Basic Hydraulic Principle

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Chapter Two Basic Hydraulics Principles

• 2.1 Geometry of Open channel
• Geometric Properties Necessary for Analysis
• For artificial channels these can usually be
  defined using simple algebraic equations given b
  y the depth of flow.
• The commonly needed geometric properties are
  shown in the figure below and defined as
• Depth (y) the vertical distance from the lowest
  point of the channel section to the free surface.
  
• Stage (z) the vertical distance from the free
  surface to an arbitrary datum

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Geometric Properties Necessary
for Analysis

• Area (A) the cross-sectional area of flow,
  normal to the direction of flow
• Wetted perimeter (P) the length of the wetted
  surface measured normal to the direction of flow.
  
• Surface width (B) width of the channel section
  at the free surface

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Geometric Properties Necessary
for Analysis

• Hydraulic radius (R) the ratio of area to
  wetted perimeter (A/P)
• Hydraulic mean depth (D) the ratio of area to
  surface width (A/B)

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Elements of hydraulic channel section geometry
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Example1

• Given rectangular channel calculate Hydraulic
  radius and hydraulic mean depth of the following
  figure.

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solution
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QUESTIONSOpen channel flow

• 1.0 Compute Hydraulic Radius and hydraulic mean
  depth for trapezoidal channel.

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2.2Fundamental
Equations

• The equations which describe the flow of fluid
  are derived from three fundamental laws of
  physics
• Conservation of matter
• Conservation of energy
• Conservation of momentum

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2.2Fundamental
Equations

• Although first developed for solid bodies they
  are equally applicable to fluids.
• A brief descriptions of the concepts are given
  below.
• Conservation of matter -
• This says that matter cannot be created nor
  destroyed, but it may be converted (e.g. by a
  chemical process.)
• In fluid mechanics we do not consider chemical
  activity so the law reduces to one of
  conservation of mass.

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2.2Fundamental
Equations

• Conservation of energy
• This says that energy cannot be created nor
  destroyed, but may be converted form one type to
  another (e.g. potential may be converted to
  kinetic energy).
• When engineers talk about energy "losses" they
  are referring to energy converted from mechanical
  (potential or kinetic) to some other form such as
  heat.
• A friction loss, for example, is a conversion of
  mechanical energy to heat.
• The basic equations can be obtained from the
  First Law of Thermodynamics but a simplified
  derivation will be given below.

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2.2Fundamental Equations

• Conservation of momentum-
• The law of conservation of momentum says that a
  moving body cannot gain or lose momentum unless
  acted upon by an external force.
• This is a statement of Newton's Second Law of
  Motion
• Force rate of change of momentum

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2.2Fundamental Equations

• In solid mechanics these laws may be applied to
  an object which is has a fixed shape and is
  clearly defined.
• In fluid mechanics the object is not clearly
  defined and as it may change shape constantly.
• To get over this we use the idea of control
  volume.
• These are imaginary volumes of fluid within the
  body of the fluid.
• To derive the basic equation the above
  conservation laws are applied by considering the
  forces applied to the edges of a control volume
  within the fluid.

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The
Continuity Equation (conservation of mass)

• For any control volume during the small time
  interval dt the principle of conservation of mass
  implies that the mass of flow entering the
  control volume minus the mass of flow leaving the
  control volume equals the change of mass within
  the control volume.
• If the flow is steady and the fluid
  incompressible the mass entering is equal to the
  mass leaving, so there is no change of mass
  Within the control volume.

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The Continuity
Equation (conservation of mass)
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The Continuity
Equation (conservation of mass)

• Considering control volume above which is a short
  length of open channel of arbitrary cross section
  then,if ? is the fluid density and Q is the
  volume of flow rate then mass flow rate is ?Q
  and the continuity equation for steady
  incompressible flow can be written
• ?Qentering ?Qleaving

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The Continuity Equation
(conservation of mass)

• As Q, the volume of flow rate is the product of
  the area and the mean velocity then at the
  upstream face (face1) where the mean velocity u1
  and the cross sectional area is A1 then
• Qentering U1A1
• Similarly at the down stream face ,face2,where
  mean velocity is U2 and cross-sectional area is
  A2 then

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The Continuity Equation
(conservation of mass)

• Qleaving U2A2
• Therefore, the continuity equation can be written
  as
• U1A1 U2A2

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E nergy
Principle in channel

• Consider forms of energy available for the above
  control volume.
• If the fluid moves from the upstream face 1,to
  to the down stream face 2 in time ?t over the
  length L.
• The work done in moving the fluid through face 1
  during this time is

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E nergy
Principle in channel

