Unit: IV-Fluid Dynamic

Fluid Dynamics

Types of fluid forces - Euler equation of motion - Bernoulli’s equations

Application -Bernoulli’e equation -Venturimeter - Orifice meter - Pitot-tube - Momentum equation - Moment of Momentum eq. - Application of momentum

Section I

Fluid Dynamic:It is the branch of applied science of fluid which deals with fluid in motion condition under the force.

Types of fluid forces:1) Fg, gravity force.2) Fp, the pressure force3) Fv, force due to viscosity4) Ft, force due to turbulence.5) Fc, force due to compressibility

Reynol’d Equation of MotionNavier-Stokes Equation

Euler’s Equation of motion

Euler’s equation of motion:

The equation of motion in which the forces due to gravity and pressure are taken into consideration.

The forces acting on the cylindrical element are:1. Pressure force pdA in the direction of flow2. Pressure force opposite to the direction

3. Weight of element

The resultant force on the fluid elements

as is the acceleration in the direction of s and t

dsdz

Ө

Bernoulli’s equation from Euler’s equation:

= Pressure energy per unit weight of fluid or pressure Head

= Kinetic energy per unit weight or kinetic head

= Potential energy per unit weight or potential head

Assumptions:

1) The fluid is ideal, i.e. viscosity is zero2) The flow is steady3) The flow is incompressible4) The flow is irrotational

Bernoulli’s equation for real fluid:

Ex Water is flowing through a pipe of 5 cm diameter under a pressure of 29.43 N/cm2 (gauge) and with mean velocity of 2.0 m/s. find the total head or total energy per unit weight of the water at a cross-section, which is 5 m above the datum line.

Ans Total head = 35.204 m

Total Head = Pressure Head + Velocity Head + Datum Head

Ex A Pipe, through which water is flowing, is having diameter, 20 cm and 10 cm at the cross-section 1 and 2 respectively. The velocity of water at section 1 is given 4.0 m/s. Find the velocity head at section 1 and 2 and also rate of discharge.

Ans 1) Velocity head1= 0.815 m, velocity head at 2 = 83.047 m, 3) rate of discharge = 0.1256 m3/s

A1 V1 = A2 V2

D1 = 20 cmV1 =4.0 m/s D2 = 10 cm

Ex The water is flowing through a pipe having diameter 20 cm and 10 cm at section 1 and 2 respectively. The rate of flow through pipe is 35 lit/s. the section 1 in 6 m above datum and section 2 is 4 m above datum. If the pressure at section 1 is 39.24 N/cm2, find the intensity of pressure at section 2.

Ans v1=1.14 m/s, v2= 4.45 m/s, p2 = 40.27 N/cm2

A1 V1 = A2 V2

Ex Water is flowing through a pipe having diameter 300 mm and 200 mm at the bottom and upper end respectively. The intensity of pressure at the bottom end is 24.525 N/cm2 and the pressure at the upper end is 9.81 N/cm2. Determine the difference in datum head if the rate of flow through pipe is 40 lit/s..

Ans Difference in datum head= z2-z1= 13.70m

2

1

Z2

Z1

Practical application of Bernoulli’s equation:

1) Venturimeter: It is device used for measuring the rate of flow of a fluid flowing through a pipe

Expression:

Let d1 = diameter at inlet or at section p1 = Pressure at section v1 = velocity of fluid at section a = area at section Applying Bernoulli’s equation at section

Value of ‘h’ given by differential U-tube manometer:

Case I: Diff. Manometer contains liquid which is heavier than the liquid flowing through the pipe

Case II: Diff. Manometer contains liquid which is lighter than the liquid flowing through the pipe

Case III: Inclined Venturimeter with Differential U-tube manometer

Case IV: Inclined Venturimeter with Differential manometer contains a liquid which is lighter than the liquid flowing through the pipe

Ex A horizontal venturimeter with inlet and throat diameters 30 cm and 15 cm respectively is used to measure the flow of water. The reading of differential manometer connected to the inlet and the throat is 20 cm of mercury. Determine the rate of flow.Take Cd = 0.98

Ans h = 252 cm, Q = 125.756 lit/s

Ex A oil of sp.gr. 0.8 is flowing through a venturimeter having inlet diameter 20 cm and throat diameter 10 cm. The oil-mercury differential manometer shows a reading 0f 25 cm. calculate the discharge of oil through the horizontal venturimeter. Take Cd =0.98

