Basics: Fluid Mechanics - KNTU

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Mechanical Energy

• A pump transfers mechanical energy to a fluid by raising its pressure, & a turbine extracts mechanical energy from a fluid by dropping its pressure.

• mechanical energy of a flowing fluid PER unit-mass:

Mechanical Energy

𝒆_{𝒎𝒆𝒄𝒉} = 𝑷

𝝆 + 𝑽^𝟐

𝟐 + 𝒈𝒛 flow energy: a pressure force

acting on a fluid through a distance produces work

Kinetic energy Potential energy

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Mechanical Energy

• mechanical energy change of a fluid during incompressible flow:

• mechanical energy of a fluid does not change during flow if its pressure, density, velocity, and elevation remain constant.

∆𝒆_{𝒎𝒆𝒄𝒉}= 𝑷_𝟐 − 𝑷_𝟏

𝝆 + 𝑽_𝟐^𝟐 − 𝑽_𝟏^𝟐

𝟐 + 𝒈 𝒛_𝟐 − 𝒛_𝟏 𝒌𝑱 𝒌𝒈

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Bernoulli Equation

• The Bernoulli equation is an approximate relation between pressure, velocity, and elevation, and is valid in regions of steady, incompressible flow where net frictional forces are negligible .

Bernoulli Equation

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Bernoulli Equation

Elevation Velocity

Pressure

Bernoulli Equation

key approximation in the derivation of the Bernoulli equation:

Viscous effects are

negligibly small

compared to inertial, gravitational, and pressure effects: Inviscid Fluid

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Bernoulli Equation

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Bernoulli Equation

• Derive Bernoulli equation?

• Consider the motion of a fluid particle in a flow field in steady flow:

Derivation of the Bernoulli Equation

𝑑𝑉 = 𝜕𝑉

𝜕𝑠 𝑑𝑠 + 𝜕𝑉

𝜕𝑡 𝑑𝑡 → 𝑑𝑉

𝑑𝑡 = 𝜕𝑉

𝜕𝑠 𝑑𝑠

𝑑𝑡 + 𝜕𝑉

𝜕𝑡 = 𝜕𝑉

𝜕𝑠 𝑉 + 0 = 𝑉 𝑑𝑉 𝑑𝑠 Steady Flow

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Bernoulli Equation

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Bernoulli Equation

෍ 𝑭_𝒔 = 𝒎𝒂_𝒔

𝑷𝒅𝑨 − 𝑷 + 𝒅𝑷 𝒅𝑨 − 𝑾𝒔𝒊𝒏𝜽 = 𝒎𝑽𝒅𝑽 𝒅𝒔

𝒘𝒉𝒆𝒓𝒆 𝒎 = 𝝆𝑽 = 𝝆 𝒅𝑨 𝒅𝒔 𝒂𝒏𝒅 𝑾 = 𝒎𝒈 = 𝝆𝒈 𝒅𝑨 𝒅𝒔

−𝒅𝑷 𝒅𝑨 − 𝝆𝒈 𝒅𝑨 𝒅𝒔𝒅𝒛

𝒅𝒔 = 𝝆 𝒅𝑨 𝒅𝒔 𝑽𝒅𝑽 𝒅𝒔 𝒅𝑷

𝝆 + 𝟏

𝟐𝒅 𝑽^𝟐 + 𝒈 𝒅𝒛 = 𝟎

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Bernoulli Equation

• Bernoulli equation, which is commonly used in fluid mechanics for steady, incompressible flow along a streamline in inviscid flow:

• between any two points on the same streamline as

𝑺𝒕𝒆𝒂𝒅𝒚 𝒇𝒍𝒐𝒘: න𝒅𝑷

𝝆 + 𝑽^𝟐

𝟐 + 𝒈 𝒛 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆)

𝑺𝒕𝒆𝒂𝒅𝒚, 𝒊𝒏𝒄𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒃𝒍𝒆 𝒇𝒍𝒐𝒘:𝑷

𝝆 + 𝑽^𝟐

𝟐 + 𝒈 𝒛 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆)

𝑷_𝟏

𝝆 + 𝑽_𝟏^𝟐

𝟐 + 𝒈 𝒛_𝟏 = 𝑷_𝟐

𝝆 + 𝑽_𝟐^𝟐

𝟐 + 𝒈 𝒛_𝟐

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Bernoulli Equation

• Bernoulli equation: mechanical energy balance

The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when the compressibility and frictional effects are negligible.

