March 2017

Problem 06 - Variation of Pressure

Problem
If the pressure in the tank of oil (sp gr 0.80) is 60 psi, what is the equivalent head: (a) in feet of oil, (b) in feet of water, and (c) in inches of mercury?
 

Problem 07 - Variation of Pressure

Problem
If the pressure in the tank of oil (sp gr 0.80) is 415 kPa, what is the equivalent head: (a) in meters of oil, (b) in meters of water, and (c) in centimeters of mercury?
 

Problem 08 - Variation of Pressure

Problem
What is the pressure in pounds per square inch 4 ft below the surface of a liquid of sp. gr. 1.50 if the gas pressure on the surface is 0.4 atmosphere?
 

Problem 09 - Variation of Pressure

Problem
The pressure in a gas tank is 2.75 atmospheres. Compute the pressure in kiloPascal and the pressure head in meter of water.
 

Problem 10 - Variation of Pressure

Problem
A gage on the suction side of a pump shown vacuum of 254 mm of mercury. Compute (a) pressure head in meters of water; (b) pressure in kiloPascal; (c) absolute pressure in kiloPascal if barometer reads 736.6 mm of mercury.
 

Problem 11 - Variation of Pressure

Problem
A pressure gage on the discharge side of a pump reads 300 kPa. Oil (sp gr 0.82) is being pumped. Compute the pressure head in meters of oil.
 

Problem 12 - Variation of Pressure

Problem
In a condenser containing air and water, the air pressure is 22.1 kPa absolute. What is the gage pressure in kiloNewton per square meter at a point 1.37 m below the water surface.
 

Manometers

Manometer is a simple and inexpensive device of measuring pressure and pressure difference. It is usually bent to form a U-tube and filled with liquid of known specific gravity. The surface of the liquid will move in proportion to changes of pressure.
 

001-piezometers.gif

 

Problem 02 - Manometer

Problem
In the piezometers of the figure shown, liquid stands 1.37 m above point M. What is the pressure at M in kiloPascal if the liquid is (a) water, (b) oil (sp gr 0.90), (c) mercury, and (d) molasses (sp gr 1.5).
 

01-002-piezometers.gif

 

Problem 03 - Manometer

Problem
How high will liquid rise in the piezometers shown in the figure if the pressure at M is 68.95 kPa and the liquid is (a) water, (b) oil (sp gr 0.85), (c) mercury, (d) brine (sp gr 1.15)?
 

01-002-piezometers.gif

 

Hydrostatic Pressure on Surfaces

  1. Total Hydrostatic Force on Plane Surfaces
    • Plane surface inside a gas chamber
    • Horizontal plane surface
    • Vertical plane surface
    • Inclined plane surface
  2. Total Hydrostatic Force on Curve Surfaces
    • Fluid is above the curve surface
    • Fluid is below the curve surface
    • Fluid is below and above the curve surface
  3. Analysis of Gravity Dam
  4. Buoyancy
  5. Statical Stability of Floating Bodies
  6. Stresses on Thin-walled Pressure Vessels

 

Total Hydrostatic Force on Surfaces

Total Hydrostatic Force on Plane Surfaces
For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by
 

$F = pA$

 

where p is the uniform pressure and A is the area.
 

002-total-hydroforce-plane-surface.gif

 

In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity.
 

$F = p_{cg}A$

 

where pcg is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be expressed in terms of unit weight γ of the liquid.
 

$F = \gamma \bar{h} A$

 

where   $\bar{h}$   is the depth of liquid above the centroid of the submerged area.
 

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

 

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
 

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

 

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

 

Stresses on Thin-walled Pressure Tanks

The circumferential stress, also known as tangential stress, in a tank or pipe can be determined by applying the concept of fluid pressure against curved surfaces. The wall of a tank or pipe carrying fluid under pressure is subjected to tensile forces across its longitudinal and transverse sections.
 

006-wood-stave-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

 

Relative Equilibrium of Liquids

Relative equilibrium of liquid is a condition where the whole mass of liquid including the vessel in which the liquid is contained, is moving at uniform accelerated motion with respect to the earth, but every particle of liquid have no relative motion between each other. There are two cases of relative equilibrium that will be discussed in this section: linear translation and rotation. Note that if a mass of liquid is moving with constant speed, the conditions are the same as static liquid in the previous sections.
 

Rectilinear Translation - Moving Vessel

Horizontal Motion
If a mass of fluid moves horizontally along a straight line at constant acceleration a, the liquid surface assume an angle θ with the horizontal, see figure below.
 

007-fluid-mass-horizontal-motion.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

 

Problem 20 - Rotating Vessel

Problem
A closed cylindrical vessel 3 m. in diameter and 6 m high is filled with water to a height of 4.5 m. The rest is filled with air, the pressure of which is 105 kPa. If the vessel is rotated at 191 rpm about its axis, determine the maximum and minimum inside pressure at the base.
 

08-001-rotating-pressure-vessel.gif

 

Fundamentals of Fluid Flow

Principles relating to fluids at rest can be obtained with no ambiguity by purely rational methods. Some natural principles which are universally true are the basis of calculations. In flowing fluid, however, the laws that govern the motion are complex and most of the time defies mathematical expressions. Thanks to experimental data combined with mathematical theories which solved countless engineering problems in the past.
 

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

 

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