Ans h = 400 cm, Q = 70.465 lit/s

Ex A horizontal venturimeter with inlet diameter 20 cm and throat diameter 10 cm is used to measure the flow of oil of sp.gr. 0.8. the discharge of oil through venturimeter is 60 lit/s. Find the reading of the oil-mercury differential manometer.Take Cd = 0.98

Ans h = 289.98 cm, x = 18.12 cm

Case : Inclined Venturimeter with Differential U-tube manometer

Ex A 30 cm x 15 cm venturimeter is inserted in a vertical pipe carrying water, flowing in the upward direction. A differential mercury manometer connected to the inlet and throat gives a reading of 20 cm. Find the discharge. Take Cd = 0.98

Ans h = 252 cm, Q = 125.75 lit/s

Ex A 20 cm x 10 cm venturimeter is inserted in a vertical pipe carrying oil of sp.gr. 0.8, the flow of oil is in upward direction. The difference of levels between the throat and inlet section is 50 cm. the oil mercury differential manometer gives a reading of 30 cm of mercury. Find the discharge of oil. Neglect losses.

Ans Q = 78.725 lit/s

2) Orifice Meter or Orifice Plate: It is device used for measuring the rate of flow of a fluid flowing through a pipe

Expression:

Let p1 = Pressure at section (1) v1 = velocity of fluid at section (1) a1 = area of pipe at section (1)P2, v2, a2 are corresponding values.Applying Bernoulli’s equation to Sec (1) and (2)

Ex An orifice meter with orifice diameter 10 cm is inserted in apipe of 20 cm dia. The pressure gauge fitted u/s and d/s of the orifice meter gives readings of 19.62 N/cm2 and 9.81 N/cm2 res. Co-effiecnt of discharge for the orifice meter is given as 0.6 find the discharge of water through pipe..

3) Pitot-tube: It is device used for measuring the velocity of flow at any point in a pipe or a channel

Expression:Consider two points (1) and (2) at the same level in such a way that

point (2) is just at the inlet of the pitot-tube and point (1) is far away from the tube

Let p1 = Pressure at section (1) v1 = velocity of fluid at section (1) p2 = pressure at point (2) v2 = Velocity at point (2) H = depth of tube in the liquid h = rise of liquid in the tube

Applying Bernoulli’s equation at points (1) and (2)

The momentum equation:

It is based on the law of conservation of momentum or on the Momentum principle, which states that the net force acting on a Fluid mass is equal to the change in momentum of flow per unit time In that direction.

Force acting on the fluid mass ‘m’ is given by Newton’s second law

Force exerted by a flowing fluid on a Pipe-bend:Consider two section (1) and (2). v1 = velocity of fluid at section (1) p1 = pressure intensity at point (1) A1 = Area of cross-section of pipe at section (1) v2, p2, A2 = Corresponding valuse of velocity, pressure and area (2)

Applying Bernoulli’s equation at points (1) and (2)Net force acting on fluid in the direction of x = Rate of change of momentum in x-dir.

Resultant force (FR) acting on bend:

Angle made by the resultant force:

Moment of Momentum Equation:Moment of momentum equation is derived fom moment of momentum principle which states that the resulting torque acting on rotating fluid is equal to the rate of change of moment of momentum. V1 = velocity of fluid at section (1)r1 = Radius of curvature at section 1Q = rate of flow of fluidP = density of fluidV2 and r2 = Velocity and radius at section 2

Moment of momentum per second at section 1

Moment of momentum per second at section 2

Rate of change of moment of momentum

This equation is known as moment of momentum.This equation is applied:1. For Analysis flow problem in turbines and centrifugal pumps2. For finding torque exerted by water on sprinkler

Free liquid Jets: Free liquid jet is defined as the jet of water coming out from the nozzle in atmosphere. The path travelled by the free jet is parabolic.

x = Velocity component in x-direction x t = U cosθ x t

1) Maximum Height attained by the jet:

2) Time of flight:

3) Time reach highest point:

4) Horizontal range of the jet:

5) Value of θ for maximum range:

Prepared by, Dr Dhruvesh Patel

Prepared by, Dr Dhruvesh Patel