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Bernoulli Equation

• P: Static pressure/ Thermodynamic pressure of the fluid

• 𝝆^𝑽^𝟐

𝟐 : Dynamic pressure; it represents the pressure rise when the fluid in motion is brought to a stop isentropically.

• 𝝆𝒈𝒛: hydrostatic pressure, which is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., of fluid weight on pressure.

Static, Dynamic, and Stagnation Pressures

𝑷 + 𝝆𝑽^𝟐

𝟐 + 𝝆𝒈 𝒛 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆)

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Bernoulli Equation

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Bernoulli Equation

• The sum of the static and dynamic pressures is called the stagnation pressure,

• The stagnation pressure represents the pressure at a point where the fluid is brought to a complete stop isentropically.

• Fluid velocity at stagnation point:

𝑷_{𝒔𝒕𝒂𝒈} = 𝑷 + 𝝆𝑽^𝟐

𝟐 (𝒌𝑷𝒂)

𝑉 = 2(𝑃_{𝑠𝑡𝑎𝑔} − 𝑃) 𝜌

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Bernoulli Equation

• The flow streamline that extends from far upstream to the stagnation point is called the stagnation streamline.

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Bernoulli Equation

Steady Flow

Frictionless flow

Incompressible flow No heat transfer

(density variation)

Flow along a streamline

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HGL & EGL

• ^𝑷

𝝆𝒈: pressure head; it represents the height of a fluid column that produces the static pressure P

• ^𝑽^𝟐

𝟐𝒈: velocity head

• 𝒛: elevation head

Hydraulic Grade Line (HGL) &

Energy Grade Line (EGL)

𝑷

𝝆𝒈 + 𝑽^𝟐

𝟐𝒈 + 𝒛 = 𝑯 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆)

H is the total head for the flow

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HGL & EGL

• If a piezometer (measures static pressure) is tapped into a pipe, the liquid would rise to a height of _𝝆𝒈^𝑷 above the pipe center.

• The hydraulic grade line (HGL) is obtained by doing this at several locations along the pipe and drawing a line through the liquid levels in the piezometers.

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HGL & EGL

• if a Pitot tube (measures static+dynamic pressure) is tapped into a pipe, the liquid would rise to a height of _𝝆𝒈^𝑷 + ^𝑽^𝟐

𝟐𝒈 above the pipe center, or a distance of _𝟐𝒈^𝑽^𝟐 above the HGL.

• The energy grade line (EGL) is obtained by doing this at several locations along the pipe and drawing a line through the liquid levels in the Pitot tubes.

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HGL & EGL

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HGL & EGL

• Noting that the fluid also has elevation head z (unless the reference level is taken to be the centerline of the pipe), the HGL and EGL can be defined as follows: The line that represents the sum of the static pressure and the elevation heads, _𝝆𝒈^𝑷 + 𝒛, is called the HGL.

• The line that represents the total head of the fluid, _𝝆𝒈^𝑷 + ^𝑽^𝟐

𝟐𝒈 + 𝒛,is called the energy grade line.

• The difference between the heights of EGL and HGL is equal to the dynamic head, _𝟐𝒈^𝑽^𝟐.

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HGL & EGL

• For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid.

• The EGL is always a distance _𝟐𝒈^𝑽^𝟐 above the HGL. These two lines approach each other as the velocity decreases, and they diverge as the velocity increases.

• In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant.

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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Bernoulli Equation

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General Energy Equation

• first law of thermodynamics, also known as the conservation of energy principle.