• Work done ?1A1L
• Where p1 is pressure at face 1
• The mass entering through face 1 is
• Mass entering p1a1l
• Therefore the kinetic energy of the system is

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E nergy
Principle in channel

• KE 1/2MU2 1/2P1A1LU12
• If z1 is the height of centroid of face 1, then
  the potentail energy of the fluid entering the
  control volume is
• PE mgz P1A1Lgz1
• The total energy entering the control volume is
  the sum of the work done ,the potentail and the
  kinetic energy
• Total energy P1A1L 1/2P1A1LU12 P1A1Lgz1

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E nergy
Principle in channel

• At the exit to control volume ,face 2 similar
  consideration
• The total energy per unit weight
• P2/?2g u22/2g z
• If no energy is applied to control volume between
  the inlet and the outlet then the energy entering
  energy leaving if fluid is incompressible ?1
  ?2 ?.
• so

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E nergy
Principle in channel

This is the Bernoulli equation
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Energy-Depth Relationships

• Energy in open channel Flow
• It is known in elementary hydraulics that the
  total energy in foot-pound per pound of water in
  any streamline passing through ,a channel section
  may be expressed as the total head in feet of
  water, which is equal to the sum of the elevation
  above a datum, the pressure head, and the
  velocity head.

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Energy-Depth Relationships
Fig. Energy in gradually varied 'open-channel
flow.
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Energy-Depth Relationships

• For example, with respect to the datum plane, the
  total head H at a section 0 containing point A on
  a streamline of flow in a channel of large slope
  (Fig. ) may be written as
• H ZA dA cos VA2/2g
• where ZA is the elevation of point A above the
  datum plane, dA is the depth of point A below the
  water surface measured along the channel section,
  ? is the slope angle of the channel bottom, and
  VA2/2g is the velocity head of the flow in the
  streamline passing through A.

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Energy-Depth Relationships

• In general, every streamline passing through a
  channel section will have different velocity
  head, owing to the non uniform velocity
  distribution in actual flow.
• Only in un ideal parallel flow of uniform
  velocity distribution can only the velocity head
  be truly Identical for all points on the cross
  section.
• In the case of gradually varied flow,
  however, it may be assumed, for practical
  purposes, that the velocity heads for all points
  on the channel section are equal , and the energy
  coefficient may be used to correct for the
  over-all effect of the non uniform velocity
  distribution.

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Energy-Depth Relationships

• Thus, the total energy at the channel section is
• H Z d cos V2/2g
• For channels of small slope, O.
• Thus, the tots'! energy at the channel section is
• H Z d V2/2g
• Consider prismatic channel of large slope (Fig.
  3-1).
• The line Representing the elevation of the total
  head of flow is the energy line.
• The slope of the line is known as the energy
  gradient, denoted by Sf.
• The slope of the water surface is denoted by Sw
  and the slope of the channel bottom by So Sin.
• In uniform flow, Sf Sw So Sin?.

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Energy-Depth Relationships

• According to the principle of conservation of
  energy, the total energy head at the upstream
  section 1 should be equal to the total energy
  head at the downstream section 2 plus the loss of
  energy hf between the two sections or
• Z1d1cos ? ?1 V12/2g Z2 d2COS? ?2V22/2g hf
• This equation applies to parallel or gradually
  varied flow.
• For a channel of small slope, it becomes Z1y1
  ?1 V12/2g Z2 y2 ?2V22/2g hf

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Energy-Depth Relationships

• Either of these two equations is known as the
  energy equation.
• When ?1 ?2 1 and hf 0, Eq. above becomes
• Z1y1 V12/2g Z2 y2 V22/2g constant
• This is the well-known Bernoulli energy equation.

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Specific Energy and Critical Depth

• Specific energy in a channel section is defined
  as the energy per weight of water at any section
  of a channel measured with respect to the channel
  bottom.
• Thus, according to Energy equation above with Z
  0, the specific energy becomes
• E d cos ? ? V2/2g
• or, for a channel of small slope and? 1,
• E y V2/2g
• which indicates that the specific energy is equal
  to the sum of the depth of water and the velocity
  head.

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Specific Energy and Critical Depth

• For a channel of small slope, since V Q/A the
  specific energy can be written as
• E y Q2/2A2g
• where, A By .
• It can be seen that, for a given channel section
  and discharge Q, the specific energy in a channel
  Section is a function of the depth of flow only.
• E y Q2/2B2Y2g
• When the depth of flow is plotted against the
  specific energy for a given
• channel section and discharge, a specific-energy
  Curve of Fig. below is obtained.

