1. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
Lesson 4:
Bernoulli's Theorem
Objective
In this lesson we will discuss the relationships of "fluid mechanics" and how it APPLIES to flow and
pressure or head. The flow of ground water is controlled by the laws of physics and thermodynamics. We
will discuss a variety of concepts that apply to ground water flow and hydro-geology, and also flow in a
pipe as well as, the concept of pressure or "head." We will also discuss the Bernoulli equation or theorem.
Reading Assignment
Read the online lesson as well as Chapter 3 in your textbook.
Lecture
Introduction
The flow of groundwater is controlled by the laws of physics, thermodynamics and hydro-geology. In this
lesson we will assume that the water temperature is nearly constant and is not very warm. Most drinking
water sources are not very warm even in the summer months. Thermodynamics is more important in
industrial WATER TREATMENT , where boiler water is often treated, and is usually very hot to begin
the process. We will study the process of mechanical energy. Water contains energy due to 1- elevation, 2-
pressure and 3- velocity. Head is the amount of energy possessed by a unit quantity of water expressed in
feet. (Example: A 30 ft. tank will have a head of 30 ft.) Simply put head is the vertical distance between
two levels in a fluid. Another term for velocity pressure is dynamic pressure.
We will briefly discuss some terms and relationships, which will help you to better understand the rest of
this lesson.
Velocity pressure = total pressure - static pressure.
Velocity Head - the energy of motion; the vertical distance or height through which water must fall
freely, under the force of gravity, to acquire the velocity that it possesses. It is theoretically equal to the
square of the velocity divided by twice the acceleration of gravity. V2/ 2g
Static Head - Head due to elevation: the actual elevation difference between water surfaces or a water
surface and some other reference point. Example: If one WATER TANK is at the elevation of 2300ft.,
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2. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
while another is at the elevation of 2100 ft., the static head would be 200 ft.
Pressure Head - the head represented by the expression of pressure over weight (p/y ) where p is
pressure and y is weight. Where p is in pounds /sq. ft. and y is the weight of the liquid per cubic feet, h is
head in feet. 1 psi = 2.31 ft. and 1 ft. = 0.433 psi. These are very helpful conversions to use.
For a unit volume of fluid, the mass, m, is numerically equal to the density, p, since density is defined as
mass per unit volume. The total energy of the unit volume of fluid is the three components; 1- kinetic, 2-
gravitational, 3- fluid pressure energy.
E tv = ½ pv2 + pgz + P
g = the acceleration of gravity
z = the elevation of the center of gravity of the fluid
v = velocity above the reference elevation.
where Etv is the total energy per unit volume. If this equation is divided by p, the result is the total energy
per unit mass.....
E tm = v2/2 + gz + P/p )which is the BERNOULLI equation).
The Bernoulli Theorem says that total pressure = the same at any two points along a streamline, in a
flow.
Total Pressure - static pressure + ½ × density × velocity squared.
Total pressure = static pressure + dynamic pressure
Static pressure = pressure that is measured when water is at rest.
The Theory of Bernoulli
This theory, which forms part of the study of hydraulic principles, states that the static pressure of a
moving fluid varies inversely as its velocity, which means that as velocity increases, the static pressure
decrease.
The principle of the relationship between velocity and pressure is illustrated below. Two vessels (a) and
(b), of the same shape and cross-sectional area, are joined together by a small-bore pipe, (c). If fluid is
added to vessel (b) and gradually find its own level, this level will still be maintained if two positions (d)
and (e) of the same size and weight are placed on the liquid in the vessels (a) and (b), respectively.
If a force of 1N is exerted on piston (d), in a downward direction, fluid in (a) will flow through the pipe (c)
into vessel (b). The velocity of the fluid will increase as it passes through (c), because the cross-sectional
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3. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
area of the pipe (c) is much less than that of (a), and the same volume of fluid must pass through (c) as
leaves (a) in the same time.
Some of the 1N pressure in (a) will be converted into velocity pressure in pipe (c), and a pressure gauge
situated at (f) would REGISTER a pressure less than that equivalent to 1N. When the fluid reaches the
vessel (b), its velocity will reduce to its original rate and some of the kinetic energy is reconverted into
potential energy."
Total Pressure: In flowing are, it is the sum of the static and velocity pressures.
Velocity Pressure: In flowing air, it is the pressure due to velocity and density of air.
