Chapter 2

REVIEW OF RELATED STUDIES AND LITERATURE

Review of Related Literature:

* Development of Contour Lines

A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the

function has a constant value. It is a cross-section of the three-dimensional graph of the function f(x, y)

parallel to the x, y plane. In cartography, a contour line (often just called a "contour") joins points of

equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated

with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness

of slopes. The contour interval of a contour map is the difference in elevation between successive contour

lines.

More generally, a contour line for a function of two variables is a curve connecting points where the

function has the same particular value. The gradient of the function is always perpendicular to the contour

lines. When the lines are close together the magnitude of the gradient is large: the variation is steep.

A level set is a generalization of a contour line for functions of any number of variables.

Contour lines are curved, straight or a mixture of both lines on a map describing the intersection of a real

or hypothetical surface with one or more horizontal planes. The configuration of these contours allows

map readers to infer relative gradient of a parameter and estimate that parameter at specific places.

Contour lines may be either traced on a visible three-dimensional model of the surface, as when a

photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from estimated

surface elevations, as when a computer program threads contours through a network of observation

points of area centroids. In the latter case, the method of interpolation affects the reliability of individual

isolines and their portrayal of slope, pits and peaks.

Contour lines are often given specific names beginning "iso-" (Ancient Greek: ἴσος isos "equal") according

to the nature of the variable being mapped, although in many usages the phrase "contour line" is most

commonly used. Specific names are most common in meteorology, where multiple maps with different

variables may be viewed simultaneously. The prefix "iso-" can be replaced with "isallo-" to specify a

contour line connecting points where a variable changes at the same rate during a given time period.

The words isoline and isarithm (ἀριθμός arithmos "number") are general terms covering all types of

contour line. The word isogram (γράμμα gramma "writing or drawing") was proposed by Francis Galton in

1889 as a convenient generic designation for lines indicating equality of some physical condition or

quantity; but it commonly refers to a word without a repeated letter.

An isogon (from γωνία or gonia, meaning 'angle') is a contour line for a variable which measures

direction. In meteorology and in geomagnetics, the term isogon has specific meanings which are

described below. An isocline (from κλίνειν or klinein, meaning 'to lean or slope') is a line joining points with

equal slope. In population dynamics and in geomagnetics, the terms isocline and isoclinic line have

specific meanings which are described below.

The idea of lines that join points of equal value was rediscovered several times. In 1701, Edmond

Halley used such lines (isogons) on a chart of magnetic variation. The Dutch engineer Nicholas

Cruquius drew the bed of the river Merwede with lines of equal depth (isobaths) at intervals of 1 fathom in

1727, and Philippe Buache used them at 10-fathom intervals on a chart of the English Channel that was

prepared in 1737 and published in 1752. Such lines were used to describe a land surface (contour lines)

in a map of the Duchy of Modena and Reggio by Domenico Vandelli in 1746, and they were studied

theoretically by Ducarla in 1771, and Charles Hutton used them in the Schiehallion experiment. In 1791, a

map of France by J. L. Dupain-Triel used contour lines at 20-metre intervals, hachures, spot-heights and

a vertical section. In 1801, the chief of the Corps of Engineers, Haxo, used contour lines at the larger

scale of 1:500 on a plan of his projects for Rocca d'Aufo.

By around 1843, when the Ordnance Survey started to regularly record contour lines in Great

Britain and Ireland, they were already in general use in European countries. Isobaths were not routinely

used on nautical charts until those of Russia from 1834, and those of Britain from 1838.

When maps with contour lines became common, the idea spread to other applications. Perhaps the latest

to develop are air quality and noise pollution contour maps, which first appeared in the US, in

approximately 1970, largely as a result of national legislation requiring spatial delineation of these

parameters. In 2007, Pictometry International was the first to allow users to dynamically generate

elevation contour lines to be laid over oblique images.

*MANNING’S FORMULA

The Manning formula is also known as the Gauckler–Manning formula, or Gauckler–Manning–Strickler

formula in Europe. In the United States, in practice, it is very frequently called simply Manning's Equation.

The Manning formula is an empirical formulaestimating the average velocity of a liquid flowing in a conduit

that does not completely enclose the liquid, i.e., open channel flow. All flow in so-called open channels is

driven by gravity. It was first presented by the French engineer Philippe Gauckler in 1867, and later re-

developed by the Irish engineer Robert Manning in 1890

The Gauckler–Manning formula states:

where:

V is the cross-sectional average velocity (L/T; ft/s, m/s);

n is the Gauckler–Manning coefficient. Units for values of n are often left off, however it is not

dimensionless, having units of: (T/[L1/3]; s/[ft1/3]; s/[m1/3]).

Rh is the hydraulic radius (L; ft, m);

S is the slope of the hydraulic grade line or the linear hydraulic head loss (L/L), which is the same

as the channel bed slope when the water depth is constant. (S = hf/L).

k is a conversion factor between SI and English units. It can be left off, as long as you make sure

to note and correct the units in your "n" term. If you leave "n" in the traditional SI units, k is just

the dimensional analysis to convert to English. k=1 for SI units, and k=1.49 for English units.

