The document discusses control volume analysis and the conservation laws applied to control volumes, including:
- Conservation of mass for fixed, moving, and deforming control volumes
- Linear momentum (Newton's second law) for fixed control volumes
- The energy equation for fixed control volumes, including work, heat transfer, and applications to steady, incompressible flow
- An example problem calculating the work output of steam passing through a turbine using the energy equation
1. The background of Fluid Mechanics
2. Fields of Fluid mechanics
3. Introduction and Basic concepts
4. Properties of Fluids
5. Pressure and fluid statics
6. Hydrodynamics
1. The background of Fluid Mechanics
2. Fields of Fluid mechanics
3. Introduction and Basic concepts
4. Properties of Fluids
5. Pressure and fluid statics
6. Hydrodynamics
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
Topics:
1. Introduction to Fluid Dynamics
2. Surface and Body Forces
3. Equations of Motion
- Reynold’s Equation
- Navier-Stokes Equation
- Euler’s Equation
- Bernoulli’s Equation
- Bernoulli’s Equation for Real Fluid
4. Applications of Bernoulli’s Equation
5. The Momentum Equation
6. Application of Momentum Equations
- Force exerted by flowing fluid on pipe bend
- Force exerted by the nozzle on the water
7. Measurement of Flow Rate
a). Venturimeter
b). Orifice Meter
c). Pitot Tube
8. Measurement of Flow Rate in Open Channels
a) Notches
b) Weirs
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
Topics:
1. Introduction to Fluid Dynamics
2. Surface and Body Forces
3. Equations of Motion
- Reynold’s Equation
- Navier-Stokes Equation
- Euler’s Equation
- Bernoulli’s Equation
- Bernoulli’s Equation for Real Fluid
4. Applications of Bernoulli’s Equation
5. The Momentum Equation
6. Application of Momentum Equations
- Force exerted by flowing fluid on pipe bend
- Force exerted by the nozzle on the water
7. Measurement of Flow Rate
a). Venturimeter
b). Orifice Meter
c). Pitot Tube
8. Measurement of Flow Rate in Open Channels
a) Notches
b) Weirs
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
thermodynamics introduction & first lawAshish Mishra
Review of Fundamental Concepts and Definitions: Introduction- Basic Concepts: System, Control Volume, Surrounding, Boundaries, Universe, Types of Systems
Macroscopic and Microscopic viewpoints, Concept of Continuum, Thermodynamic Equilibrium, State, Property, Process, Exact & Inexact Differentials
, Cycle Reversibility Quasi – static Process, Irreversible Process, Causes of Irreversibility Energy and its forms, Work and heat (sign convention),
Gas laws, Ideal gas, Real gas, Law of corresponding states, Property of mixture of gases
electrical, magnetic, gravitational, spring and shaft work. Zeroth law of thermodynamics: Concept of Temperature and its’ measurement, Temperature scales
First law of thermodynamics: First Law for Flow Processes
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
4. Basic Laws for a System
• Momentum Equation for
Inertial Control Volume
4
5. Basic Laws for a System
• The Angular Momentum Principle
5
6. Basic Laws for a System
• The First Law of Thermodynamics
6
7. Basic Laws for a System
• The Second Law of Thermodynamics
7
8. Control Volume Analysis: Introduction
• Practical problems involve finite regions
• We call these regions control volumes
• Physical laws govern these regions
• We Apply Conservation Laws• We Apply Conservation Laws
• We look at Mass, Momentum, and Energy
of the Region
8
9. Conservation of Mass: Fixed Control Volume
Apply the Reynold’s Transport Theorem to the System of Mass:
With B = Mass, and b = 1, for a fixed non-deforming control volume:
9
10. Conservation of Mass: Fixed Control Volume
Time rate of change of
the mass of the
coincident system
Time rate of change of
the mass of the
contents of the
coincident control
volume
Net rate of flow of
mass through the
control surface
volume
Recall:
“Coincident Condition”
Time = t
Time = t -
Time = t +
10
11. Conservation of Mass: Fixed Control Volume
Recall,
Then, Conservation of Mass in Control Volume Form:
If the flow is steady:
And, we sum up all the differential elements for mass flow through the surface:
= 0
where the control surface has the area A, is the
density of the fluid, and Q is the volumetric flow rate.
11
12. Conservation of Mass: Fixed Control Volume
“outflow across the surface”
“inflow across the surface”
“no flow across the surface”
Mass flow rate:
“no flow across the surface”
The Average Velocity:
If the velocity, is uniformly distributed:
Control Volume
12
13. Conservation of Mass: Fixed Control Volume
If the flow is steady and incompressible, then:
Q is the volumetric flow rate.
If the flow is unsteady:
is important.is important.
(+) means mass is being added to the C.V.
