SINGLE COLUMN MANOMETER

 SINGLE COLUMN MANOMETER .

           Single column manometer is a modified form of a U -tube manometer in which a reservoir, having a large cross -sectional area (about 100 times) as compared to the area of the tube is connected to one of the limbs (say left limb) of the manometer as shown in figure. Due to large cross -sectional area of the reservoir, for any variation in pressure, the change in liquid level in the reservoir will be very small which may be neglected and hence the pressure is given by the height of the liquid in the other limb. The other limb may be vertical or inclined. Thus there are two types of single column manometer as :

    1. Vertical Single Column Manometer. 

    2. Inclined Single Column Manometer. 

VERTICAL SINGLE COLUMN MANOMETER

VERTICAL SINGLE COLUMN MANOMETER

  Above figure shows the vertical single column manometer. Let X-X be the datum line in the reservoir and in the right limb of manometer, when it is not connected to the pipe. When the manometer is connected to the pipe, due to high pressure at A, the heavy liquid in the reservoir will be pushed downward and will rise in the right limb. 

Vertical Single Column Manometer

2.  Inclined Single Column Manometer. 

     

Inclined Single Column Manometer

  

  Figure shows the inclined single column manometer. This manometer is more sensitive. Due to inclination the distance moved by the heavy liquid in the right limb will be more. 

Inclined Single Column Manometer

MEASUREMENT OF PRESSURE

MEASUREMENT OF PRESSURE. 

          The pressure of a fluid is measured by the following devices :

1. Manometers 

2. Mechanical Gauges 

1. MANOMETERS. 

              Manometers are defined as the devices used for measuring the pressure at a point in a fluid by balancing the fluid column by the spring or dead weight. The commonly used mechanical pressure gauges are : 

 (a) Diaphragm pressure gauge, 

 (b) Bourdon tube pressure gauge, 

 (c) Dead -weight pressure gauge, and 

 (d) Bellows pressure gauge. 

SIMPLE MANOMETERS. 

            A simple manometer consists of a glass tube having one of its ends connected to a point where pressure is to be measured and other end remains open to atmosphere. Common types of simple manometers are : 

(1) Piezometer, 

(2) U -tube Manometer, and

(3) Single Column Manometer. 

(1) PIEZOMETER. 

              It is the simplest form of manometer used for measuring gauge pressures. One end of this manometer is connected to the point where pressure is to be measured and other end is open to the atmosphere as shown in figure. The rise of of liquid gives the pressure head at that point. If at a point A, the height of liquid say water is h in piezometer tube, then pressure at A

                           =  ⍴ × g × h  N/m²

Piezometer

(2)  U -tube Manometer. 

                        It consists of glass tube bent in U-shape, one end of which is connected to a point at which pressure is to be measured and other end remains open to the atmosphere as shown in figure. The tube generally contains mercury or any other liquid whose specific gravity is greater than the specific gravity of the liquid whose pressure is to be measured. 

(a) For Gauge Pressure. 

(a) For gauge pressure

For gauge pressure

(b) For Vacuum Pressure. 

(b) For vacuum pressure

For vacuum pressure

ABSOLUTE, GAUGE, ATMOSPHERIC AND VACUUM PRESSURES

ABSOLUTE, GAUGE, ATMOSPHERIC AND VACUUM PRESSURES. 

            The pressure on a fluid is measured in two different systems. In one system, it is measured above the absolute zero or complete vacuum and it is called the absolute pressure and in other system, pressure is measured above the atmospheric pressure and it is called gauge pressure. Thus :

1. Absolute pressure is defined as the pressure which is measured with reference to absolute vacuum pressure. 

2. Gauge pressure is defined as the pressure which is measured with the help of a pressure measuring instrument, in which the atmospheric pressure is taken as datum. The atmospheric pressure on the scale is marked as zero. 

