PHAROS UNIVERSITYME 259

FLUID MECHANICS FOR ELECTRICAL STUDENTS

Basic Equations for a Control Volume

Main TopicsFlow ClassificationBasic Laws for a SystemRelation of System Derivatives

to the Control Volume FormulationConservation of MassBernoulli Equation

Flow Classification Classification of Fluid Dynamics

Inviscidµ = 0

Viscous

Laminar

Turbulent

Compressible Incompressibleϱ = constant

Internal External

Basic Laws for a System

Conservation of Mass

Relation of System Derivatives to the Control Volume Formulation

Extensive and Intensive Properties

Relation of System Derivatives to the Control Volume Formulation

Reynolds Transport Theorem

• Continuity Equation

• Bernoulli’s Equation

• Momentum Equation

• Energy Equation

FLUID FLOW

Basic Laws Conservation of mass: dM/dt=0 for

system∂/∂t ∫ϱ d 𐐏 +∫ ϱ ⊽. d Ᾱ=0 for control

vol.

Newton’s 2nd law Σ F = ma Σ F = ∂/∂t ∫ ⊽ ϱ d 𐐏 + ∫ ⊽ ϱ ⊽. d Ᾱ

First Law of Thermo: Q - W = dE/dt

Q-w=∂/∂t∫e ϱ d𐐏+∫(p/ ϱ+0.5V2+gz)ϱ⊽. dᾹ

Relation of System Derivatives to the Control Volume Formulation

Interpreting the Scalar Product

Conservation of Mass

Basic Law, and Transport Theorem

Conservation of Mass

Conservation of Mass

Basic Law for a System

Conservation of Mass

Incompressible Fluids

Steady, Compressible Flow

DefinitionsVolume (Volumetric) Flow Rate• Q = Cross Sectional Area*Average

Velocity of the fluid• Q = A*v cms

Weight Flow Rate• W = *Q N/s

Mass Flow Rate• M = *Q kg/s

Volume vQ = Volume/Unit time

Q = Area*Distance/Unit Time

Flow in non-circular sectionsFlow rate is determined by:• Q = A*v

Where,

A = Net flow area

v = average velocity Example:Dlarge, i = 0.5 mDsmall, o = 0.25 mDsmall, i = 0.2 mVsmall, i = 1 m/secVlarge, i = 1 m/secFind Qlarge and Qsmall

Continuity Equation

Continuity for any fluid (gas or liquid)• Mass flow rate In = Mass Flow Rate out

• M1 = M2

• 1*A1*v1 = 2*A2*v2

Continuity for liquids• Q1 = Q2

• A1*v1 = A2*v2

M1 M2

17/27

Equation of continuity

1 1 2 2(volume flow rate) constantvR Av A v

Volume flow rate has units m3/s

Mass flow rate has units kg/s

Units and Conversion Factors

Q: m3/secM: kg/sec,Volume Flow Rate:• 1 L/min = 0.06 m3/h• 1 m3/sec = 60,000 L/min• 1 gal/min = 3.785 L/min

Example #1If d1 and d2 are 50 mm and 100 mm,

respectively, and water at 70° C is flowing at 8 m/sec in section 1, determine: v2, Q, W, M.

1 2d1d2

v1 v2

© Pritchard

Example # 2

If d1, d2 and d3 are 10 cm, 20 cm, and 50 cm, determine Q and the velocities, v2 and v3 if v1 = 1 m/sec.

V1 = 1 m/sec

d1 = 10 cm d2 = 20 cm d3 = 50 cm

Example # 3

Determine the required size standard Schedule 40 steel pipe to carry 192 m3/hr with a maximum velocity of 6.0 m/sec.

Example # 4The tank is being filled with water by two 1-D inlets. Air is at the top of the tank. The water height is h. (a) Find an expression for the change in water height dh/dt. (b) Compute dh/dt if D1 = 1 cm, D2 = 3 cm, V1 = 3 m/s, V2 = 2 m/s and At = 2 m2.

Tank Area At

a

w

h

12

Example # 5

Consider the entrance region of a circular pipe for laminar flow. What is mean velocity of the fluid.

© PritchardApr 19, 2023 24/27

Assume:

the flow of fluids is laminar (not turbulent) or steady flow

- the fluid has no viscosity (no friction).

Ideal Fluids in Motion:Continuity & Bernoulli’s equation

A fluid element traces out a streamline as it moves. The velocity vector of the element is tangent to the streamline at every point.

The steady flow of a fluid around an air foil, as revealed by a dye tracer that was injected into the fluid upstream of the airfoil

Conservation of EnergyBernoulli’s Equation

Energy cannot be created or destroyed, just transformed

Three forms of energy in fluid system:• Potential• Kinetic• Flow energy

Potential Energy

Due to the elevation of the fluid element

Where,

w = weight of fluid element

z = elevation with respect to a reference level

zwPE

Kinetic Energy

Due to the velocity of the fluid element

Where,

v = average velocity of the fluid element

g

vwKE

2

2

Flow Energy

Flow work or pressure energyAmount of energy necessary to move a fluid

element across a certain section against pressure

Where,p = pressure on the fluid element

P

wPE

Total Energy and Conservation of Energy Principle

E = FE + PE + KE

Two points along the same pipe: E1 = E2

Bernoulli’s Equation:

Pz

v

g

Pz

v

g1

112

22

22

2 2

g

vwzw

PwE

2

2

g

wvwz

wp

g

wvwz

wp

22

22

22

21

11

Heads

Pessure H ead

P r _

z E leva tion H ead _

v

gVelocity H ead

2

2 _

Pz

v

gTo ta l H ead

2

2_

Assumption: No energy is added or lost

Assumption: Energy level remains constant

Restrictions on Bernoullis’ Equation

Valid only for incompressible fluidsNo energy is added or removed by pumps,

brakes, valves, etc.No heat transfer from or to liquidNo energy lost due to friction

Application of Bernoulli’s Equation

Write Bernoulli’s equation in the direction of flow,

Label diagramSimplify equation by canceling terms that

are zero, or equal on both sides of the equation

Solve equation and find desired result(s)

Example

A hose carries water at a flow rate of 0.01 m3/sec. The hose has an internal diameter of 12 mm, and the gauge pressure at faucet is 100 kPa. Determine the pressure at the end of the hose Z = 10 m

Torricelli’s Theorem

For a liquid flowing from a tank or reservoir with constant fluid elevation, the velocity through the orifice is given by:

where, h is the difference in elevation between the orifice and the top of the tank

Example: If h = 3.00 m, compute v2

v gh2 2

h

Take Home ExperimentA reservoir of water has the surface at 310m above the outlet nozzle

of a pipe with diameter 15mm. What is the a) velocity, b) the discharge out of the nozzle and c) mass flow rate.

Water Velocity = (2gh)0.5

= (2x 9.81 x 310)0.5 = 78 m/s

Individual Experiment Pipe Flow: Ideal flow Assumption and Energy Equation

The aim is to study Continuity equation and Bernoulli equation as will as pressure losses due to viscous ( frictional) effects in fluid flows through pipes

Flow meter

Differential Pressure Gauge- measure ΔP

LValve

H

Reservoir

PipeD

Schematic of experimental Apparatus

• Pipes with different Diameter and Length will be used later for the experiments to study Energy Equation and pressure losses