Transcript Lecture 01 units w

Hydrology Basics
• We need to review fundamental
information about physical properties
and their units.
http://www.engineeringtoolbox.com/average-velocity-d_1392.html
Scalars and Vectors
• A scalar is a quantity with a size, for
example mass or length
• A vector has a size (magnitude) and a
direction.
http://www.engineeringtoolbox.com/average-velocity-d_1392.html
Velocity
• Velocity is the rate and direction of
change in position of an object.
• For example, at the beginning of the
Winter Break, our car had an average
speed of 61.39 miles per hour, and a
direction, South. The combination of these
two properties, speed and direction, forms
the vector quantity Velocity
Vector Components
• Vectors can be broken down into
components
• For example in two dimensions, we can
define two mutually perpendicular axes in
convenient directions, and then calculate
the magnitude in each direction
• Vectors can be added
• The brown vector plus
the blue vector equals
the green vector
Vectors 2: Acceleration.
• Acceleration is the change in Velocity
during some small time interval. Notice
that either speed or direction, or both, may
change.
• For example, falling objects are
accelerated by gravitational attraction, g.
In English units, the speed of falling
objects increases by about
g = 32.2 feet/second every second, written
g = 32.2 ft/sec2
SI Units: Kilogram, meter, second
• Most scientists and engineers try to avoid
English units, preferring instead SI units. For
example, in SI units, the speed of falling
objects increases by about 9.81
meters/second every second, written
g = 9.81 m/sec2
• Unfortunately, in Hydrology our clients are
mostly civilians, who expect answers in
English units. We must learn to use both.
Système international d'unités
pron dooneetay
http://en.wikipedia.org/wiki/International_System_of_Units
What’s in it for me?
• Hydrologists will take 1/5th of Geol. jobs.
• Petroleum Geologists make more money,
127K vs. 80K, but have much less job
security during economic downturns.
• Hydrologists have much greater
responsibility.
• When a petroleum geologist makes a
mistake, the bottom line suffers. When a
hydrologist makes a mistake, people
suffer.
http://www.bls.gov/oco/ocos312.htm
http://www.issaquahpress.com/tag/howard-hanson-dam/
Issaquah Creek Flood, WA
What does a Hydrologist do?
• Hydrologists provide numbers to
engineers and civil authorities. Clients
ask, for example:
• “When will the crest of the flood arrive, and
Trenton, Bound Brook, Rahway, Pompton,
how high will it be?” Wayne, Paterson after Hurricane Irene
• “When will the contaminant plume arrive at
our municipal water supply?
Dupont and Pompton Lakes, Syncon Resins and Passaic River
http://www.weitzlux.com/dupont-plume_1961330.html
Data and Conversion Factors
• In your work as a hydrologist, you will be
scrounging for data from many sources. It
won’t always be in the units you want. We
convert from one unit to another by using
conversion factors.
http://waterdata.usgs.gov/nj/nwis/current/?type=flow
http://climate.rutgers.edu/njwxnet/dataviewernetpt.php?yr=2010&mo=12&dy=1&qc=&hr=10&element_id%5B%5D=24&states=NJ&newdc=1
• Conversion Factors involve multiplication by
one, nothing changes
• 1 foot = 12 inches so 1 foot = 1
12 “
Example
• Water is flowing at a velocity of 30 meters per
second from a spillway outlet. What is this speed
in feet per second?
• Steps: (1) write down the value you have, then
(2) select a conversion factor and write it as a
fraction so the unit you want to get rid of is on
the opposite side, and cancel. Then calculate.
• (1)
(2)
• 30 meters x 3.281 feet
= 98.61 feet
second
meter
second
Flow Rate Q = V . A
• The product of velocity and area is a flow rate
•
V [meters/sec] x A [meters2] = Flow Rate [m3/sec]
• Notice that flow rates have units of Volume/ second
• It is very important that you learn to recognize which
units are correct for each measurement or property.
Example Problem
• Water is flowing at a velocity of 30 meters
per second from a spillway outlet that has
a diameter of 10 meters. What is the flow
rate?
Chaining Conversion Factors
• Water is flowing at a rate of 3000 meters cubed per
second from a spillway outlet. What is this flow rate in
feet3 per hour?
• Let’s do this in two steps
• 3000 m3 x 60 sec x 60 min
sec
min
hour
10800000 m3
hour
x
(3.281 feet)3
( 1 meter) 3
3/hour
10800000
m
=
= 381454240. ft3/hr
Momentum (plural: momenta)
• Momentum (p) is the product of velocity
and mass, p = mv
• In a collision between two particles, for
example, if there is no friction the total
momentum is conserved.
• Ex: two particles collide and m1 = m2, one
with initial speed v1 ,
the other at rest v2 = 0,
• m1v1 + m2v2 = constant
Force
• Force is the change in momentum with
respect to time.
• A normal speeds, Force is the product of
Mass (kilograms) and Acceleration
(meters/sec2),
• So Force must have SI units of kg . m
sec2
• 1 kg . m
sec2
is called a Newton (N)
Statics and Dynamics
• If all forces and Torques are balanced, an
object doesn’t move, and is said to be
static
• Discussion Torques, See-saw
F=2
• Reference frames
The forces are balanced in the
y direction. 2 + 1 force units
(say, pounds) down are
balanced by three pounds
directed up.
The torques are also balanced
around the pivot: 1 pounds is
2 feet to the right of the pivot
(= 2 foot-pounds)
and 2 pounds one foot to the
left = -2 foot - pounds
F=1
-1
0
+2
F=3
• Discussion Dynamics
Dynamics is the study of moving objects. Fluid Dynamics is the study of fluid flow.