3. Vacuum pressure is defined as the pressure below the atmospheric pressure. 

    The relationship between the absolute, gauge pressure and vacuum pressure are shown in figure:

Relationship between pressures

Note. (1) The atmospheric pressure at sea level at 15°C is 101.3 kN/m² or 10.13 N/cm² in SI unit. In case of MKS units, it is equal to 1.033 kgf/cm².

(2) The atmospheric pressure head is 760 mm of mercury or 10.33 m of water. 

PRESSURE VARIATION IN A FLUID AT REST

Pressure Variation In A Fluid At Rest. 

                      The pressure at any point in a fluid at rest is obtained by the Hydrostatic Law which states that the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point. This is proved as : 

    Consider a small fluid element as shown in figure :

Forces on a fluid element.

Let    ∆A = Cross -sectional area of element

          ∆Z = Height of fluid element

            p = Pressure on face AB

            Z = Distance of fluid element from free surface. 

Derivation of Hydrostatic Law. 

FLUID PRESSURE AT A POINT & PASCAL'S LAW

FLUID PRESSURE AT A POINT. 

                       Consider a small area dA in large mass of fluid. If the fluid is stationary, then the force exerted by the surrounding fluid on the area dA will always be perpendicular to the surface dA. Let dF is the force acting on the area dA in the normal direction. Then the ratio of (dF/dA) is known as the intensity of pressure or simply pressure and this ratio is represented by p. Hence mathematically the pressure at a point in a fluid at rest is 

                                      p = dF /dA.

 If the force (F)  is uniformly distributed over the area (A)  , then the pressure is given by

                            p = F /A = Force /Area  . 

So,      Force or pressure force, 

                                     F =  p ×A  . 

The units of pressure are : (1) kgf /m² and kgf /cm² in MKS units, (2)  Newton /m² and N /mm² in SI units. N /m² is known as Pascal and is represented by Pa. Other commonly used units of pressure are MPa ( Mega pascal) , kPa ( kilo pascal) and bar . 

PASCAL'S LAW. 

                          It states that the pressure or intensity of pressure at a point in a static fluid is equal in all directions. This is proved as :

       The fluid element is of very small dimensions i.e. , dx, dy and ds. 

Page -1 ( PASCAL'S law) 

and the rest part is : 

 

Page -2 (PASCAL'S law)

VAPOUR PRESSURE AND CAVITATION

VAPOUR PRESSURE AND CAVITATION.

                                 A change from the liquid state to the gaseous state is known as vaporization. The vaporization (which depends upon the prevailing pressure and temperature condition) occurs because of continuous escaping of the molecules through the free liquid surface.
Consider a liquid (say water) which is confined in a closed vessel. Let the temperature of liquid is 20°C and pressure is atmospheric. This liquid will vaporise at 100°C . When vaporization takes place, the molecules escape from the free surface of the liquid. These vapour molecules gets accumulated in the space between the free liquid surface and top of the vessel. These accumulated vapour exert a pressure on the liquid surface. This pressure is known as vapour pressure of the liquid or this is the pressure at which the liquid is converted into vapours.
          Again consider the same liquid at 20°C at atmospheric pressure in the closed vessel. If the pressure above the liquid surface is reduced by some means, the boiling temperature will also reduce. If the pressure is reduced to such an extent that it becomes equal to or less than the vapour pressure, the boiling of the liquid will start, though the temperature of liquid is 20°C. Thus a liquid may boil even at ordinary temperature, if the pressure above the liquid surface is reduced so as to be equal or less than the vapour pressure of the liquid at that temperature.
               Now consider a flowing liquid in a system. If the pressure at any point in this flowing liquid becomes equal to or less than the vapour pressure, the vaporization of the liquid starts. The bubbles of these vapours are carried by the flowing liquid into the region of high pressure where they collapse, giving rise to high impact pressure. The pressure developed by the collapsing bubbles is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation.
                       Hence the cavitation is the phenomenon of formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below the vapour pressure and sudden collapsing of these vapour bubbles in a region of higher pressure. When the vapour bubbles collapse, a very high pressure is created. The metallic surfaces, above which the liquid is flowing, is subjected to these high pressures which cause pitting action on the surface. Thus cavities are formed on the metallic surface and hence the name is cavitation.

