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Unformatted text preview: HES 2340: Fluid Mechanics 1 Week 4 & 5
Mass, Bernoulli, and Energy
Equations School of Engineering, Computing & Science
Sarawak Campus Introduction
• The conservation laws, which are the:
– Conservation of mass,
– Conservation of momentum, and
– Conservation of energy.
• The Bernoulli equation is concerned with the conservation of
kinetic, potential, and flow energies of a fluid stream and their
conversion to each other.
• The energy equation is a statement of the conservation of
energy principle (mechanical energy balance). Conservation of mass
• Conservation of mass principle is one of the most fundamental
principles in nature.
• Mass, like energy, is a conserved property, and it cannot be created
or destroyed during a process.
• Example:
– when 16kg of oxygen reacts with 2 kg of H2, 18 kg of water is
formed. In an electrolysis process, the water will separated back
to 2 kg of H2 and 16 kg of oxygen.
• The conservation of mass principle can be expressed as where
and
the CV, and are the total rates of mass flow into and out of
/ is the rate of change of mass within the CV. Conservation of momentum
• The product of mass and the velocity of the body is called linear
momentum.
• What is conservation of momentum?
• For a collision occurring between object 1 and object 2 in an isolated
system, the total momentum of the two objects before the collision is
equal to the total momentum of the two objects after the collision.
That is, the momentum lost by object 1 is equal to the momentum
gained by object. Conservation of energy
• Energy can be transferred to or from a closed system by heat or
work, and the conservation of energy principle requires that the net
energy transfer to or from a system during a process be equal to the
change in the energy content of the system.
• Conservation of energy . Continuity
The general form of the continuity equation is obtained by substituting
the properties for mass into the Reynolds transport theorem Let and ∀ ∙ 1 , resulting in / ∀
However, ∙ 0 (conservation of mass), so the general, or integral, form of the continuity equation is
∀ ∙ 0 Continuity ∀ ∙ 0 This equation can be expressed in words as
The accumulation rate
of mass in the
control volume The net outflow rate
of mass through
the control surface 0 If the mass crosses the control surface through a number of inlet and
exit ports, the continuity equation simplifies to
0 Steady flow processes
• For steady flow, the total amount of
mass contained in CV is constant,
that is, total amount of mass
entering must be equal to total
amount of mass leaving,
0 0
• For singlestream steadyflow
systems, outflow inflow Incompressible flows
• For incompressible flows ( = constant), • The is called the volume flow passing through the given cross section. The volume flow
meters per second (m3/s). will have units of cubic • If the cross section is not onedimensional, we have to integrate
∙ • The above equation allows us to define an average velocity
which,
when multiplied by the section area, gives the correct volume flow
1
∙ • This could be called the volumeaverage velocity.
• If the density varies across the section, we can define an average
density in the same manner:
1 • But the mass flow would contain the product of density and velocity,
and the average product
would in general have a different
value from the product of the averages
1
∙ EXAMPLE
Write the conservationofmass relation for steady flow
through a streamtube (flow
everywhere parallel to the
walls) with a single onedimensional exit 1 and inlet 2:
Solution:
For single flow applies with the single inlet and exit
constant
Therefore, in a streamtube in steady flow, the mass flow is constant
across every section of the tube. If the density is constant, then
constant or The volume flow is constant in the tube in steady incompressible flow, and
the velocity increases as the section area decreases. EXAMPLE
The tank is being filled with water
by two onedimensional inlets. Air
is trapped at the top of the tank.
The water height is . Find an
expression for the change in
water height / .
Solution
0 The flow within is unsteady, and the system has no outlets and two inlets: ∀
If 0 is the tank crosssectional area, the unsteady term can be evaluated as follows:
∀ EXAMPLE
0 The above term vanishes because it is the rate of change of air mass and is
zero because the air is trapped at the top. Therefore, ∀ For water, Euler’s Equation
• The hydrostatic equations were derived by equating the sum of the forces on a fluid element equal to zero. • The same ideas are applied in this section to a moving fluid by equating the sum of the forces acting on a fluid element to the element's acceleration, according to Newton's second law. • The resulting equation is Euler's equation, which can be used to predict pressure variation in moving fluids. Euler’s Equation
• Consider the
cylindrical element
oriented in an
arbitrary direction ℓ
with crosssectional
area ∆A in a flowing
fluid.