Static Pressure: In flowing air, it is the total pressure minus velocity pressure, pushing equally in all
directions.
Pressure Static measures elevation change in feet.
Static Pressure
Static pressure is the difference between the air pressure on the inside of your barn and the air pressure
outside your barn.
The static pressure is an indicator of how much negative pressure the fans are creating and how effective
they will be at drawing fresh air in through your inlets. An overly high static pressure can also indicate
that you do not have enough inlets in your barn.
In terms of your barn, the static pressure should be around 0.04" wc (inches of water column) to 0.10"
wc. When it is warm outside, such as during the summer, and your fans are running on or near their
maximum speed, you want your static pressure to be around 0.05". In the winter, you want the static
pressure higher so that the air pulled through the inlets enters the room faster, promoting thorough mixing
of incoming air with the air already in the barn.
To measure static pressure in a barn you would use a manometer.
The following diagrams indicate the changes in static pressure when your fans are off and when they are
running.
Fig. 4-1: This diagram shows a u-tube manometer measuring an equal pressure between the outside of the barn and the inside of
the barn. This happens when the fans are off.
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4. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
Fig. 4-2: This diagram shows a u-tube manometer measuring a decreased pressure in the barn compared to outside due to the fans
running. Note that compared to figure one the tube closest to the wall is lower because the greater pressure is pushing the liquid
in the tube downward.
Reference: BSM Agri Ltd.
Velocity Pressure
Tank Gauge - 20 psi = Line Gauge
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5. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
Swap gauges
Tank Gauge - 20 psi = Line Gauge
Velocity pressure is then 20 psi
Velocity of water would be 57 feet/sec
The faster the molecules flow, the harder it is to make a turn into the line gauge.
Calculation of Velocity of a Fluid Through a Pipeline
The speed or velocity of a fluid flowing through a channel or pipeline is related to the cross-sectional area
of the pipeline and the quantity of water moving through the line.
For example, if the diameter of a pipeline is reduced, then the velocity of the water in the line must
increase to allow the same amount of water to pass through the line.
SAMPLE Problem:
If the flow through a 2 foot diameter pipe is 8 MGD, the velocity is:
If this same 8 MGD flow is transferred to a pipe with a 1 foot diameter, the velocity would be:
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6. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
Based upon this example problem, we can see that if the cross-sectional area decreased, the velocity of
the flow must increase. Mathematically we can say that the velocity and cross-sectional area are
inversely proportional when the amount of flow (Q) is constant. (Fig. 4-3).
This is extremely important in the operation of a centrifugal pump.
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7. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
Differential Pressure
Molecular velocity energy is transferred directly to gauge.
Much higher pressure.
Review
In this lesson we studied the concepts involved with fluid mechanics, such as total pressure, velocity
pressure, and static pressure. Simply put HEAD is the vertical distance between two levels in a fluid. Total
pressure is the sum of the static and velocity pressures, and velocity pressure is the pressure due to
velocity and density of air, and static pressure is the total pressure minus velocity pressure, pushing
equally in all direction. As well as, velocity head, static head, pressure head. Velocity head is the energy of
motion, while static head is the actual head due to elevation, while pressure head is the ratio of pressure
over weight, if pressure is in pounds per square foot and weight is in cubic feet, and head is in feet. We
also discussed the calculation of velocity of a fluid through a pipeline, and conversions from feet of water
to head in psi.
Sources
Water Distribution Operator Training Handbook - AWWA
APPLIED Hydro-geology - Merrill Publishing Co., C.W. Fetter
Operator Certification Study Guide - AWWA
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8. Lesson 4: Bernoulli's Theorem http://water.me.vccs.edu/courses/CIV240/lesson4_print.htm
LEGO Design & PROGRAMMING Systems - "Ask an Engineer"
BSM Agri Ltd.
Assignment
Answer the following questions and either email or fax to the instructor.
The total dynamic head against which 1. a pump must operate is ______________.
A well is pumping into a reservoir at 300 gpm while the service pumps are pumping from the
reservoir at 400 gpm. Which direction is the reservoir (tank) level going:
2.
In a well the static water level is 15 feet. The pumping water level is 45 feet. The drawdown is how
much?
3.
A venturi tube increases the velocity and decreases the pressure as water flows through it. This type
of tube is used to ______________.
4.
5.
If one tank in a system is at 2100 ft. elevation, and another tank, same size, is at 2300 ft. elevation,
what is the static head?
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