Lectures 3 & 4: Pressure and Head

Lecture 2: Pressure and Head Lecture outcomes:Solve fluid problems using the concept of pressure and the hydrostatic equationMeasure pressure in manometers and tubes

Activity: Explain why the candles are floating and the flowers are not crushed by the water.2Statics of Fluid SystemsThe general rules of statics apply to fluids at rest, such that there will be no shearing forces acting on it. All forces exerted between the fluid and a solid boundary must at right angles to the boundary.If the boundary is curved, it can be considered to be composed of a series of chords on each of which a force F1, F2,, Fn acts perpendicular at the surface at the section concerned.

PressurePressure p of the fluid is defined as the force exerted per unit area. Pressure = Force exerted / Area of Boundary p = F/A (N/m2)Pascals law for pressure at a pointThe Pascals Law states as follows: The intensity of pressure at any point in a liquid at rest, is the same in all directions. Such that Ps = Px = Py

Variation of Pressure Vertically in a Fluid under gravitySuppose that the pressure is p1 is on the underside at level z1 and p2 on the top at level z2.

Since the fluid is at rest the element must be in equilibrium and the sum of all the vertical forces must be zero.Force due to p1 on area A acting up =p1 AForce due to p2 on area A acting up =p2 AForce due to the weight of the element = mg = gA(z2 - z1)

Thus, p1 A p2 A gA(z2 - z1) = 0p1 p2 = g(z2 - z1) Since g(z2 - z1) is +ve, therefore, p1 > p2

It is concluded that the pressure increase when the depth of fluid increaseQuiz 1A diver descends from the surface of the sea to a dept of 30m. What would be the pressure under which the diver would be working above that at the surface? Assume that the density of sea water is 1025kg/m3 and remains constant.Equality of pressure at the same level in static fluidIf P and Q are two points at same level in a fluid at rest, a horizontal prism of fluid of constant cross-sectional area A will be in equilibrium.

For static equilibrium, p1A = p2A p1= p2

Thus, the pressure at two points at the same level in a body of fluid at rest will be the same.

General Equation for the Variation of pressure due to gravity from point to point in a static fluid

Pressure and Head

The hydrostatic paradoxFrom equation p = hg, it can be seen that the pressure exerted by a fluid is dependent only on the vertical head of fluid and its mass density , it is not affected by the weight of the fluid present.All have the same base area A and are filled to the same height h with the same liquid of density . Pressure on bottom in each case = hg

Pressure Measurement Pressure Tube or PiezometerU-tube GuagePressure Tube or Peizometer

Quiz 2What is the maximum gauge pressure of water that can be measured by means of a piezometer tube 2m high? (Mass density of water water = 1000kg/m3 )U-Tube Gauge (Manometer)

Quiz 3A U-tube manometer as shown below is used to measure the gauge pressure of a fluid P of density = 800kg/m3 . If the density of the liquid Q is 13600 kg/m3 , what will be the gauge pressure at A if (a) h1 = 0.5m and D is 0.9m above BC, (b) h1 = 0.1m and D is 0.2m below BC

Quiz 4A U-tube manometer is used to measure the pressure of oil of specific gravity 0.85 flowing in a pipe line. Its left end is connected to the pipe and the right limb is open to the atmosphere. The centre of the pipe is 100mm below the level of mercury (specific gravity = 13.6) in the right limb. If the different of mercury level in the two limbs is 160mm, determine the absolute pressure of the oil in the pipe.

U-tube gauge is arranged to measure the pressure difference between two points in a pipeline. The principle involved in calculating the pressure difference is that the pressure at the same level CD in the two limbs must be the same, since the fluid in the bottom of the U-tube is at rest.

Quiz 5A U-tube manometer is arranged as shown below, to measure pressure difference between two points A and B in a pipeline conveying water of density = 103kg/m3 . The density of manometric liquid Q is 13.6 103kg/m3, and point B is 0.3m higher than point A. Calculate the pressure difference when h = 0.7m

Inverted-U TubeThe inverted U-tube is used to measuring pressure differences in liquids.

Quiz 6An inverted U-tube of the form shown in Figure below is used to measure the pressure difference between two points A and B in an inclined pipeline through which water is flowing (water = 103kg/m3). The difference of level h=0.3m, a=0.25m and b=0.15m. Calculate the pressure difference pB pA if the top of the manometer is filled with (a) air, (b) oil of relative density 0.8.

Quiz 7A U-tube differential manometer connecting two pressure pipes at A and B. The pipe A contains a liquid of specific gravity 1.6 under a pressure of 110kN/m2. The pipe B contains oil of specific gravity 0.8 under a pressure of 200kN/m2. Find the difference of pressure head measured by mercury as fluid filling U-tube.

Quiz 8Calculate the pressure head difference h if the U-tube manometer in example 7 is inverted as shown below.