• The element is
oriented at an angle
with respect to the
horizontal plane (the
xy plane).
• The element has been isolated from the flow field and can be treated as a
“free body” where the presence of the surrounding fluid is replaced by
pressure forces acting on the element. • Assume that the viscous forces are zero. Euler’s Equation
• Here the element is
being accelerated in
the ℓdirection.
• Note that the
coordinate axis z is
vertically upward
and that the
pressure varies
along the length of
the element.
• Applying Newton's second law in the ℓdirection results in
ℓ ℓ
ℓ • The mass of the fluid element is ∆ ∆ℓ Euler’s Equation • The net force due to pressure in the ℓdirection is ∆ ∆ ∆ ∆∆ • Note that any pressure forces acting on the side of the cylincer
element will not contribute to a force in the ℓdirection. Euler’s Equation
• The force due to
gravity is the
component of weight
in the ℓdirection
∆ℓ
∆ sin
where the minus
sign occurs because
the component of
weight is in the
negative ℓdirection.
• From the diagram, it showing the relationship for angle α with
respect to ∆ℓ, and ∆z, one notes that sin
∆ /∆ℓ, so the force due
to gravity can be expressed as
∆
∆
∆
∆ℓ∆
∆ℓ
∆ℓ
• Note that the weight of the element is ∆
∆ℓ∆ . ℓ ∆
∆∆
∆ℓ∆
∆ℓ
Dividing through by ∆ ∆ℓ results in ∆ ∆ℓ ℓ ∆
∆
ℓ
∆ℓ
∆ℓ
Taking the limit as ∆ℓ approaches zero (element shrinks to a point)
leads to the differential equation for acceleration in the ℓdirection
ℓ
For an incompressible flow, ℓ
ℓ
is constant and thus ℓ
ℓ
The above equation is Euler's equation for motion of a fluid. ℓ ℓ • Euler’s equation shows that the acceleration is equal to the
change in piezometric pressure with distance, and the minus sign
means that the acceleration is in the direction of decreasing
piezometric pressure.
• In a static body of fluid, Euler's equation reduces to the
hydrostatic differential equation.
– In a static fluid, there are no viscous stresses, which is a
condition required in the derivation of Euler's equation.
– Also there is no motion, so the acceleration is zero in all
directions.
– Thus, Euler's equation reduces to ℓ = 0. EXAMPLE
A column water in a vertical tube is
being accelerated by a piston in
the vertical direction at 100 m/s2.
The depth of the water column is
10 cm. Find the gage pressure on
the piston. The water density is
103 kg/m3. Solution
Because the acceleration is constant there is no dependence on
time so the partial derivative in Euler’s equation can be replaced by
an ordinary derivative. Euler’s equation in zdirection: EXAMPLE Integrate between sections 1 and 2: ∆ Gage pressure
∆ Gage pressure
10.9 kPa Gage pressure The Bernoulli Equation
• From the dynamics of particles in solid‐body mechanics, one knows that integrating Newton's second law for particle motion along a path provides a relationship between the change in kinetic energy and the work done on the particle. • Integrating Euler's equation along a pathline in the steady flow of an incompressible fluid yields an equivalent relationship called the Bernoulli equation. Frictionless flow: the 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.
• Equation is useful in flow
regions outside of boundary
layers and wakes, where the
fluid motion is governed by
the combined effects of
pressure and gravity forces. The Bernoulli equation
• The Bernoulli equation is
developed by applying Euler's
equation along a pathline with
the direction , ℓ replaced by s,
the distance along the
pathline, and the acceleration
, the
ℓ replaced by
direction tangent to the
pathline.