• total energy consists of internal, kinetic, and potential energies General Energy Equation

𝐸_𝑖𝑛 − 𝐸_𝑜𝑢𝑡 = ∆𝐸 ሶ𝑄_{𝑛𝑒𝑡 𝑖𝑛} + ሶ𝑊_{𝑛𝑒𝑡 𝑖𝑛} = 𝑑𝐸_𝑠𝑦𝑠

𝑑𝑡 𝑜𝑟 ሶ𝑄_{𝑛𝑒𝑡 𝑖𝑛} + ሶ𝑊_{𝑛𝑒𝑡 𝑖𝑛} = 𝑑 𝑑𝑡 න

𝑠𝑦𝑠

𝜌𝑒 𝑑𝑉 ሶ𝑄_{𝑛𝑒𝑡 𝑖𝑛} = ሶ𝑄_𝑖𝑛 − ሶ𝑄_𝑜𝑢𝑡 𝑊ሶ_{𝑛𝑒𝑡 𝑖𝑛} = ሶ𝑊_𝑖𝑛 − ሶ𝑊_𝑜𝑢𝑡

𝑒 = 𝑢 + 𝑘𝑒 + 𝑝𝑒 = 𝑢 + 𝑉²

2 + 𝑔𝑧

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General Energy Equation

• Transfer of thermal energy from one system to another as a result of a temperature difference is called heat transfer.

• A process during which there is no heat transfer is called an adiabatic process.

• There are two ways a process can be adiabatic:

– Either the system is well insulated so that only a negligible amount of heat can pass through the system boundary,

– or both the system and the surroundings are at the same temperature and therefore there is no driving force (temperature difference) for heat transfer.

Energy Transfer by Heat, Q

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General Energy Equation

• An energy interaction is work if it is associated with a force acting through a distance.

• The time rate of doing work is called power: 𝑾ሶ

• A system may involve numerous forms of work, and the total work can be expressed as:

Energy Transfer by Work, W

𝑾_{𝒕𝒐𝒕𝒂𝒍} = 𝑾_{𝒔𝒉𝒂𝒇𝒕} + 𝑾_{𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆} + 𝑾_{𝒗𝒊𝒔𝒄𝒐𝒖𝒔} + 𝑾_{𝒐𝒕𝒉𝒆𝒓}

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General Energy Equation

• Many flow systems involve a machine such as a pump, a turbine, a fan, or a compressor whose shaft protrudes through the control surface, and the work transfer associated with all such devices is simply referred to as shaft work.

• The power transmitted via a rotating shaft:

Shaft Work

ሶ𝑾_{𝒔𝒉𝒂𝒇𝒕} = 𝝎 𝑻_{𝒔𝒉𝒂𝒇𝒕} = 𝟐𝝅 ሶ𝒏𝑻_{𝒔𝒉𝒂𝒇𝒕}

Angular speed of shaft (rad/s) Number of revolutions (rev/min) or (rpm)

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General Energy Equation

• piston-cylinder devices:

Work Done by Pressure Forces: Flow Work

𝜹 ሶ𝑾_{𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆} = 𝜹 ሶ𝑾_{𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚} = 𝑷𝑨𝑽_{𝒑𝒊𝒔𝒕𝒐𝒏}

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General Energy Equation

• Pressure always acts inward and normal to the surface;

𝜹 ሶ𝑾_{𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆} = −𝑷 𝒅𝑨 𝑽_𝒏 = −𝑷 𝒅𝑨 (𝑽. 𝒏) ሶ𝑾𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆,𝒏𝒆𝒕 𝒊𝒏 = − න

𝑨

𝑷 𝑽. 𝒏 𝒅𝑨

ሶ𝑾_{𝒏𝒆𝒕 𝒊𝒏} = ሶ𝑾𝒔𝒉𝒂𝒇𝒕, 𝒏𝒆𝒕 𝒊𝒏 + ሶ𝑾𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆,𝒏𝒆𝒕 𝒊𝒏 = ሶ𝑾𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 − න

𝑨

𝑷 𝑽. 𝒏 𝒅𝑨

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• For a closed system:

ሶ𝑸_{𝒏𝒆𝒕 𝒊𝒏} + ሶ𝑾𝒔𝒉𝒂𝒇𝒕, 𝒏𝒆𝒕 𝒊𝒏 + ሶ𝑾𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆,𝒏𝒆𝒕 𝒊𝒏 = 𝒅𝑬_𝒔𝒚𝒔 𝒅𝒕

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یروآدای

• conservation of energy for a system:

• ሶ𝑄 (the rate of heat transfer) is positive when heat is added to the system from the surroundings;

• 𝑊ሶ (the rate of work) is positive when work is done by the system on its surroundings.

