Flow computation Formula

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Chapter Three Flow computation
Formula

• 3.1 Specific Energy and Critical depth
• Specific Energy
• Consider the following figure

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Chapter Three Flow computation
Formula
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Chapter Three Flow computation
Formula
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Chapter Three Flow computation
Formula
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Chapter Three Flow computation
Formula
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3.2 Critical Depth

• A minimum specific energy occurs at E Ec.
• The flow at this condition is known as the
  critical state of flow and the depth
  corresponding to this is known as the critical
  depth.

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3.2 Critical Depth
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3.2 Critical Depth
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3.3 Critical flow

• The characteristics of critical flow are
• The specific energy and specific force are
  minimum for the given discharge.
• The Froude number is equal to unity.
• For a given specific energy the discharge is
  maximum at the critical flow.
• The velocity head is equal to half the hydraulic
  depth in a channel of small slope.
• The velocity of flow in a channel of small slope
  with uniform velocity distribution, is equal to
  the celerity of small gravity waves ( ) C is
  shallow water caused by local disturbance.
• Flow at the critical state is unstable.
•  

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3.3 Critical flow

• Critical flow may occur at a particular section
  or in the entire channel, then the flow in the
  channel is called "Critical flow".
• Yc f(A,D )for a given discharge.
• For a prismatic channel for a given discharge the
  critical depth is constant at all sections of a
  channel.
• The bed slope which sustains a given discharge at
  a uniform and critical depth is called "Critical
  slope Sc".
• A channel slope causing slower flow in sub
  critical state for a given discharge is called
  "sub critical slope or mild slope". A slope
  greater than the critical slope is called steep
  slope or super critical slope.

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3.3 Critical flow

• For a given specific energy and discharge per
  unit width q, there are two possible (real)
  depths of flow, and that transition from one
  depth to the other can be accomplished under
  certain situations.
• These two depths represented on the two different
  limbs of the E-y curve separated by the crest c,
  are characteristic of two different kinds of
  flow a rational way to understand the nature of
  the difference between them is to consider first
  the flow represented by the point c.
• Here the flow is in a critical condition, poised
  between two alternative flow regimes, and indeed
  the word critical " is used to describe this
  state of flow it may be defined as the state at
  which the specific energy E is a minimum for a
  given q.

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3.3 Critical flow

• Analytical Properties of Critical Flow
• Consider the Specific energy equation

in which y is the depth of flow andq is the
discharge per unit width.
Differentiating the above equation with respect
to y and equating to zero t can be written as
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Analytical Properties of Critical Flow
The subscript c indicates critical flow
conditions. Thus the critical depth yc is a
function of discharge per unit width alone.
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Analytical Properties of Critical Flow

• Further, the above equation it can be written as

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Analytical Properties of Critical Flow

• The second derivative should be negative i.e

• The above equations are established by
  considering the variation of specific energy with
  y for a given q.
• Clearly the curve will be of the general form as
  shown in Figure.

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Analytical Properties of Critical Flow
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Analytical Properties of Critical Flow

• Variation of the Discharge with depth for a given
  specific energy value
• How q varies with y for a given E Eo?
• When y?E0and then q? 0. Similarly when y ?0,
  q?0 and there will clearly be a maximum value of
  q for some value of y between 0 and E0(y cannot
  be greater than E0).
• The relationship can be written as ( and
  differentiating the above equation with respect
  to y,

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Analytical Properties of Critical Flow
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Analytical Properties of Critical Flow
Alternative approach
Show that the flow is maximum when it is critical
flow for a given specific energy plot the graph "
E0 verses q
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Alternative approach
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Alternative approach
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Alternative approach

• Which is essentially equation representing the
  critical flow.
• Thus critical flow cannotes not only minimum
  specific energy for a given discharge per unit
  width, but also maximum discharge per unit width
  for given specific energy.
• Any one of the above three equations may be used
  to define critical flow

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Alternative approach
Any one of the above three equations may be used
to define critical flow
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Alternative approach
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Quiz(10)

• Show that for a trapezoidal channel the mininimum
  specific energy EC is related to critical depth
  Yc as

Where,
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1 The section factor for critical
flow computation

• The section factor for critical flow computation
  (Z) is the product of the water area and the
  square root of the hydraulic depth.