• Euler's equation becomes: The tangential component of acceleration is The Bernoulli equation
0
Since the properties along a streamline depend only on the distance s Steady flow • Euler’s equation now becomes
2
• Moving all the terms to one side yields
2 0 ⇒ 2 constant Bernoulli equation states that the sum of the piezometric pressure and kinetic pressure /2 is constant along a streamline for the steady flow of an incompressible, inviscid fluid. Illustration
of the
concept
underlying
the
Bernoulli
equation • The constant in the Bernoulli equation is the same at all three locations 2 2 2 • Even though the elevation, pressure head, and velocity head vary
through the venturi section, the sum of the three heads is the same. Assumptions of the Bernoulli equation
• Steady flow ( / =0) – It should not be used during the transient startup and
shutdown periods, or during periods of change in the flow
conditions.
• Incompressible flow ( ∙ 0) – Acceptable if the Mach number is less than 0.3.
• Frictionless flow
– Very restrictive, solid walls introduce friction effects. Assumptions of the Bernoulli equation
• Flow along a single streamline
– Different streamlines may have different “Bernoulli constant”
, depending on the flow conditions.
• No shaft work ( 0) – The Bernoulli equation can still be applied to a flow section
prior to or past a machine (with different Bernoulli constants)
• No heat transfer (qnet,in 0) Illustration of the regions of validity
and invalidity of the Bernoulli equation For windtunnel model test, the
Bernoulli equation is only valid in
the core flow of the tunnel but not
in the tunnelwall boundary
layers, the model surface
boundary layers, or the wake of
the model, all of which are
regions with high friction. Illustration of the regions of validity
and invalidity of the Bernoulli equation In propeller flow, Bernoulli’s equation
is valid both upstream and
downstream, but with a different
constant , cause by the work addition of the propeller. Illustration of the regions of validity
and invalidity of the Bernoulli equation For windtunnel model test, the
Bernoulli equation is only valid in
the core flow of the tunnel but not
in the tunnelwall boundary
layers, the model surface
boundary layers, or the wake of
the model, all of which are
regions with high friction. Static, dynamic, and stagnation pressures
2
• constant along a streamline is the static pressure; it represents the actual
thermodynamic pressure of the fluid.
• This is the same as the pressure used in
thermodynamics and property tables. • is the dynamic pressure; it represents the pressure
rise when the fluid in motion. • is the hydrostatic pressure, depends on the
reference level selected. • • The sum of the static, dynamic, and
hydrostatic pressures is called the
total pressure (a constant along a
streamline).
The sum of the static and dynamic
pressures is called the stagnation
pressure, kPa • 2
Therefore, the fluid velocity at that
location can be calculated from
2 Pitot‐static probe 2
, , The Bernoulli equation
2 constant Dividing the equation by specific gravity 2
where
and 2 is the piezometric head and /2 is the velocity head, is a constant. Pressure
head Elevation
head 2 Velocity
head Constant along
streamline It is often convenient to plot mechanical energy graphically using
heights.
constant 2
Dividing the equation by specific gravity , constant 2
Constant along
Velocity
Pressure
Elevation
head
head
head
streamline
• / is the pressure head; it represents the height of a fluid
column that produces the static pressure .
• /2 is the velocity head; it represents the elevation needed for
a fluid to reach the velocity V during frictionless free fall. • is the elevation head; it represents the potential energy of the
fluid. • is the total head. HGL and EGL • Hydraulic Grade Line (HGL) • Energy Grade Line (EGL) (or total head) Tips for Drawing HGLs and EGLs
• For stationary bodies such as reservoirs or lakes, the EGL and HGL
coincide with the free surface of the liquid, since the velocity is zero and
the static pressure (gage) is zero. •
• For steady flow in a Pipe of constant diameter and wall roughness, the
slope Δ /Δ for the EGL and the HGL will be constant.
The EGL is always a distance /2 above the HGL. • A pump causes an abrupt rise in the EGL and HGL by adding energy
to the flow.