The First Law of Thermodynamics

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یروآدای

The First Law of Thermodynamics

• u is the specific internal energy, V the speed, and z the height (relative to a convenient datum) of a particle of substance having mass dm.

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یروآدای

• For a system of fluid,

Relation of System Derivatives to the Control Volume Formulation: RTT

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یروآدای

Reynolds Transport Theorem

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General Energy Equation

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• In the case of a deforming control volume:

𝑽_𝒓 = 𝑽 − 𝑽_𝑪𝑺

fluid velocity C. S. velocity

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General Energy Equation

• For a fixed control volume (no motion or deformation of control volume)

• This equation is not in a convenient form for solving practical engineering problems because of the integrals, and thus it is desirable to rewrite it in terms of average velocities and mass flow rates through inlets and outlets.

• Approximated by:

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• Or:

• Used definition of enthalpy:

• general expressions of conservation of energy, but their use is still limited to fixed control volumes, uniform flow at inlets and outlets, and negligible work due to viscous forces and other effects.

• Also, the subscript “net in” stands for “net input,” and thus any heat or work transfer is positive if to the system and negative if from the system.

ℎ = 𝑢 + 𝑃 𝜌

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General Energy Equation

• For steady flows, the time rate of change of the energy content of the control volume is zero,

• for such single-stream devices:

Energy Analysis of Steady Flows

ሶ𝑸_{𝒏𝒆𝒕 𝒊𝒏} + ሶ𝑾𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 = ሶ𝒎(𝒉_𝟐 − 𝒉_𝟏 + 𝑽_𝟐^𝟐 − 𝑽_𝟏^𝟐

𝟐 + 𝒈 𝒛_𝟐 − 𝒛_𝟏 )

ሶ𝑸_{𝒏𝒆𝒕 𝒊𝒏} + ሶ𝑾𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 = ෍

𝒐𝒖𝒕

ሶ𝒎 𝒉 + 𝑽^𝟐

𝟐 + 𝒈𝒛 − ෍

𝒊𝒏

ሶ𝒎 𝒉 + 𝑽^𝟐

𝟐 + 𝒈𝒛

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General Energy Equation

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• unit-mass basis:

• If the flow is ideal with no irreversibilities such as friction, the total mechanical energy must be conserved,

mechanical energy input mechanical energy output 𝒒_{𝒏𝒆𝒕 𝒊𝒏} + 𝒘𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 = 𝒉_𝟐 − 𝒉_𝟏 + 𝑽_𝟐^𝟐 − 𝑽_𝟏^𝟐

𝟐 + 𝒈 𝒛_𝟐 − 𝒛_𝟏 𝒘𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 + 𝑷_𝟏

𝝆_𝟏 + 𝑽_𝟏^𝟐

𝟐 + 𝒈𝒛_𝟏 = 𝑷_𝟐

𝝆_𝟐 + 𝑽_𝟐^𝟐

𝟐 + 𝒈𝒛_𝟐 + (𝒖_𝟐 − 𝒖_𝟏 − 𝒒_{𝒏𝒆𝒕 𝒊𝒏})

𝒊𝒅𝒆𝒂𝒍 𝒇𝒍𝒐𝒘 𝒏𝒐 𝒎𝒆𝒄𝒉𝒂𝒏𝒊𝒄𝒂𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝒍𝒐𝒔𝒔 𝒒_{𝒏𝒆𝒕 𝒊𝒏} = 𝒖_𝟐 − 𝒖_𝟏

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• For single-phase fluids (a gas or a liquid):

𝒎𝒆𝒄𝒉𝒂𝒏𝒊𝒄𝒂𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝒍𝒐𝒔𝒔 𝒆_{𝒎𝒆𝒄𝒉, 𝒍𝒐𝒔𝒔} = 𝒖_𝟐 − 𝒖_𝟏 − 𝒒_{𝒏𝒆𝒕 𝒊𝒏}

𝒖_𝟐 − 𝒖_𝟏 = 𝒄_𝒗(𝑻_𝟐 − 𝑻_𝟏)