(Note: (1 m)1/3/s = (3.2808399 ft) 1/3/s = 1.4859 ft1/3/s)

NOTE: Ks strickler = 1/n manning. The coefficient Ks strickler varies from 20 (rough stone and rough

surface) to 80 m1/3/s (smooth concrete and cast iron).

The discharge formula, Q = A V, can be used to manipulate Gauckler–Manning's equation by

substitution for V. Solving for Q then allows an estimate of the volumetric flow rate (discharge)

without knowing the limiting or actual flow velocity.

The Gauckler–Manning formula is used to estimate the average velocity of water flowing in an open

channel in locations where it is not practical to construct a weir or flume to measure flow with greater

accuracy. The friction coefficients across weirs and orifices are less subjective than n along a natural

(earthen, stone or vegetated) channel reach. Cross sectional area, as well as n', will likely vary along

a natural channel. Accordingly, more error is expected in estimating the average velocity by

assuming a Manning's n, than by direct sampling (i.e., with a current flowmeter), or measuring it

across weirs, flumes or orifices. Manning's equation is also commonly used as part of a

numerical step method, such as the Standard Step Method, for delineating the free surface profile of

water flowing in an open channel.

The formula can be obtained by use of dimensional analysis. Recently this formula was derived

theoretically using the phenomenological theory of turbulence.

*FLOW AND VELOCITY

In physics and engineering, in particular fluid dynamics and hydrometry, the volumetric flow rate, (also

known as volume flow rate, rate of fluid flow or volume velocity) is the volume of fluid which passes per

unit time; usually represented by the symbol Q. The SI unit is m3/s (cubic metres per second). Another

unit used is sccm (standard cubic centimeters per minute).

In US Customary Units and British Imperial Units, volumetric flow rate is often expressed as ft3/s (cubic

feet per second) or gallons per minute (either U.S. or imperial definitions).

Volumetric flow rate should not be confused with volumetric flux, as defined by Darcy's law and

represented by the symbol q, with units of m3/(m2·s), that is, m·s−1. The integration of a flux over an area

gives the volumetric flow rate.

Volumetric flow rate can also be defined by:

where:

 = flow velocity

 = cross-sectional vector area/surface

The above equation is only true for flat, plane cross-sections. In general, including curved surfaces,

the equation becomes a surface integral:

This is the definition used in practice. The area required to calculate the volumetric flow rate is

real or imaginary, flat or curved, either as a cross-sectional area or a surface. The vector area is

a combination of the magnitude of the area through which the volume passes through,  A, and

a unit vector normal to the area,  . The relation is  .

The reason for the dot product is as follows. The only volume flowing through the cross-section is

the amount normal to the area; i.e., parallel to the unit normal. This amount is:

where θ is the angle between the unit normal   and the velocity vector v of the substance

elements. The amount passing through the cross-section is reduced by the factor  .

As θ increases less volume passes through. Substance which passes tangential to the area,

that is perpendicular to the unit normal, does not pass through the area. This occurs

when θ = π⁄2 and so this amount of the volumetric flow rate is zero:

These results are equivalent to the dot product between velocity and the normal

direction to the area.

When the mass flow rate is known, and the density can be assumed constant, this is an

easy way to get  .

Where:

 = mass flow rate (kg/s).

 = density (kg/m3).

*PRISMOIDAL FORMULA

The prismoidal formula for approximating the value of a definite integral is given in following

theorem:

Theorem 1. If f(x) is a polynomial of degree 3 or less, then

In this equation f(a), of course, represents the value of the integrand when x = a, f(b) is its value

when x = b, and   is its value when the value of x is half-way between a and b.

Proof of Theorem

The geometrical interpretation is given by Fig. 3. Assuming that f(x) is a polynomial of degree 3

or less, then the area under the curve y = f(x) in the interval from x = a to x = b is given exactly

by the formula

If f(x) is not a polynomial in x or if it is not of degree 3 or less, then 4) gives an approximation to

the area by giving exactly the area under the parabola

 

            y = Ax2 + Bx + C 

that passes through the points P, Q and R (Fig. 3). It may be remarked that the area under any

cubic curve through P, Q, and R, in the interval from x = a to x = b, is equal to that under the

parabola.

*STADIA METHOD

The Stadia is a method of measuring distances rapidly with a telescope (usually on an

engineer's transit or an alidade) and a graduated rod. When the telescope is focused on

the rod, the distance s intercepted on the

vertically-held rod between two stadia

hairs seen in the eyepiece gives the

distance D as D = ks, where k, the stadia constant is often made to be 100. Therefore,

if 6 ft is intercepted on the rod, then the distance from the telescope to the rod is 600

ft. There are small corrections to this that will be mentioned below. If the line of sight

is inclined, the vertical angle is also measured and can be used to reduce the results to

horizontal and vertical distances. Stadia can give results correct to about 1 ft under the

best conditions, which is often sufficient, and can also serve as a check on more

precise measurements.