• Height of the EGL decreases in the flow direction unless a pump is
present. • A turbine causes an abrupt drop in the EGL and HGL by removing
energy from the flow.
• Power generated by a turbine can be increased by using a gradual
expansion at the turbine outlet. The expansion converts kinetic
energy to pressure •
•
• If the outlet to a reservoir is an abrupt expansion, the kinetic energy is lost.
When a flow passage changes diameter, the distance between the EGL and
the HGL will change because velocity changes.
In addition, the slope on the EGL will change because the head loss per
length will be larger in the conduit with the larger velocity. • When a pipe discharges into the atmosphere the HGL is coincident
with the system because / = 0 at these points.
• For example, the HGL in the liquid jet is drawn through the jet itself. • If the HGL falls below the pipe, then / is negative, indicating
subatmospheric pressure and a potential location of cavitation EXAMPLE
Water is flowing from a hose attached to a water main at
400 kPa gage. A child places his thumb to cover most of
the hose outlet, causing a thin jet of highspeed water to
emerge. If the hose is held upward, what is the
maximum height that the jet could achieve?
Assumptions: The flow exiting into the air is steady,
incompressible, and irrotational (so that the Bernoulli
equation is applicable). The velocity inside the hose is
relatively low (V1 = 0) and we take the hose outlet as the
reference level (z1 = 0). At the top of the water trajectory
V2 = 0, and atmospheric pressure applied. 40.8 m EXAMPLE
A piezometer and a Pitot tube are tapped into
a horizontal water pipe to measure static and
stagnation pressures. For the indicated water
column heights, determine the velocity at the
center of the pipe. EXAMPLE
Find a relation between
nozzle discharge
velocity
and tank
freesurface height .
Assume steady
frictionless flow. Solution:
• Note that mass conservation
is usually a vital part of
Bernoulli analyses.
• If
is the tank cross section
and
the nozzle area, this
approximation a onedimensional flow with constant
density 1 Bernoulli’s equation gives
2
2
But since sections 1 and 2 are both exposed to atmospheric pressure
, the pressure terms cancel, leaving
2
2 2
Combining Eqs. (1) and (2)
2
1
/
Generally the nozzle area
is very much smaller than the tank area
, so that the ratio / is negligible, and an accurate approximation
for the outlet velocity is
2
The above equation states that the discharge velocity equals the
speed which a frictionless particle would attain if it fall freely from point
1 to point 2; that is the potential energy of the surface fluid is entirely
converted to kinetic energy. Euler’s equation
ℓ ℓ The Bernoulli equation
constant along a streamline 2 2 constant along a streamline The Energy Equation
The energy equation involves energy, work, and power as well as machines that interact with flowing fluids. Energy, Work and Power
Energy
• When matter has energy, the matter can be used to do work.
• A fluid can have several forms of energy. For example a fluid jet
has kinetic energy, water behind a dam has gravitational
potential energy, and hot steam has thermal energy.
Work
• An energy interaction is work if it is associated with a force
acting through a distance.
• For example wind passing over the blades of a wind turbine.
• The wind exerts a force on the blades; this force produces a
torque and work is given by
Work = force X distance = torque X angular displacement Energy, Work and Power
• A system may involve numerous forms of work, and the total
work can be expressed as When air passes across the rotor of a
wind turbine, the air exerts forces that
result in a net torque. This torque
does work on the blades Energy, Work and Power
• Work and energy both have the same primary dimensions, and the
same units, and both characterize an amount or quantity.
– For example, 1 calorie is the amount of thermal energy needed
to raise the temperature of 1 gram of water by 1°C. Power
• The time rate of doing work is called power,
quantity of work or energy
≡
interval of time , which is defined by: ∆
lim
∆→ ∆ • A derivative is used because power can vary with each instant in time. • To derive an equation, let the amount of work be given by the product
of force and displacement ∆ where ∆: ∆
lim
∆→ ∆
is the velocity of a moving body. • When a shaft is rotating, the amount of work is given by the product of
torque and angular displacement ∆ ∆ . In this case, the power equation is
∆
lim
∆→ ∆
• Work Done by Pressure Forces: the work done
by the pressure forces on the control surface The associated power is Turbine and Pump
• In fluid mechanics, a
turbine is a machine
that is used to extract
energy from a
flowing fluid.