𝒘𝒔𝒉𝒂𝒇𝒕,𝒏𝒆𝒕 𝒊𝒏 + 𝑷_𝟏

𝝆_𝟏 + 𝑽_𝟏^𝟐

𝟐 + 𝒈𝒛_𝟏 = 𝑷_𝟐

𝝆_𝟐 + 𝑽_𝟐^𝟐

𝟐 + 𝒈𝒛_𝟐 + 𝒆_{𝒎𝒆𝒄𝒉, 𝒍𝒐𝒔𝒔} 𝒆_{𝒎𝒆𝒄𝒉, 𝒊𝒏} = 𝒆_{𝒎𝒆𝒄𝒉, 𝒐𝒖𝒕} + 𝒆_{𝒎𝒆𝒄𝒉, 𝒍𝒐𝒔𝒔}

𝒘𝒔𝒉𝒂𝒇𝒕, 𝒏𝒆𝒕 𝒊𝒏 = 𝒆_{𝒔𝒉𝒂𝒇𝒕, 𝒊𝒏} − 𝒆_{𝒔𝒉𝒂𝒇𝒕,𝒐𝒖𝒕} = 𝒘_{𝒑𝒖𝒎𝒑} − 𝒘_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆}

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where w_pump is the mechanical work input (due to the presence of a pump, fan, compressor, etc.) and w_turbine is the mechanical work

output.

𝒘_{𝒑𝒖𝒎𝒑} + 𝑷_𝟏

𝝆_𝟏 + 𝑽_𝟏^𝟐

𝟐 + 𝒈𝒛_𝟏 = 𝑷_𝟐

𝝆_𝟐 + 𝑽_𝟐^𝟐

𝟐 + 𝒈𝒛_𝟐 + 𝒘_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆} + 𝒆_{𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔}

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General Energy Equation

• Multiplying by the mass flow rate:

• 𝑾ሶ _{𝒑𝒖𝒎𝒑}: shaft power input through the pump’s shaft

• 𝑾ሶ _{𝒕𝒖𝒓𝒃𝒊𝒏𝒆}: shaft power output through the turbine’s shaft

• ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔}: total mechanical power loss (turbine losses+ pump losses+ frictional losses in the piping network)

ሶ𝑾_{𝒑𝒖𝒎𝒑} + ሶ𝒎(𝑷_𝟏

𝝆_𝟏 + 𝑽_𝟏^𝟐

𝟐 + 𝒈𝒛_𝟏) = ሶ𝒎(𝑷_𝟐

𝝆_𝟐 + 𝑽_𝟐^𝟐

𝟐 + 𝒈𝒛_𝟐) + ሶ𝑾_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆} + ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔}

ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔} = ሶ𝑬𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔,𝒑𝒖𝒎𝒑+ ሶ𝑬𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔,𝒕𝒖𝒓𝒃𝒊𝒏𝒆+ ሶ𝑬𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔,𝒑𝒊𝒑𝒊𝒏𝒈

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• Thus the energy equation can be expressed in its most common form in terms of heads as:

• useful head delivered to the fluid by the pump 𝒉_{𝒑𝒖𝒎𝒑,𝒖} + 𝑷_𝟏

𝝆_𝟏𝒈 + 𝑽_𝟏^𝟐

𝟐𝒈 + 𝒛_𝟏 = 𝑷_𝟐

𝝆_𝟐𝒈 + 𝑽_𝟐^𝟐

𝟐𝒈 + 𝒛_𝟐 + 𝒉_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆} + 𝒉_𝑳

𝒉_{𝒑𝒖𝒎𝒑,𝒖} = 𝒘_{𝒑𝒖𝒎𝒑,𝒖}

𝒈 = 𝑾ሶ _{𝒑𝒖𝒎𝒑,𝒖}

ሶ𝒎𝒈 = 𝜼_{𝒑𝒖𝒎𝒑} ሶ𝑾_{𝒑𝒖𝒎𝒑} ሶ𝒎𝒈

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General Energy Equation

• Where:

∆ ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒇𝒍𝒖𝒊𝒅} = ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒐𝒖𝒕} − ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒊𝒏}

∆ ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒇𝒍𝒖𝒊𝒅} • :

خرن شیازفا یژرنا

یکیناکم لایس

اب لداعم

Useful pumping power supplied to the fluid ( ሶ𝑾_{𝒑𝒖𝒎𝒑,𝒖})

(55)

• extracted head removed from the fluid by the turbine.