The term stadia comes from the plural of the Greek stadion, the word for a distance of

185 to 192 metres (607-630 ft). A very similar length is the modern furlong, or eighth

of a mile, 660 ft. A "stadion" was also an athletic venue, with lengths laid out for

competition and seats for spectators. The Latin stadium, stadia was a direct borrowing

with the same meaning.

Distance over the ground was traditionally measured by long poles or rods laid

successively end to end. The ancient Egyptians used rope for the same purpose. This

practice is reflected in the traditional rod, pole or perch of 16.5 feet. This odd length

came from dividing down an English mile of 5280 ft, first into furlongs of 1/8 mile or

660 ft, then into tenths, or chains, of 66 ft, and finally into quarters of this, or 16.5

feet. Four rods make a chain, ten chains a furlong, and 80 chains a mile. The Gunter's

chain of 100 iron links and length 66 ft was much easier to use and carry than an

ungainly pole, and gave more accurate results. 10 square chains is an acre, so Gunter's

chain was closely related to traditional measures of distance and land areas. The

engineer's chain of 100 links, each 1 ft long, replaced Gunter's chain, and was itself

replaced by the 100 ft steel tape, which is an excellent and easily handled way to

measure distances. Doing this is still called "chaining," however, and the people who

do it are called chainmen. Accurate chaining is subject to many errors, which are

largely systematic, but with care they can be overcome. These errors include thermal

expansion and elasticity of the tape, as well as ground irregularities.

Distances are now conveniently measured by timing modulated laser beams returned

by retroreflectors. Large distances can be covered at one leap, and the intervening

ground does not have to be traversed on foot. Stadia shares these advantages.

Microwaves were first used for this purpose, but have now been superseded by lasers.

The main errors are in estimating propagation conditions, temperature and humidity,

which affect the velocity of light, and are often poorly known or vary over the path.

Even without consideration of these uncertainties, laser ranging is more accurate than

stadia, but is also much more expensive. We also have Global Positioning System

location, which is accurate to roughly 1 metre (with special care, centimetre accuracy

is possible, but it requires work). In spite of these excellent alternatives, it is still

interesting to know the stadia method, which is often applicable in unusual

circumstances.

The stadia method is an application of paraxial optics. The telescope consists of an

objective (usually one achromatic lens, but sometimes more) that produces an image

of the distant scene close to its focal plane, which is then examined by the eyepiece.

We will be concerned only with the objective. The action of the telescope objective is

described by principal planes, nodal planes and focal lengths. Since the final and

initial media are the same, the nodal planes coincide with the principal planes, and the

primary and secondary focal lengths are equal. The telescope is mounted so that the

outer principal plane of the lens is a distance c from the axis of the instrument, that is

vertically over the occupied location. If the distance of the stadia rod from the

instrument axis is D, then the object distance is D - c. The corresponding image

distance d behind the other principal plane is then given by 1/d + 1/(D - c) = 1/f.

Fine lines are etched on a glass reticle placed approximately at the focal point of the

telescope objective. These were once crosshairs made of spider web, and are still

called crosshairs for that reason. There are vertical and horizontal crosshairs for

sighting purposes, and two shorter stadia hairs at equal distances above and below the

horizontal crosshair. The separation of the stadia hairs is denoted by i. The eyepiece is

adjusted so that the crosshairs are sharp, and the telescope then focused so that the

object viewed is also sharp, so that their images occur at the same point.

We now make use of the unit angular

magnification property of the nodal points to

establish that the angle s/(D - c) of the rod intercept as seen from the outer nodal point

is equal to the angle i/d at the inner nodal point, or 1/d = s/i(D - c). The relations are

illustrated in the diagram. When this is substituted in the lens equation, the result is f +

fs/i = D - c, or D = (f/i)s + (f + c), which is the fundamental stadia formula. The

derivation is confused in Breed and Hosmer; the principles are not clearly stated, and

reference is made to a different diagram than the one appearing on the page, possibly

one from an earlier edition. I hope that this derivation will make things clear, since

they really aren't very difficult. Now, f/i will be a constant determined by the

construction of the telescope and reticle, and is usually called k, the stadia constant of

the instrument. It is commonly 100, but a more accurate value can be established by

experiment if necessary. k = 100 corresponds to an angle of 0.01 radian, or 0.573°.

The correction (f + c) is to be added to ks to find D. If f = 200mm and c = 100 mm,

then (f + c) is 30 cm, or about 1 ft. This correction is sometimes ignored.

The formula just derived applies to a horizontal sight on a vertical rod, or to an

inclined sight on a rod held perpendicular to the direction of view. It is not easy to

hold a rod perpendicular to the line of sight, so it is held accurately vertical. If s is the

intercept on a vertical rod, then s cos α would be the intercept, approximately, if the

rod were held perpendicular to the line of sight. The slant distance is, then, D' = ks cos

α = (f + c). Now it is easy to find the horizontal distance D = D' cos α and the vertical

distance V = D' sin α. At one time, tables were prepared for performing these

calculations, but with pocket calculators they are no longer necessary. A pocket

calculator can reduce the data quickly and accurately, including the correction (f + c)

without any approximation.

*WATER BALANCE