• Similarly, a pump is
a machine that is
used to provide
energy to a flowing
fluid. Energy transfer/change
• One of the most fundamental laws in nature is the 1st law of
thermodynamics, which is also known as the conservation
of energy principle.
• It states that energy can be neither created nor destroyed
during a process; it can only change forms
• Falling rock, picks up speed as PE
is converted to KE.
• If air resistance is neglected,
PE + KE = constant
• The conservation of energy
principle: • We frequently refer to the
sensible and latent forms of
internal energy as heat, or
thermal energy.
• For single phase substances, a
change in the thermal energy is
equivalent to a change in
temperature.
• The transfer of thermal energy
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: insulated
or same temperature
• An adiabatic process an
isothermal process. Energy transfer General energy equation
• The energy equation for a system is: • The above equation is also called the first law of thermodynamics,
can be stated in words:
net rate of
thermal energy
entering system •
•
• net rate at which
system does work
on environment rate of change of
energy of the mater
within the system A system is a body of matter that is under consideration.
A system always contains the same matter.
An imaginary boundary separates the system from all other matter, which is called the environment. General energy equation
• The energy equation involves sign conventions:
• Thermal energy is positive when there is an addition of thermal
energy to the system and negative when there is a removal.
• Work is positive when the system is doing work on the
environment and negative when work is done on the system.
• To extend the energy equation to a control volume, apply the
Reynolds Transport Theorem ∀ ∙ • Let the extensive property be energy
and let
energy per
unit mass, we obtain following for a fixed control volume: ∀
The net rate of energy
transfer into a CV by
heat and work transfer The time rate of
change of the energy
content in the CV ∙ The net flow rate of
energy out of the control
surface by mass flow ∀ The ∙ term: • In the field of heat transfer,
/ can be broken down into
conduction, convection, and radiation. In this fluid mechanics
course, we consider it only occasionally. The term: • The work term can be divided into three parts: • The shaft work isolates that portion of the work which is deliberately
done by a machine (pump impeller, fan blade, piston, etc)
protruding through the control surface into the control volume. • According to the sign convention , pump work is negative. Similarly, turbine work is positive, thus • The rate of work done on pressure forces occurs at the surface only; all work on internal portions of the material in the control volume is by
equal and opposite forces and is selfcancelling.
• The pressure work equals the pressure force on a small surface element
times the normal velocity component into the control volume ∙ ,
• The total pressure work is the integral over the control surface
∙ If part of the control surface is
the surface of a machine part,
we prefer to delegate that
portion of the pressure to the
shaft work
not to • The shear due to viscous stresses occurs at the control surface and
consists of the product of each viscous stress and the respectively velocity
component ∙ ⇒ ∙ where is the stress vector on the elemental surface
.
• This term may vanish or be negligible according to the particular type of
surface at that part of the control volume:
• Solid surface: for all parts of the control surface which are solid
0.
confining walls,
0 from the viscous noslip condition; hence
• Surface of a machine: here the viscous work is contributed by the
machine, and so we absorb this work in the term .
• An inlet or outlet: at an inlet or outlet, the flow is approximately
normal to the element ; hence the only viscouswork term comes
. Since viscous normal stress are
from the normal stress
extremely small in all cases, it is customary to neglect viscous work in
inlets and outlets of the control volume. • • This term may vanish or be negligible according to the particular type of
surface at that part of the control volume:
• Streamline surface (SS): if the control surface is a streamline such as
the upper curve in the boundarylayer, the viscouswork term must be
evaluated and retained if shear stresses are significant along this line.