CAPILLARITY

CAPILLARITY 

                      Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjecent general level of liquid when the tube is held vertically in the liquid. The rise of liquid surface is known as capillary rise while the fall of liquid surface is known as capillary depression. It is expressed in terms of cm or mm of liquid. Its value depends upon the specific weight of the liquid, diameter of the tube and surface tension of the liquid.

CAPILLARITY RISE 

EXPRESSION FOR CAPILLARY RISE. 

                     Consider a glass tube of small diameter ‘d’ opened at both ends and is inserted in a liquid, say water. The liquid will rise in the tube above the level of the liquid. 

      Let, h  = height of the liquid in the tube

                        Under a state of equilibrium, the weight of liquid of height ‘h’ is balanced by the force at the surface of the liquid in the tube is due to surface tension. 

      Let  𝛔 = Surface tension of liquid

              θ = Angle of contact between liquid and glass tube.

EXPRESSION FOR CAPILLARITY RISE

   

                                   The End. 

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SURFACE TENSION

SURFACE TENSION. 

                         Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrane under tension. The magnitude of this force per unit length of the free surface will have the same value as the surface energy per unit area. It is denoteby Greek letter 𝛔 (called sigma) . In MKS units, it is expressed as kgf/m while in SI units as N/M.

SURFACE TENSION

               The phenomenon of surface tension is explained by above Figure. Consider three molecules of A, B, C of a liquid in a mass of liquid. The molecule A is attracted in all directions equally by the surrounding molecules of the liquid. Thus the resultant force acting on the molecule A is zero . But the molecule B, which is situated near the free surface, is acted upon by upward and downward forces which are unbalanced. Thus a net resultant force on molecule B is acting in downward direction. The molecule C situated on the free surface of the liquid does experience a resultant downward force. Thus the free surface of the liquid acts like a very thin film under tension of the surface of the liquid acts as though it is an elastic membrane under tension. 

SURFACE TENSION ON LIQID DROPLET.

                               Consider a small spherical droplet of a liquid of radius ‘r’ . On the entire surface of droplet, the tensile due to surface tension will be acting. 
   Let 𝛔 = Surface tension of the liquid
           p = Pressure intensity inside the droplet (in excess of the outside pressure intensity)
           d = Dia. of the droplet.

SURFACE TENSION ON LIQUID DROPLET

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Newton's Law of Viscosity

NEWTON'S LAW OF VISCOSITY. 

             It states that the shear stress(τ ) on a fluid element layer is directly proportional to the rate of shear strain. The constant of proportionality is called the co-efficient of viscosity. Mathematically, it is expressed as given by equation 

       
                               τ  =  μ (du/dy)

                Fluid which obey the above relation are known as Newtonian fluids and the fluids which do not obey the above relation are called Non-Newtonian fluids.

 VARIATION OF VISCOSITY WITH TEMPERATURE. 

                                Temperature affects the viscosity. The viscosity of liquids decreases with the increase of temperature while the viscosity of gases increases with the increase of temperature. This is due to reason that the viscous forces in a fluid are due to cohesive forces and molecular momentum transfer. In liquids, the cohesive forces predominates the molecular momentum transfer, due to closely packed molecules and with the increase in temperature, the cohesive decreases with the result of decreasing viscosity. But in case of gases the cohesive forces are small and molecular momentum transfer predominates. With the increase in temperature, molecular momentum transfer increases and hence viscosity increases.

TYPES OF FLUIDS. 

                              The fluid may be classified into the following five types :

1. Ideal fluid, 

2. Real fluid, 

3. Newtonian fluid

4. Non-Newtonian fluid, and 

5. Ideal plastic fluid. 

1.  IDEAL FLUID.  

                       A fluid, which is incompressible and is having no viscosity, is known as an ideal fluid. Ideal fluid is only an imaginary fluid as all the fluids, which exists, have some viscosity. 