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1 The section factor for critical
flow computation

• For critical flow ,
• by substituting

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1 The section factor for critical
flow computation
Where Qc represent the discharge that would make
the depth y critical and know as the critical
discharge. When the energy coefficient is not
assumed to be unity
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1 The section factor for critical
flow computation

• Eq. (3.1 and 3.2) are very useful tool for the
  computation and analysis of critical flow in open
  channel, when the discharge is given the
  equations will give the critical section factor
  (Zc), and hence the critical Depth yc.
• Section factor (Z) for different channel section
  shape is given as

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1 The section factor for critical
flow computation

• To simplify the computation of critical flow,
  dimensionless curves showing the relation between
  the depth and the section factor Z have been
  prepared for different type of channels sections.
  
• These self-explanatory curves will help to
  determine the depth y for a given section factor
  Z and vice versa.

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1 computationComputation of
critical flow

• Computation of critical flow involves the
  determination of critical depth and velocity when
  the discharge and channel section are known.
• The methods illustrated by examples are given
  below. On the other hand ,if critical depth and
  channel section are known, the critical discharge
  can be determined from the relation

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1 computationComputation of
critical flow

• Algebraic Method.

For geometrically simple channel sections, the
uniform-flow condition may be determined by an
algebraic solution, as illustrated by the
following example
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1 computationComputation of
critical flow

• Example For a trapezoidal channel with base
  width b6.0m, side slope x 2(i.e. 1vertical2
  horizontal) and Mannings n 0.02, calculate the
  critical velocity, criticaldepth and critical
  slope if its discharge Q17m3/s.
• Solution

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1 computationComputation of
critical flow
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1 computationComputation of
critical flow

• By trial and error for value of YC
• YC 0.87m

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1 computationComputation of
critical flow
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QUESTIONS
CRITICAL FLOW
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QUESTIONS
CRITICAL FLOW
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B.Graphical method

• For channels of complicated cross section and
  variable flow conditions, a graphical solution of
  the problem is found to be convenient.
• Method of design chart
• The design chart for determining the critical
  depth can be used with great expediency.

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The Section Factor for Uniform-flow Computation

• '11h expression AR2/3 is called the section
  factor for uniform-flow computation It IS an
  important element in the computation of uniform
  flow.
• this factor may be expressed as

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The Section Factor for Uniform-flow Computation

• This equation applies to a channel section when
  the flow is uniform.
• The right side of the equation contains the
  values of n, Q, and S but the left side depends
  only on the geometry of the water area.
• Therefore, it shows that, for a given condition
  of n,Q, and S, there is only one possible depth
  for maintaining a uniform flow, provided that the
  value of AR2/3 is always increases with increase
  in depth, which is true in most cases

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The Section Factor for Uniform-flow Computation

• In order to simplify the computation,
  dimensionless curves showing the relation between
  depth and section fuctor AR2/3 (Fig. 6-1) have
  been prepared for rectangular,. trapezoidal,
  and cirpular channel' sections.
• These self-explanatory curves will help to
  determine, the depth for a given section factor
  AR2/3, and vice versa.

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The Section Factor for Uniform-flow Computation
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Control section

• Control of flow in open channel or at structure
  means the establishment of explicit relationship
  (one to one relationship) between the stage
  (water level) and the discharge of flow.
• When the control of channel is achieved at
  certain part of channel or structure, this
  section is called control section.

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Control section

• Holding fixed stage-discharge relationship,
  control section is always suitable for gauging
  station since it is always suitable site for
  developing the discharge rating curve, a curve
  representing depth-discharge relationship.
• At critical state of flow a definite stage
  discharge relationship can be established and
  represented by the equation

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Control section

• This section equation is theoretically
  independent of the channel roughness and other
  uncontrolled circumstances.
• Therefore, critical flow section is a control
  section.
• For Further knowledge read the open channel
  hydraulics books.

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Flow measurement

• It was mentioned in the preceding article that,
  at critical control section, the relationship
  between the depth and discharge is explicit
  independent of the channel roughness and other
  uncontrolled circumstances such explicit
  relationship between stage-discharge
  relationship offers a theoretical basis for the
  measurement of discharge in open channels.

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Flow measurement

• Based on the principle of critical flow, various
  devices for flow measurement have been developed.
  
• In such devices the critical depth is created by
  developing low hump on the channel bottom such as
  a weir or by contraction in the cross section,
  such as the critical flow flume(venture flume
  or par shall flume) in the transition part.
• The use of weir is a simple method, but it causes
  high head loss.
• If water contains suspended particles, some will
  be deposited in the pool at upstream of the weir,
  resulting in gradual

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Flow measurement

• change in discharge coefficient. These
  difficulties, however, can be overcome by at
  least partially by the use of critical flow
  flume.