In summary, the net result of the rateofwork term consists essentially of The
• • ∙ ∙ term: The system energy per unit mass may be of several types: where
could encompass chemical reactions, electrostatic or magnetic
field effects.
If we neglect
here and consider only the first three terms, with
defined as “up”
1
2 Recalling the Reynolds transport theorem for a fixed control volume: ∀ ∙ Substituting ∙ ∀ ∙ ∙ Rearrange the equation ∀ Next, substitute
1
2
Using the definition of enthalpy
1
2 ∀ 1
2
/ , we obtain ∀ 1
2 ∙ ∙ • 1
1 ∀ ∙ 2
2
If the control volume has a series of onedimensional inlets and outlets, the
surface integral reduces to a summation of outlet fluxes minus inlet fluxes:
1
2 ∙ 1
2 Finally, the simplified form for the energy
equation for a fixed control volume:
1
2
1
2
1
2 ∀ 1
2 The steadyflow energy equation
• For steady flow, time rate of change of the energy content of the CV
is zero.
1
2 1
2 • This equation states that the net rate of energy transfer to a CV by
heat and work transfers during steady flow is equal to the difference
between the rates of outgoing and incoming energy flows with mass.
• For steadyflow with one inlet and one outlet, the equation reduces to
a relation used in many engineering analyses.
• Let section 1 be the inlet and section 2 the outlet, then
1
2 1
2 The net rate of energy transfer into a CV by heat and work transfer The time rate of change of the energy content in the CV 1
2
1
2 ∀ 1
2 The net flow rate of energy out of the control surface by mass flow The net rate of energy transfer into a CV by heat and work transfer The time rate of change of the energy content in the CV 1
2
1
2 ∀ Steady‐flow
1
2 The net flow rate of energy out of the control surface by mass flow Steady‐flow
1 inlet and 1 outlet
The net rate of energy transfer into a CV by heat and work transfer 1
2 Flow rate of energy in to the control surface by mass flow 1
2 Flow rate of energy out of the control surface by mass flow The steadyflow energy equation
1
2 1
2
, the equation can rearrange as follows: • From continuity,
1
2 1
2 where
, , Recall that is positive if heat is added to the control volume and that and are positive if work is done by the fluid on the surroundings. 1
1
2
2
• Note that each term in the equation has the dimensions of energy per
unit mass, which is a form commonly used by mechanical engineers.
• If we divide through by , each term becomes a length, or head,
which is a form preferred by civil engineers.
• The traditional symbol for heard is , in order to avoid the confusion / ), we use internal energy in rewriting the with enthalpy ( head form of the energy relation:
2 2 where
, , • A very common application of the steadyflow energy equation is for
lowspeed flow with negligible viscous work ( → 0). 2 2 This is the difference between the available head upstream and downstream and is normally positive, representing the loss in head due to friction, denoted as . • Therefore, in lowspeed (nearly incompressible) flow with one inlet
and one exit, we may write
2 2 • The terms are all positive; that is, friction loss is always positive in real
flows, a pump adds energy (increases the lefthand side), and a turbine
extracts energy from the flow.
,
are included, the pump and/or turbine must lie between
• If
points 1 and 2. Kineticenergy correction factor
• Often the flow entering or leaving a port is not
strictly onedimensional, the velocity may vary
over the cross section.
• In this case the kineticenergy term for a given
port should be modified by a dimensionless
correction factor so that the integral can be
proportional to the square of the average
velocity through the port
1
2
• By letting
1
2 ∙ 1
≡ 2 where 1 be the velocity normal to the port, for incompressible flow,
1
1 or 2 • The term is the kineticenergy correction factor, having a value of
about 2.0 for fully developed laminar pipe flow and from 1.04 to 1.11 for
turbulent pipe flow. Incompressible steadyflow energy equation
• The complete incompressible steadyflow energy equation,
including pumps, turbines, and losses, would generalize to: 2 2
• The terms on the right ( , , numerically positive.
• All additive terms have dimensions of length ) are all Power equation
• Here we discuss how to relate head to power and efficiency.