( - ) means mass is being subtracted from the C.V.
If the flow is one dimensional (uniform flow):
If the flow is not uniform:
For steady flow with one stream in and out:
For steady and incompressible flow with one stream:
13
14. Conservation of Mass: Fixed Control Volume
For steady flow, involving more than one stream:
14
15. There are cases where it is convenient to have the control volume move. The
most convenient is when the control volume moves with a constant velocity.
Conservation of Mass: Moving Control Volume
Reynolds Transport Theorem for a Moving Control Volume
With B = Mass, and b = 1, for a moving, non-deforming control volume:
Recall,
Then,
15
16. Example
• An airplane moves forward at speed of 971 km/hr as shown in
Figure below. The frontal intake area of the jet engine is
0.80m2 and the entering air density is 0.736 kg/m3. A
stationary observer determines that relative to the earth, the jet
engine exhaust gases move away from the engine with a speed
of 1050 km/hr. The engine exhaust area is 0.558 m2, and the
exhaust gas density is 0.515 kg/m3. Estimate the mass flowrateexhaust gas density is 0.515 kg/m3. Estimate the mass flowrate
of fuel into the engine in kg/hr.
16
20. Conservation of Mass: Deforming Control Volume
The equation for the moving control volume can be used for a
deforming control volume.
is non-zero.is non-zero.
W will vary as the velocity of the control
surface varies.
20
21. Conservation of Mass: Example Control Volumes
One inlet an one outlet:
Air in a Pipe:
Steady Flow
Non-uniform velocity, V2 is an average velocity
Air Density varies at each location
Calculate:Calculate:
If we choose a control volume that excludes the
fan and the condenser coils:
Dehumidifier:
Three inlet/outlet combinations, steady state:
If we choose the a second control volume:
Gives the same answer!
Five inlet/outlet combinations:
21
22. Linear Momentum (Newtons 2nd Law): Fixed Control Volume
For “coincidence” of the system with the control volume:For “coincidence” of the system with the control volume:
Using Reynolds Transport Theorem with b = V, and B = Momentum:
Apply the Reynold’s Transport Theorem to the System of Mass:
22
23. Linear Momentum: Fixed Control Volume
Time rate of change of
the linear momentum
of the coincident
system
Time rate of change of
the linear momentum
of the contents of the
coincident control
volume
Net rate of flow of
linear momentum
through the control
surface
volume
Recall:
“Coincident Condition”
Time = t 23
24. Linear Momentum: Fixed Control Volume
Then,
•The forces that act on the control volume are body forces and surface forces
•The equation is a vector equation—linear momentum has direction.
•Uniform (1-D) flows are easiest to work with in these equations
•Momentum flow can be positive or negative out of the control volume
•The time rate of change of momentum is zero for steady flow.
24
25. Linear Momentum: Fixed Control Volume
•If the control surface is perpendicular to the flow where fluid enter or leaves
the control volume, the surface force exerted by the fluid at the control surface
will be due to pressure.
•At an open exit, the surface pressure is atmospheric pressure.
•Gage pressures may be used in certain situations.
•The external forces have an algebraic sign, either positive or negative.
•Only external forces acting on the control volume are considered.•Only external forces acting on the control volume are considered.
•If the fluid alone is considered in the control volume, the reaction forces dues
to any surfaces will need to be considered.
•If the fluid and the surface are in the control volume are in the control volume,
no reaction forces do not appear between the surface and the fluid.
•Anchoring forces are considered external forces
•Anchoring forces will generally exist in response surface stresses (shear and
pressure acting on the control surface.
25
26. Example
• Determine the anchoring force required to hold in place a
conical nozzle attached to the end of a laboratory sink faucet
when the water flow rate is 0.6 liter/s. The nozzle mass is
0.1kg. The nozzle inlet and exit diameters are 16mm and 5mm,
respectively. The nozzle axis is vertical and the axial distancerespectively. The nozzle axis is vertical and the axial distance
between section (1) and (2) is 30mm. The pressure at section
(1) is 464 kPa. to hold the vane stationary. Neglect gravity and
viscous effects.
26
30. Example
• A static thrust as sketched in Figure below is to be designed
for testing a jet engine. The following conditions are known
for a typical test: Intake air velocity = 200 m/s; exhaust gas
velocity=500 m/s; intake cross-section area = 1m2; intake
static pressure = -22.5 kPa=78.5 kPa (abs); intake static
temperature = 268K; exhaust static pressure = 0 kPa=101 kPatemperature = 268K; exhaust static pressure = 0 kPa=101 kPa
(abs). Estimate the normal trust for which to design.