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Channel with a Hump

• a) Subcritical Flow
• Consider a horizontal, frictionless rectangular
  channel of width B carrying discharge Q at depth
  y1.
• Let the flow be subcritical. At a section 2
  (Fig.3. 1) a smooth hump of heights ?Z is built
  on the floor.
• Since there are no energy losses between
  sections 1 and 2, construction of a hump causes
  the specific energy at section to decrease by
  Z.
• Thus the specific energies at sections 1 and 2
  are,

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Channel with a Hump
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Channel with a Hump

• Since the flow is subcritical, the water surface
  will drop due to a decrease in the specific
  energy.
• In Fig. (3.2), the water surface which was at P
  at section 1 will come down to point R at section
  2. The depth y2 will be given by,

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Channel with a Hump
Figure 3.2. Specific energy diagram for Fig.
(3.1)
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Channel with a Hump

• It is easy to see from Fig. (3.2) that as the
  value of ? Z is increased, the depth at section
  2, or y2 , will decrease.
• The minimum depth is reached when the point R
  coincides with C, the critical depth.
• At this point the hump height will be maximum,
  ?Zmax , y2 yc critical depth, and E2 Ec
  minimum energy for the flowing discharge Q.
• The condition at ?Zmax is given by the rela tion

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Channel with a Hump
The question may arise as to what happens when
?Z gt ? Zmax. From Fig. (3.2) it is seen that
the flow is not possible with the given
conditions (given discharge). The upstream
depth has to increase to cause and increase in
the specific energy at section 1. If this
modified depth is represented by
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Channel with a Hump
At section 2 the flow will continue at the
minimum specific energy level, i.e. at the
critical condition. At this condition, y2 yc ,
and
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Channel with a Hump
while y2 will continue to remain yc). The
variation of y1 and y2 with ?Z in the
subcritical regime can be clearly seen in Fig.3.3
Recollecting the various sequences, when 0 lt ? Z
lt ?Zmax the upstream water level remains
stationary at y1 while the depth of flow at
section 2 decreases with ?Z reaching a minimum
value of yc at ?Z ?Zmax . (Fig.3.2). with
further increase in the value of ?Z, (i.e., for
?Z gt?Zmax, y1will change to y1
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b) Supercritical Flow

• If y1 is in the supercritical flow regime Fig
  (3.2) shows that the depth of flow increases due
  to the reduction of specific energy.
• In Fig (3.2) point P corresponds to y1 and point
  R to depth at the section 2.
• Up to the critical depth, y2 increases to reach
  yc at ?Z ? Zmax For ? Z gt ?Zmax, the depth
  over the hump y2 yc will remain constant and
  the max upstream depth y1 will change.
• It will decrease to have a higher specific energy
  E1 by increasing velocity V1.
• The variation of the depths y1 and y2 with ?Z
  in the supercritical flow is shown in Fig. (3.4).
  

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Example 3.3

• A rectangular channel has a width of 2.0 m and
  carries a discharge of 4.80m /sec with a depth of
  1.60 m. At a certain cross-section a small,
  smooth hump with a flat top and a height 0.10 m
  is proposed to be built.
• a). Calculate the likely change in the water
  surface. Neglect the energy loss.
• b). If the height of the hump is 0.50 m,
  estimate the water surface elevation on the hump
  and at a section upstream of the hump

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Solution
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Solution
At section 2
it show the upstream flow is subcritical and hump
will cause a drop in the water surface elevation.

Let the suffixes 1 and 2 refer to the upstream
and downstream sections respectively as shown in
the fig
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Solution
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Solution
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solution

• The drop at water surface elevation is

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solution
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solution

• The minimum specific energy required at section 2
  is greater than E2, (Ec21.26 gtE21.215), the
  available specific energy at that section .
• Hence , the depth at section 2 will be at the
  critical depth and E2Ec21.26m.
• The upstream depth y1 will increase to a depth y1
  such that the new specific energy at the upstream
  section 1 is

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Uniform flow

• Uniform flow in open channel has the following
  main features
• The depth, water area, velocity and discharge at
  every section of the channel are constant.
• The energy line, water surface and channel
  bottom are all parallel i.e their slopes are
  all equal

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Uniform flow

• Uniform flow is considered to be steady only,
  since unsteady uniform flow is practically
  nonexistent.
• In natural streams, even steady uniform flow is
  rare, for rivers and streams in natural state
  scarcely ever experience a strict uniform flow
  condition.
• Despite this deviation from the truth, the
  uniform flow condition is frequently assumed in
  the computation of flow in natural streams.
• The results obtained from this assumption are
  understood to be appropriate and general,but they
  offer a relatively simple and satisfactory
  solution to many practical problems.

