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GUJARAT TECHNOLOGICAL UNIVERSITY, NOVEMBER , 2016

GOVERNMENT ENGINEERING COLLEGE,PALANPUR

Subject :- Fundamental of Fluid Mechanics

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Submitted by :- Bajariya Maheshkumar 150610122004Kishankumar Nagar 150610122026Patel Jigneshkumar 150610122033Roy malay 150610122046Solanki kalpesh 150610122048

BACHELOR OF ENGINEERING In

MINING

Internal guide Head of department

Prof :- G. M. Savaliya Prof :- H. B. Patel

Fluid staticsFluid Statics:- It is a study of a fluid at rest. In this fluid there will be relative motion between adjacent or neighbouring fluid layers. No shear force present force as the fluid particles do not move respect to one another. There velocity gradient is equal to zero.

Velocity gradient = change of velocity between two adjecent layers Distance between two layers = du = 0 dy

Pressure variation in static fluid – hydrostatic law

The hydrostatic law states that “the rate of increase of Pressure in a vertically downward direction is equal to the Weight density of fluid at that point.” Consider a small element of fluid, vertical column of constant cross sectional area dA, and totally surrounded byfluid of mass density p . p1 = Pressure on face AB, dz = Height of fluid element, p2 = Pressure on face CDThe forces acting on the fluid element are i) Pressure force on face AB, p1dA, downward directionii) Pressure force on face CD, p2dA, upward direction

iii) Force due to weight of fluid element, W = mg = p.Vg W = p(dA.dZ)g, downward direction.iv) Pressure forces on surfaces AC and BD, but they are equaland opposite to each other. P1.dA

A B

C D

P2.dA

dzCylindricalElement of fluid

W

dA

Fluid is at rest the element must be in equilibrium and the sum of allvertical forces must be zero. p1dA – p2dA + W = 0 (p1 – p2)dA + p(dAdz)g = 0 (p2 – p1) – pgdz = 0 p2 – p1 = pgdz dP = pgdz dP = pg = w dzWhere w = Weight density of fluid Above equation states that “rate of increase of pressure in a verticaldirection is equal to weight density of the fluid at the point” and this is known as Hydrostatic law. ʃ dp = ʃ pgdz p = pgzWhere, p= pressure above atmospheric pressure, and z is the height of point from free surfaces.

PASCAL’S LAW:- “The intensity of pressure at any point in a liquid at rest, is the same in all direction.”Consider an arbitrary fluid element of wedge shape ABC in a fluid mass atrest . The width of the element perpendicular to the plane of paper is Unity.

Let, px= Pressure acting on a face AB py= Pressure acting on a face AC pz= Pressure acting on a face BC Px= Force on a face AB Py= Force on a face AC Pz= Force on a face BCThe forces acting on the element are i) Force normal to the surface dye to fluid pressure ii) Force due to weight of fluid mass in vertical direction Px= px × area of face AB = px (dy × 1) Py= py × area of face AC = py (dx × 1) Pz= pz × area of face BC = pz (ds × 1)Weight of element = mass of element × g = p × volume ×g = p×(½×AC×AB×1) × g w= ½ p(dy×dx)g,The element of the liquid is at rest, therefore sum of horizontal and Vertical components of the forces equal to zero.

Force acting in X – direction , px × dy × 1 = pz × ds × sin(90-θ) × 1 px × py × 1 = pz ×dy × 1 px = pzForce acting in Y – direction , py ×dx ×1 = pz ×ds ×cos(90-θ)× 1+(½dx×dy×1) py ×dx ×1 = pz ×dx ×1 (½dx×dy×1 is neglecting) py = pz px = py = pzHence , at any point in a fluid rest the pressure is exerted equally in all directions.Atmospheric pressure:- It is pressure exerted by the air on theSurface of earth. The air is compressible fluid the density of air vary from time to time due to changes in its temperature, therefore atmospheric pressure is not constant. The atmospheric pressure at sea level at 15Ċ is 101.3 KN/m2 in SI unit and 1.033 kgf/cm2 in MKS units. 1 atmospheric pressure = 101.3 KN/m2 = 1.033 kgf/cm2 = 760 mm of Hg = 10.33 m of water

