumption 5: Uniform at each cross section. Or, evaluate at some reference radius. The Associated Power Equation: abj

Analysis for The Performance of Idealized Turbomachines
abj

Problem: Idealized Turbomachine Performance Analysis
1 2 The Associated Power Equation: Euler Turbomachine Equation: Need to find the two kinematical unknowns Given geometry and geometric parameters blade angles radii kinematical parameter: angular velocity mass flowrate Questions 1. Sketch the blade shape, and Find the idealized 2. shaft torque (= hydraulic torque) 3. shaft power (= hydraulic power, h = 1) 4. hydraulic head. abj

How to tell the geometry of the blade: Blade b (and Flow a) Angles Convention
For an idealized machine where the shockless entry/exit condition is applied, this is the direction of the relative velocity of fluid wrt an observer moving with the blade, CS tangent to blade at inlet/exit = solid blade tangential speed = absolute fluid velocity (relative to IFR) = relative velocity of fluid wrt moving blade = the normal (to CS) component of absolute fluid velocity = blade angle = the angle that the blade tangent at inlet/exit makes with (measured away from ) = flow angle = the angle that the absolute flow velocity makes with (measured towards ) abj

Example of Blade Angles in Axial- and Radial-Flow Machines
1 2 Axial-flow machine Turbine Cascade (From Flow r1 1 2 Radial-flow machine Centrifugal fan (From Flow abj

Example: Sketch The Blade Shape: Axial-Flow Machine
Example: Sketch an axial-flow machine blade with b1 = 30o , b2 = 60o flow flow b1 z w Slope/angle increases from 30o to 60o b1 Curvature b2 b2 Blade concave towards the direction of Example: Sketch an axial-flow machine blade with b1 = 60o , b2 = 30o flow flow b1 z w Slope/angle decreases from 60o to 30o b1 Curvature b2 b2 Blade convex towards the direction of abj

Example: Sketch The Blade Shape: Radial-Flow Machine
Example: Sketch a radial-flow machine blade with b1 = 90o, b2 = 45o q b1 Backwardly-curved blade (wrt the direction of angular rotation) w b2 - q b2 Example: Sketch a radial-flow machine blade with b1 = 90o, b2 = 135o q b1 w Forwardly-curved blade (wrt the direction of angular rotation) b2 abj

Whether it is an axial- or radial-flow machine, we can represent the blade and the kinematics of the flow through the blade by the cascade diagram 1 2 abj

Shockless-Entry/Exit Condition Further Assumption for Idealized Turbomachines
is tangent to the blade at inlet and exit. The direction of the relative velocity of the fluid with respect to the moving blade (wrt an observer moving with the blade) is tangent to the blade at inlet/exit. 1 2 Assumptions (Idealized Turbomachines) All flow properties are steady in mean. Evaluate mean properties. Incompressible flow Neglect all other torques, except shaft torque. [Neglect torques due to surface force (pressure + shear), body force (mg), frictional torque at bearings, etc.]. 5. Shockless entry/exit condition abj

Relative Velocity Relation and Velocity Diagram Recall our Problem
1 2 The Associated Power Equation: Euler Turbomachine Equation: Need to find the two kinematical unknowns Given geometry and geometric parameters blade angles radii kinematical parameter: angular velocity mass flowrate Questions 1. Sketch the blade shape, and Find the idealized 2. shaft torque (= hydraulic torque) 3. shaft power (= hydraulic power, h = 1) 4. hydraulic head. abj

1 2 1. Euler Turbomachine Equation: 2. The Associated Power Equation:
3. Relative Velocity Relation: abj

Velocity diagram can be used as a graphical/geometrical aid in solving the relative velocity vector relation. 1 2 Inlet Exit abj

Velocity Diagram for Axial-Flow Machines
1. For simplicity we evaluate the properties at the mean radius of the CV shown on the left. Thus, since 2. From C-Mass, we also find It is then recommended that the velocity diagrams at inlet and exit be superimposed on the same base and the velocity diagrams look like below. z c 1 2 1 2 Exit Inlet abj

Is it a pump or a turbine? is an external torque acting on the system.
System / MV is an external torque acting on the system. Recall that if Energy is input into the system  Pump Energy is extracted from the system  Turbine abj

z c 1 2 r1 1 2 From the associated power equation, then for an idealized machine Energy is input into the stream  Pump Energy is extracted from the stream  Turbine abj

Example: Axial-Flow Machine: Is it A Pump or A Turbine?
1 2 1 2 abj

[Ideal] Hydraulic Power and Hydraulic Head
2. The Associated Power Equation: 1. Euler Turbomachine Equation: 3. Relative Velocity Relation: If there is no me loss in the machine (h = 1), we have abj

