Discharge or Flow Rate

Discharge (also called flow rate)
The amount of fluid passing a section of a stream in unit time is called the discharge. If v is the mean velocity and A is the cross sectional area, the discharge Q is defined by Q = Av which is known as volume flow rate. Discharge is also expressed as mass flow rate and weight flow rate.

Volume flow rate, $Q = Av$

Mass flow rate, $M = \rho Q$

Weight flow rate, $W = \gamma Q$

 

009-continuous-flow.gif

 

Rotation - Rotating Vessel

When at rest, the surface of mass of liquid is horizontal at PQ as shown in the figure. When this mass of liquid is rotated about a vertical axis at constant angular velocity ω radian per second, it will assume the surface ABC which is parabolic. Every particle is subjected to centripetal force or centrifugal force CF = mω2x which produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity force W = mg and normal force N.
 

008-rotating-vessel.gif

 

Stability of Floating Bodies

Any floating body is subjected by two opposing vertical forces. One is the body's weight W which is downward, and the other is the buoyant force BF which is upward. The weight is acting at the center of gravity G and the buoyant force is acting at the center of buoyancy BO. W and BF are always equal and if these forces are collinear, the body will be in upright position as shown below.
 

005-floating-body-upright-position.gif

 

Analysis of Gravity Dam

Dams are structures whose purpose is to raise the water level on the upstream side of river, stream, or other waterway. The rising water will cause hydrostatic force which will tend the dam to slide horizontally and overturn about its downstream edge or toe. The raised water level on the upstream edge or heel will also cause the water to seep under the dam. The pressure due to this seepage is commonly called hydrostatic uplift and will reduce the stability of the dam against sliding and against overturning.
 

003-cross-section-typical-gravity-dam.gif

 

Problem 01 - Buoyancy

Problem 01
A piece of wood 305 mm (1 ft) square and 3 m (10 ft) long, weighing 6288.46 N/m3 (40 lb/ft3), is submerged vertically in a body of water, its upper end being flush with the water surface. What vertical force is required to hold it in position?
 

02-001-wood-submerged-in-water.gif           02-001-wood-submerged-in-water-english-units.gif

 

Buoyancy

Archimedes Principle

004-archimedes.gif
Archimedes (287-212 B.C.)

Any body immersed in a fluid is subjected to an upward force called buoyant force equal to the weight of the displaced fluid.
 

$BF = \gamma V_D$

Where
$BF$ = buoyant force
$\gamma$ = unit weight of fluid
$V_D$ = volume of fluid displaced by the body
 

Circular Gate with Water on One Side and Air on the Other Side

Situation
The figure below shows a vertical circular gate in a 3-m diameter tunnel with water on one side and air on the other side.
 

002-submerged-circular-gate.jpg

 

  1. Find the horizontal reaction at the hinge.
    A.   412 kN
    B.   408 kN
    C.   410 kN
    D.   414 kN
  2. How far from the invert of the tunnel is the hydrostatic force acting on the gate?
    A.   1.45 m
    B.   1.43 m
    C.   1.47 m
    D.   1.41 m
  3. Where will the hinge support be located (measured from the invert) to hold the gate in position?
    A.   1.42 m
    B.   1.46 m
    C.   1.44 m
    D.   1.40 m

 

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