𝒉_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆} = 𝒘_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆}

𝒈 = 𝑾ሶ _{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆}

ሶ

𝒎𝒈 = 𝑾ሶ _{𝒕𝒖𝒓𝒃𝒊𝒏𝒆} 𝜼_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆}𝒎𝒈ሶ

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• Where:

∆ ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒇𝒍𝒖𝒊𝒅} = ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒊𝒏} − ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒐𝒖𝒕}

∆ ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒇𝒍𝒖𝒊𝒅} • :

خرن شهاک یژرنا

یکیناکم لایس

اب لداعم

Mechanical power extracted from the fluid by turbine ( ሶ𝑾_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆})

(57)

• we use the absolute value sign to avoid negative values for efficiencies.

• A pump or turbine efficiency of 100 percent indicates perfect conversion between the shaft work and the mechanical energy of the fluid,

• this value can be approached (but never attained) as the frictional effects are minimized.

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General Energy Equation

Motor and Generator

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General Energy Equation

Motor and Generator

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General Energy Equation

• A pump is usually packaged together with its motor, and

• Aa turbine with its generator.

• Overall efficiency of pump–motor and turbine–generator combinations???

Motor and Generator

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General Energy Equation

Turbine and Generator

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General Energy Equation

Motor and Pump

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General Energy Equation

• due to all components of the piping system other than the pump or turbine.

• represents the frictional losses associated with fluid flow in piping, and it does not include the losses that occur within the pump or turbine due to the inefficiencies of these devices.

𝒉_𝑳 = 𝒆𝒎𝒆𝒄𝒉 𝒍𝒐𝒔𝒔,𝒑𝒊𝒑𝒊𝒏𝒈

𝒈 = ሶ𝑬𝒎𝒆𝒄𝒉 𝒍𝒐𝒔𝒔,𝒑𝒊𝒑𝒊𝒏𝒈

ሶ

𝒎𝒈

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General Energy Equation

• which is the Bernoulli equation derived earlier using Newton’s second law of motion.

Special Case: Incompressible Flow with No Mechanical Work Devices and Negligible Friction

𝑷_𝟏

𝝆𝒈 + 𝑽_𝟏^𝟐

𝟐𝒈 + 𝒛_𝟏 = 𝑷_𝟐

𝝆𝒈 + 𝑽_𝟐^𝟐

𝟐𝒈 + 𝒛_𝟐 𝑷

𝝆𝒈 + 𝑽^𝟐

𝟐𝒈 + 𝒛 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

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General Energy Equation

• The kinetic energy correction factors are often ignored (i.e., a is set equal to 1) in an elementary analysis since (1) most flows encountered in practice are turbulent, for which the correction factor is near unity, and

Kinetic Energy Correction Factor

ሶ𝑾_{𝒑𝒖𝒎𝒑} + ሶ𝒎(𝑷_𝟏

𝝆 + 𝜶_𝟏 𝑽_𝟏^𝟐

𝟐 + 𝒈𝒛_𝟏)

= ሶ𝒎(𝑷_𝟐

𝝆 + 𝜶_𝟐 𝑽_𝟐^𝟐

𝟐 + 𝒈𝒛_𝟐) + ሶ𝑾_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆} + ሶ𝑬_{𝒎𝒆𝒄𝒉,𝒍𝒐𝒔𝒔} 𝒉_{𝒑𝒖𝒎𝒑,𝒖} + 𝑷_𝟏

𝝆𝒈 + 𝜶_𝟏 𝑽_𝟏^𝟐

𝟐𝒈 + 𝒛_𝟏 = 𝑷_𝟐

𝝆𝒈 + 𝜶_𝟐 𝑽_𝟐^𝟐

𝟐𝒈 + 𝒛_𝟐 + 𝒉_{𝒕𝒖𝒓𝒃𝒊𝒏𝒆,𝒆} + 𝒉_𝑳

(66)

General Energy Equation

• (2) the kinetic energy terms are often small relative to the other terms in the energy equation, and multiplying them by a factor less than 2.0 does not make much difference.

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General Energy Equation

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