In Circular Cylindrical Coordinates r-q-z: Axial-Flow Machine
Recall in circular cylindrical coordinates: Right-handed coordinates: Position vector: Reference axis z c 1 2 1 2 Axial- Flow Machine abj

In the Circular Cylindrical Coordinates r-q-z: Radial-Flow Machine
Recall in circular cylindrical coordinates: Right-handed coordinates: Position vector: z c Reference axis 1 2 r1 1 2 Radial- Flow Machine abj

In the Circular Cylindrical Coordinates r-q-z
1. Euler Turbomachine Equation: 2. The Associate Power Equation: 3. Relative Velocity Relation: abj

Basic of Velocity Diagram
Decomposition into two systems of coordinates. a. b. This is a system of 2 equations in 5 unknowns: Require the knowledge of 3 to solve for 2. For example: If geometry and speed are given, are known. may be found from C-Mass (e.g., flowrate is given). tangent to blade at i b The relations between velocities can be illustrated as follows. 1. Relative velocity relation: Regardless of whether it is a radial-flow or an axial-flow machine at inlet or exit, the relative velocity relation, which is a kinematic relation, holds. Thus, we have abj

From Fox, R. W. , McDonald, A. T. , and Pritchard, P. J
From Fox, R. W., McDonald, A. T., and Pritchard, P. J., 2004, Introduction to Fluid Mechanics, Sixth Edition, Wiley, New York. abj

abj From Fox, R. W., McDonald, A. T., and Pritchard, P. J., 2004, Introduction to Fluid Mechanics, Sixth Edition, Wiley, New York.

Appendix and Review Recall: The Terminologies:
Hydraulic / Fluid Stream VS Mechanical / Shaft Recall: Free-Body-Diagram (FBD) Concept abj

Recall: The Terminologies: Hydraulic/Fluid Stream VS Mechanical/Shaft
abj Recall: The Terminologies: Hydraulic/Fluid Stream VS Mechanical/Shaft Hydraulic / Fluid Stream: The term is used to refer to the quantities that are associated with, and evaluated from, the properties of the fluid stream (fluid stream side). They are properties of the fluid stream. Examples: Hydraulic torque is the torque that is evaluated from – or equivalent to - the change in the angular momentum flux of a fluid stream. Hydraulic power is the mechanical power gained by a fluid stream (pump), or the mechanical power removed from a fluid stream (turbine). In the case of steady incompressible stream, Subscript h Mechanical / Shaft: The term is used to refer to the quantities that are associated with the mechanical / shaft side. Examples: Mechanical / shaft torque is the torque that is evaluated/measured at the shaft (resultant moment of shear stress distribution over a cross section of a solid shaft) . Mechanical power at shaft is Recall that that the hydraulic quantities and the mechanical quantities may be related but, e.g., due to friction, they are generally not equal. Subscript s abj

Recall: Free-Body-Diagram (FBD) Concept
The Free-Body-Diagram (FBD) concept is related to the Newton’s Second Law: where = resultant/net external force on MV = sum of all external forces on MV = resultant/net external moment about a fixed point c of all external forces/moments on MV = sum of all external moments about a fixed point c of all external forces/ moments on MV abj

Hence, in the application of these laws, we need to be able to find
and Hence, “FBD” here does not refer to the diagram per se, but refers to the fact that we need to be able to FBD Concept Define the system of interest clearly. Know and Recognize various types of forces (/moments, and their natures) that can act on the system. We then classify various types of forces (/moments) in order to be able to take them into account in the application of FBD and the Newton’s second law systematically and effectively. [e.g., line, surface (pressure + friction), and body forces, etc.] With 2, recognize all forces (/moments) that act on the current system of interest. Know how to find their resultant force and their resultant moment [Recall that moment is an effort of force in causing angular motion.] Note: Due to the complexity of the surface geometry of turbomachines, the surface forces may not be drawn on all surfaces in the diagram for FBD. However, the concept of the FBD in relation to the application of the Newton’s second law as mentioned above [i.e., we need to take into account and sum all the external forces/moments on MV] should be kept in mind. abj

FBD Concept Define the system of interest clearly.
Know and Recognize various types of forces (/moments). With 2, recognize all forces (/moments) that act on the current system of interest. Know how to find their resultant force , and resultant moment abj

Lecture 6.3: C-Angular Momentum and Turbomachines

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Lecture 6.3: C-Angular Momentum and Turbomachines

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