Gauge pressure:- Gauge pressure is measured with the help of a Pressure gauge. The atmospheric pressure is taken as datum. In this Pressure, atmospheric pressure is considered zero, and this pressure is Above the atmospheric pressure.Vacuum pressure:- When pressure is below the atmospheric pressure is called vacuum pressure. It is also known as negative gauge pressure. It is measured by vacuum gauge.Absolute pressure:- It is pressure which is measured with reference to absolute vacuum pressure. It is independent of the change in atmospheric pressure. It is measured above the zero of pressure.Mathematically, absolute pressure= atmo. Pressure + Gauge pressure pabs = patm +pgauge

The Hydrostatic paradox:- We know the equation of pressure at a point , p = pgh.The intensity of pressure depends only on height of the column, and density of fluid and it dose not depends on size of the column.

Consider four vessels all have the same base area A and are filled to the same height h with the same liquid of density p. since weight of fluid is different in the four vessels.We know, the forec on the base of vessel = p . A = p gh . AFor same base area A, same density p and same column height, the forceon the base of vessel is same for all four vessels and hence pressure intensity exerted on the base are same for all four vessel.Thus ,pressure intensity is independent on weight of fluid. This situation is called Hydrostatic paradox.

ManometerManometer is a device used for

measuring the pressure at a point in a fluid by balancing the column of fluid with the same column or another of the fluid.

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(1) Simple manometer:PiezometerU-tube manometersingle column manometer

Vertical single column manometer Inclined single column manometer

(2) Differential manometer :  U-tube differential manometerInverted U-tube differential

manometer

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Classification of Manometers :

A piezometer is the simplest form of the manometer. It measures gauge pressure only.

The pressure at any point in the liquid is indicated by the height of the liquid in the tube above that point, which can read on the calibrated scale on glass tube.

The pressure at point A is given by;

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1. Piezometer :

It can be measure large pressure or vacuum pressure and gas pressure.

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2. U-tube Manometer :

x x

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3. Single column Manometer:(A) Vertical single columnvmanometer

One of the limbs in double column manometer is converted into a reservoir having large cross sectional area (about 100 times) with respect to the other limb.

Pressure in left col. = pressure in right col.…………(i)

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(B) Vertical single column manometer

It is modified of vertical column manometer. This manometer is useful for the measurement of small pressure.Here,

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4. U-tube differential manometer It is used to measure pressure difference at two points in a pipe or between two pipes at different levels.Case 1 - U-tube upright differential manometer connected at two points in a pipe at same level

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Case 2 - U-tube upright differential manometer connected between two pipes at different levels and carrying different fluids

It is used for low pressure difference.

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5. Inverted U-tube differential manometer

Bourdon Tube

- The linkage is constructed so that the mechanism may be adjusted for optimum linearity and minimum hysteresis as well as compensate for wear which may develop over a period of time. - An electrical-resistance strain gauge may also be installed on the bourdon-tube to sense thr elastic deformation.

4) Diaphragm and Bellows Gauges

● Represent similar types of elastic deformation devices useful for pressure measurement applications.● Architecture and operation: Diaphragm gauge: - Consider first the flat diaphragm subjected to the differential pressure p1-p2 as shown in figure . - The diaphragm will be deflected in accordance with this pressure differential and the deflection sensed an appropriate displacement transducer. - Various types of diaphragm gauge are shown figure 4.10

(a) Diaphragm and (b) Bellows

(a) (b)

Bellows Gauge: - The bellows gauge is shown in figure (b). - A differential gauge pressure force causes displacement of the bellows, which may be converted to an electrical signal or undergo a mechanical amplification to permit display of the output on an indicator dial. - Figure shows various types of bellows gauges. ● The bellows gauge is generally unsuitable for transient measurements because of the larger relative motion and mass involved.● The diaphragm gauge which may be quite stiff, involves rather small displacements and is suit for high frequency pressure measurement.

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