Pressure
• Pressure is Force per unit Area
• So Pressure must have units of kg . m
sec2 m2
• 1 kg . m is called a Pascal (Pa)
sec2 m2
Density
• Density is the mass contained in a unit
volume
• Thus density must have SI units kg/m3
• The symbol for density is r, pronounced
“rho”
• Very important r is not a p, it is an r
• It is NOT the same as pressure
Chaining Conversion Factors
Suppose you need the density of water in
kg/m3. You may recall that 1 cubic centimeter
(cm3) of water has a mass of 1 gram.
1 gram water x (100 cm)3 x 1 kilogram = 1000 kg / m3
(1 centimeter)3
(1 meter)3
1000 grams
r water = 1000 kg / m3
Don’t forget to cube the 100
Mass Flow Rate
• Mass Flow Rate is the product of the
Density and the Flow Rate
• i.e. Mass Flow Rate = rAV
elocity
• Thus the units are kg m2 m
m3 sec
= kg/sec
Conservation of Mass – No Storage
Conservation of Mass : In a confined system “running full” and
filled with an incompressible fluid, the same amount of mass that
enters the system must also exit the system at the same time.
r1A1V1(mass inflow rate) = r2A2V2( mass outflow rate)
A pipe full of water
What goes in, must come out.
Notice all of the conditions/assumptions confined (pipe), running full (no compressible
air), horizontal (no Pressure differences) incompressible fluid.
Conservation of Mass for a horizontal Nozzle
Consider liquid water flowing in a horizontal pipe where the
cross-sectional area changes.
r1A1V1(mass inflow rate) = r2A2V2( mass outflow rate)
Liquid water is incompressible,
so the density does not change
and r1= r2. The density cancels
out, r1A1V1 = r2A2V2
so A1V1 =A2V2
V1 ->
A2 V2 ->
A1
Notice If A2 < A1 then V2 > V1
In a nozzle, A2 < A1 .Thus, water
exiting a nozzle has a higher
velocity than at inflow
Q2 = A2V2
Q =A V
A1V1 = A2V2
The water exiting is called a JET
Spillway Outlet. Here is Hoover Dam, a hydroelectric plant
that provides tremendous amounts of electricity to the west.
Notice the jets of water at the outlets. These are produced
by horizontal nozzles. The water must be going fast enough
to reach the center of the river where it strikes an opposing
jet. The opposing momenta nearly cancel, slowing both
flows. This is easier on the life in the river.
Example Problem
Water enters the inflow of a horizontal nozzle at a velocity of
V1 = 10 m/sec, through an area of A1 = 100 m2
The exit area is A2 = 10 m2. Calculate the exit velocity V2.
V1 ->
A2 V2 ->
A1
Solve the equation for V2, plug in the
numbers and state the answer and units.
V2 = A1/A2 x V1 = 100/10 x 10m/sec
= 100m/sec
Q2 = A2V2
Q1 = A1V1
A1V1 = A2V2
The Equation
Energy
• Energy is the ability to do work, and work and
energy have the same units
• Work is the product of Force times distance,
• W = Fd
Distance has SI units of meters
• 1 kg . m2 is called a N.m or Joule (J)
sec2
•
•
Energy in an isolated system is conserved
KE + PE + Pv + Heat = constant
N.m is pronounced Newton meter, Joule sounds like Jewel.
KE is Kinetic Energy, PE is Potential Energy, Pv is Pressure Energy,
v is unit volume
An isolated system, as contrasted with an open system, is a physical
system that does not interact with its surroundings.
Pressure Energy is
Pressure x volume
• Energy has
units kg . m2
sec2
So pressure energy must have the same units,
and Pressure alone is
kg . m
sec2 m2
So if we multiply Pressure by a unit volume m3 we
get units of energy
Kinetic Energy
• Kinetic Energy (KE) is the energy of
motion
• KE = 1/2 mass . Velocity 2 = 1/2 mV2
• SI units for KE are 1/2 . kg . m . m
•
sec2
Note the use of m both for meters and for mass. The context will tell you which.
That’s the reason we study units.
Note that the first two units make a Newton (force) and the remaining unit is meters,
so the units of KE are indeed Energy
Potential Energy
• Potential energy (PE) is the energy
possible if an object is released within an
acceleration field, for example above a
solid surface in a gravitational field.
• The PE of an object at height h is
PE = mgh Units are kg . m . m
sec2
Note that the first two units make a Newton (force) and the
remaining unit is meters, so the units of PE are indeed Energy
Note also, these are the same units as for KE
KE and PE exchange
• An object falling under gravity loses
Potential Energy and gains Kinetic Energy.
• A pendulum in a vacuum has potential
energy PE = mgh at the highest points,
and no kinetic energy because it stops
• A pendulum in a vacuum has kinetic
energy KE = 1/2 mass.V2 at the lowest
point h = 0, and no potential energy.
• The two energy extremes are equal
Stops v=0 at high point, fastest but h = 0 at low point.
Without friction, the kinetic energy at the lowest spot (1) equals
the potential energy at the highest spot, and the pendulum will
run forever.
Conservation of Energy
• We said earlier “Energy is Conserved”
• This means
KE + PE + Pv + Heat = constant
• For simple systems involving liquid water without
friction heat, at two places 1 and 2
1/2 mV12 + mgh1 + P1v = 1/2 mV22 + mgh2 + P2v
If both places are at the same pressure (say both
touch the atmosphere) the pressure terms are
identical
Example Problem
• A tank has an opening h = 1 m
below the water level. The opening
has area A2 = 0.003 m2 , small
compared to the tank with area A1 =
3 m2. Therefore assume V1 ~ 0.
1/2mV12 + mgh1 = 1/2mV22 + mgh2
• Calculate V2.
Method: only PE at 1, KE at 2
mgh1=1/2mV22 V2 = 2gh