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Establishment of uniform flow

• When flow occurs in open channel resistance is
  encountered by the water as it flows downstream.
• This resistance generally counteracted by the
  components of gravity forces acting on the body
  of lthe water in the direction of motion .
• A uniform flow will be developed if the the
  resistance is balanced by the gravity forces,
  example the head loss due to turbulent flow is
  exactly balanced by the reduction of in potential
  energy due to the uniform decrease in the
  elevation of the channel.
• The magnitude of the resistance, when other
  physical factors of the channel are kept
  unchanged, depends on the velocity of flow.

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Establishment of uniform flow

• If the water enters the channel slowly,the
  velocity and hence the resistance are small and
  the resistance is out balanced by gravity
  forces,resulting in an accelerating flow in the
  upsteam reach.
• The velocity and the resistance gradually
  increase until a balance between resistance and
  garavity fdorce is reached.
• At this moment and afterwards the flow become
  uniform.
• The upstream rreach isrequired for establishment
  of uniform flow is known as the the transitory
  zone.In this zone the flow is accelerating and
  varied.
• .

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Establishment of uniform flow

• If the channel is shorter than the transitory
  length required by the given conditions,
• uniform flow could not be attained.
• Towards the end of the channel the resistance may
  again exceeded by the gravity forces and the flow
  become varied.
• Ingeneral,uniform flow can acquire only in very
  long ,straight,prismatic channels where terminal
  velocity of can be achieved.For purpose of
  explanation a long channel is channel is shown
  with three different slopes (figures given during
  your lecture or refer ventechow)subcritical,criti
  cal and supercritical

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Establishment of uniform flow

• At the subcritical slope the water surface in the
  transitory zone appears adulatory.
• The flow is uniforn in the middle reach of
  channel but varied at the two ends.At the
  critidcal slope middle sketch).
• The water surface of critical flow is un stable.
  Possible undulation may occur in the middle reach
  but on the average the depth is constant and the
  flow may be considered uniform.

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Establishment of uniform flow

• At the super critical slope the transitory waater
  surface passess from the subcritical stage t o
  the super critical stage through gradual
  hydraulic drop.
• Beyond the transitory zone the flow is
  approaching uniformidy.
• The depth of uniform flow is called normal depth.
• In the figures the long dashed lines represents
  the normal depth line abbreviated as N.D.L and
  the short dashed lines represents the dcritical
  depth line or C.D.L.

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Expressing the Velocity of a Uniform Flow.

• For hydraulic computations the mean velocity of a
  turbulent uniform flow in open channels is
  usually expressed approximately by a so-called
  uniform-flow formula.
• Most practical uniform-flow formula can be
  expressed in the following general form

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Expressing the Velocity of a Uniform Flow.

• where
• V is t,he mean velocity in m/s R is the
  hydraulic radius m S is the energy slope, x and
  yare exponents and, C is a factor of flow
  resistance varying with the mean velocity,
  hydraulic channel roughness, viscosity, and many
  other factors.
•  

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READING ASSIGNMENT

• Refer reference books chow for other resistance
  formula.

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Computation of uniform flow
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Computation of uniform flow
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Computation of uniform flow
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Computation of uniform flow

• In computation of uniform flow there are two
  common problems to solve

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solution
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solution
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QUESTIONS
UNIFORM FLOW
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QUESTIONS
UNIFORM FLOW
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Best( economic) hydraulic section 
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Best( economic) hydraulic section 
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  Best( economic) hydraulic section
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  Best( economic) hydraulic section
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  Best( economic) hydraulic section
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  Best( economic) hydraulic section
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trapezoidal

• Example

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Exercise

• A trapezoidal, concrete lined channel (Mannings
  n0.015) is to be constructed to carry flood
  water. The slope of the channel bed slope is 1 in
  500.The design discharge is 10m3/s.
• calculate the proportion of the trapezoidal
  channel that will minimize excavation and result
  in optimum hydraulic section.
• If the cross sectional area kept the same as that
  of part (a) but for safety reasons depth of flow
  is limited to1m,what will be the discharge now
  take side slope1v2H.

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The end!

• Thank you!