• These parameters are used for applications such as selecting a
motor for operating a centrifugal pump, calculating the amount of
power that can supplied by a proposed hydroelectric plant, and
estimating the pump size for a piping system.
• Using the definition of shaft head: ⇒ ⇒ Eq. 1 Eq. 2 • From Eq. (1) and Eq. (2), the generalized power equation is: • Both pumps and turbines lose energy due to factors such as mechanical
friction, viscous dissipation, and leakage. These losses are accounted for
by the efficiency , which is defined as the ratio of power output to power
input:
power output from a machine or system
power input to a machine or system
• If the mechanical efficiency of the pump is , the power output delivered by the pump to the flow is:
where is the power supplied to the pump, usually by a rotating shaft that is connected to a motor.
• If the mechanical efficiency of the turbine is the output power supplied by the turbines is:
where is the power input to the turbine from the flow. EXAMPLE A hydrostatic power plant
takes in 30 m3/s of water
through its turbine and
discharge it to the
atmosphere at
2
m/s. The head loss in the
turbine and penstock
system is
20 m.
Assuming turbulent flow,
1.06, estimate the
power in MW extracted
by the turbine. We neglect the viscous and heat
transfer and take section 1 at the
reservoir surface where
0,
, and
100 m.
Section 2 is at the turbine outlet.
The steadyflow equation becomes
2
0 2
100 m 1.06 2 m/s
2 9.81 m/s 0 20 m 79.8 m
The turbine extracts about 79.8 percent of the 100m head available from
the dam. The total power extracted may be evaluated from the water
mass flow:
23.4 MW EXAMPLE
A pump delivers water (
1000 kg/m3) at 0.085 m3/s to a machine
at section 2, which is 6.096 m higher than the reservoir surface. The
/ 2 , where
7.5
losses between 1 and 2 are given by
is a dimensionless loss coefficient. Take
1.07. Find the power
required for the pump if it is 80 person efficient. If the reservoir is large,
the flow is steady, with
0.
0.085
/4 0.0762
18.64 m/s The steadyflow energy equation becomes
2 2
2 101350 68950
1000 9.81 6.096 1.07 7.5 18.64
2 9.81 154 m The pump head is negative, indicating work done on the fluid.
The power delivered is computed from
1000 0.085 9.81 154 128 kW We drop the negative sign when merely referring to the “power” required.
If the pump is 80 percent efficient, the input power required to drive it is
efficiency 128 kW
0.8 160 kW Incompressible steady‐flow energy equation
1 inlet and 1 outlet 2
2 Power Efficiency Pump
Turbine Contrasting the Bernoulli Equation and the Energy Equation The Bernoulli Equation The Energy Equation • Was derived by applying
• Was derived by starting with the
Newton’s second law to a
first law of thermodynamics and
particle and then integrating the
then using the Reynolds transport
resulting equation along a
theorem.
streamline.
• The Bernoulli equation involves • The energy equation includes both
only mechanical energy.
mechanical and thermal energy.
• The Bernoulli equation is
applied by selecting two points
on a streamline then equating
terms at these points: 2 2 • The energy equation is applied by
selecting an inlet section and an
outlet section in a pipe and then
equating terms as they apply to the
pipe:
2
2 The Bernoulli Equation The Energy Equation • Applies to steady, incompressible,
and inviscid flow. • Applies to steady, viscous,
incompressible flow in a pipe with
additional energy being added
through a pump or extracted
through a turbine. • Under special circumstances the energy equation can be reduced to the
Bernoulli equation.
• If the flow is inviscid, there is no head loss.
• If the “pipe” is regarded as a small stream tube enclosing a streamline,
then 1. • There is no pump or turbine along a streamline, so
• In this case the energy equation is identical to the Bernoulli equation.
• Note that the energy equation cannot be developed starting with the
Bernoulli equation. 0. ...
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This note was uploaded on 11/19/2011 for the course HES 2340 taught by Professor Tomedwards during the Three '09 term at Swinburne.
 Three '09
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