2.  REAL FLUID. 

                    A fluid, which possesses, is known as real fluid. All the fluids, in actual practice, are real fluids.

3. NEWTONIAN FLUID. 

                   A real fluid, in which the shear stress is directly proportional to the rate of shear strain (or velocity gradient) , is known as a Newtonian fluid. 

TYPES OF FLUID

4.  NON-NEWTONIAN FLUID. 

                  A real fluid, in which the shear stress is not proportional to the shear strain (or velocity gradient) , known as a Non-Newtonian fluid. 

5.  IDEAL PLASTIC FLUID. 

                   A fluid, in which shear stress is more than the yield value and shear stress is proportional to the rate of shear strain ( or velocity gradient) , is known a ideal plastic fluid. 

VISCOSITY

VISCOSITY

        Viscosity is defined as the property of a fluid which offers resistance to the movement of one layer of fluid over another adjacent layer of the fluid. When two layers of a fluid, a distance ‘dy’ apart, move one over  the other at different velocities, say  ‘u’ and  ‘u + du’ as shown in below figure, the velocity together with relative velocity causes a shear stress acting between the fluid layers. 

         The top layer causes a shear stress on the adjacent lower layer while the lower layer causes a shear stress on the adjacent top layer. This shear stress is proportional to  ‘y’ . It is denoted by symbol ‘tau’.

   

  Units of Viscosity 

          

    MKS unit of viscosity = kgf -sec/m²

     CGS unit of  viscosity = dyne -sec/cm²

      SI unit of viscosity = Newton - sec/ m²

    

                  The unit of viscosity in CGS is also called Poise which is equal to  dyne -sec/cm².

                Note - 

                          1. If viscosity is given in poise, it must be divided by 10 to get its equivalent numerical value in SI units. 

            The viscosity of water at 20°C is 0.01 poise or 1.0 centipoise. 

KINEMATIC VISCOSITY 

                          It is defined as the ratio between the dynamic viscosity and density of fluid. It is denoted by the Greek symbol called ‘nu’.

UNITS OF KINEMATIC VISCOSITY

                                 In MKS and SI, the unit of kinematic viscosity is ‘metre²/sec’ while in CGS units it is written as ‘cm²/sec’. In CGS units, kinematic viscosity is also known as ‘stoke’.

                                            The End. 

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PROPERTIES OF FLUIDS

INTRODUCTION

                        Fluid mechanics is that branch of science which deals with the behavior of the fluids (liquids or gases) at rest as well as in motion. 

PROPERTIES OF FLUIDS

    DENSITY OR MASS DENSITY : 

                 It is defined as the ratio of the mass of a fluid to its volume. 

                OR

   Mass per unit volume is called density. 

                 It is denoted by symbol 'rho'. 

   The value of density of water is 1000kg/m³ or       1gm/cm³.

SPECIFIC WEIGHT OR WEIGHT DENSITY

                It is the ratio between the weight of a fluid to its volume. 

  The value of specific weight or weight density for water is 9.81 × 1000 Newton/m³.

SPECIFIC VOLUME

            It is defined as the volume of a fluid occupied by a unit mass or volume per unit mass is called specific volume. 

    Thus specific volume is the reciprocal of mass density. It's is expressed as m³/kg. It is commonly applied to gases. 

  

SPECIFIC GRAVITY

           It is defined as the ratio of the weight density (or density)of a fluid to the weight density (or density) of a standard fluid. For liquids, the standard fluid is taken water at 4°C and for gases, the standard fluid is taken air. 

 Specific gravity is also called relative density. 

 If the specific gravity of a fluid is known, then the density of the fluid will be equal to specific gravity of fluid multiplied by the density of water. 

For example, the specific gravity of mercury is 13.6, hence density of mercury = 13.6×1000 = 13600 kg/m³.

                                   The End

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