30
32. Linear Momentum: Moving Control Volume
Reynolds Transport Theorem for a Moving Control Volume
With B = Momentum, and b = V, for a fixed non-deforming control volume:
Then, substituting the above equation:
Substitute for V:
32
33. Linear Momentum: Moving Control Volume
For steady flow in the control volume reference frame and VCV is constant:
And, then for an inertial frame, VCV is constant :
For steady flow (on a time average basis), “Mass conservation”:
Then,
33
34. The Energy Equation: Fixed Control Volume
Heat Transfer Rate Work RateEnergy
Rewriting,
Also, noting that energy, e, can be rewritten (all per unit mass):
Internal Energy
Kinetic Energy
Potential Energy
34
35. The Energy Equation: Fixed Control Volume
Now, invoking “coincidence” of the control volume and the system:
Using Reynolds Transport Theorem with b = e, and B = Total Energy:
Apply the Reynold’s Transport Theorem to the System of Mass:
Using Reynolds Transport Theorem with b = e, and B = Total Energy:
35
37. The Energy Equation: Work and Heat
represents heat transfer, conduction, convection, and radiation.
Heat transfer into the control volume is positive, heat transfer
out is negative.
If the process is adiabatic, there is no heat transfer.
If the heat transfer in equals the heat transfer out, the net is zero:
Heat:
Work:
Work transfer rate, power, is positive when the work is done on the
contents of the control volume, by the surroundings.
Work includes shaft work such as turbines, fans, propellers, and other
rotating equipment.
Other types of work are due to normal stresses and tangential
stresses acting on fluid particles.
37
38. The Energy Equation: Work and Heat
Work (continued):
Shaft Work:
Normal Stress:
Shear Stress:
Only non-zero at the control surface.
The tangential stress exists at the boundary, but due to
“no-slip” condition, zero velocity, it is not transferred
typically, and we consider it negligible if the appropriate
control volume is chosen.
38
39. The Energy Equation: Fixed Control Volume
Now, the Energy Equation take the following form:
+ =
Rearranging, and Substituting,
Then,
39
40. The Energy Equation: Applications
(1) (2) (3)
(1) Assume Steady Sate then, = 0
(2)
Assume properties are uniformly distributed over the flow cross-section,
Assume one inlet and one outlet:
40
41. The Energy Equation: Applications
However, the previous assumption of uniform 1D flow is often an
oversimplification for control volumes, but its ease of use justifies it’s
application to these situations.
The previous assumption is fairly
good for a fluid particle following a
stream tube in a steady state flow.
application to these situations.
Now, we can introduce shaft work. We note that shaft work is unsteady locally,
but its effects downstream are steady.
One Dimensional Energy Equation for Steady-in-the-Mean Flow:
41
42. The Energy Equation: Applications
Now, Introduce Enthalpy:
Then the 1D energy equation becomes the following:
With no shaft work—the fluid stream is constant throughout:
Or, the steady flow energy equation:
42
43. Example
• Steam enters a turbine with a velocity of 30m/s and enthalpy,
h1, of 3348 kJ/kg. The steam leaves the turbine as a mixture of
vapor and liquid having a velocity of 60 m/s and an enthalpy
of 2550 kJ/kg. If the flow through the turbine is adiabatic and
changes in elevation are negligible, determine the work output
involved per unit mass of steam through-flow.involved per unit mass of steam through-flow.
43
45. The Energy Equation: Compare to Bernoulli’s
If the flow is incompressible, in addition to being 1D and steady,
Divide the mass flow rate out:
Where,
If the flow is inviscid (frictionless), we obtain Bernoulli’s equation:
or, per unit mass,
Thus, the friction terms are the following:
45
46. The Energy Equation: 1D, Steady, Incompressible, Friction Flow
For steady, incompressible, frictional flow:
Useful or available energy:
Loss terms:
Then we can rewrite the energy equation for 1D, Steady, incompressibleThen we can rewrite the energy equation for 1D, Steady, incompressible
Frictional flow:
46
47. The Energy Equation: 1D, Steady-in-Mean Flow,
Incompressible, Friction Flow
For Steady-in-Mean Flow, we introduce shaft work again:
Divide the mass flow rate out:
Where,
Then,
47
48. The Energy Equation: In Terms of Heads
Multiply by density:
Then, divide by specific weight:
Where,
Turbine:
Pump:
is all other losses not associated with pumps or turbines
can be due to a turbine or pump
If we only have a pump or turbine, the
terms on the R.H.S become these.
48
49. Example
• An axial-flow ventilating fan driven by a motor that delivers
0.4 kW of power to the fan blades produces a 0.6-m-diameter
axial stream of air having a speed of 12 m/s. The flow
upstream of the fan involves negligible speed. Determine how
much of the work to the air actually produces a useful effects,
that is, a rise in available energy and estimate the fluidthat is, a rise in available energy and estimate the fluid
mechanical efficiency of this fan.
49