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Specific Energy and Critical Depth
Fig. Specific-energy curve.
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Specific Energy and Critical Depth

• This curve has two limbs AC and BC.
• The limb AC approaches the horizontal axis
  asymptotically toward the right.
• The limb BC approaches the line OD as it extends
  Upward and to the right.
• Line OD is a line that passes through the origin
  and has an angle of inclination equal to 450.
• For a channel of large slope, the angle of
  inclination of the line OD will be different from
  45. (Why?)
• At any point P on this curve, the ordinate
  represents the depth, and the abscissa
  represents the specific energy, which is equal
  to the sum of the pressure head y and the
  'velocity head V2/2g.

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Specific Energy and Critical Depth

• The curve shows that, for a given specific
  energy, there are tWo possible depths,. for
  instance, the low stage YI and the high stage
  y2.
• The low stage is called the alternate depth of
  the high stage, and vice versa. At point C the
  specific energy is a Minimum.
• It will be proved Iater that this condition of
  minimum specific energy corresponds to the
  critical state of flow .
• Thus, at the critical state the two alternate
  depths apparently become one, which is known as
  the critical depth Yc.
• When the depth of flow is greater than the
  critical depth, the velocity of flow is less than
  the critical velocity for the given discharge
  and, hence, the flow is subcritical.

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Specific Energy and Critical Depth

• When the depth of flow is less than the critical
  depth, the flow is supercritical. Hence, YI is
  the depth of a supercritical flow, and Y2 is the
  depth of a subcritical flow.
• If the discharge changes, the specific energy
  will be changed accordingly.
• The two curves A' B' and A" B" (Fig. 3-2)
  represent positions of, the specific-energy curve
  when the discharge is less 'and greater,
  respectively, than the discharge used for the
  construction of the curve AB.

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Critical state of flow

• The critical state of flow ' has been defined
  (Art. 1-3) as the condition for which the Froude
  number is equal to unity.
• A more common definition is that it is the state
  of flow at which the specific energy is a minimum
  for a given discharge.
• theoretical criterion for critical flow may be
  developed from this definition as follows
• Since V Q/ A, the equation for specific
  energy in a channel of small slope with ? 1,
  may be written

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Critical state of flow

• Differentiating with respect to y and noting
  that Q is a constant,

• The differential water area dA near the free
  surface (Fig. 3-2) is equal to T dy. Now dA/dy
  T, and the hydraulic depth D A/T so the above
  equation becomes

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Critical state of flow

• At the critical state of flow the specific energy
  is a minimum, or dE/dy 0. The above equation,
  therefore, gives

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Critical state of flow

• If the above criterion is to be used in any
  problem, the following conditions must be
  satisfied
• (1) flow parallel or gradually varied,
• (2) channel 0f small slope, and
• (3) energy coefficient assumed to be unity,

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Critical state of flow

• If the energy coefficient is not assumed to be
  unity, the critical-flow criterion is

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Critical state of flow

• For a channel of large slope angle and energy
  coefficient , the criterion for critical flow can
  easily be proved to be

where D is the hydraulic depth of the water area
normal to the channel bottom. In this case, the
Froude number may be defined as
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Critical state of flow

• It should be noted that the coefficient of a
  channel section actually varies with depth. In
  the above derivation, however, the coefficient is
  assumed to be constant therefore, the resulting
  equation is absolutely exact.

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Minimum Specific energy for different channel
section
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Minimum Specific energy for different channel
section
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Minimum Specific energy for different channel
section

• Generally, for non rectangular section for
  critical condition.
• Differentiating with respect to y and noting that
  Q is constant.

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Minimum Specific energy for different channel
section

• The differential water area dA near the free
  surface (Fig. 3-2) is equal to T dy.
• Now dA/dy T, and for minimum specific energy
  dE/dy 0 the the general equation

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Minimum Specific energy for different channel
section

• is applicable for critical condition.
• Exercise Show that for triangular channel
  section the critical depth may be represented by
• Where m is the side slope (m1) and specific
  energy at critical depth represented by E1.25YC

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Minimum Specific energy for different channel
section

• Example Calculate the critical depth and
  corresponding specific energy for a discharge of
  5.0m3/sec in the following channels
• a) Rectangular channel B2.0m
• b) Triangular channel m0.5
• c) Circular channel D2.0m and ?60

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Momentum Principle (Moment
Equation)

• Again consider the control volume above during
  the time

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Momentum Principle (Moment Equation)
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ENERGY AND MOMENTUM COEFFICIENT
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ENERGY AND MOMENTUM COEFFICIENT
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ENERGY AND MOMENTUM COEFFICIENT
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THE END!