Lecture-1

ChE-202 Fluid Dynamics

Compressibility Mass per unit volume is defined as density of a gas.

If the density of the fluid remains constant not only with regard to the time but also with

regard to the coordinates, fluid is said to be incompressible. If density varies then the fluid

is compressible.

Density may vary by the following ways

Drastic temperature change

Appreciable pressure change

Both Pressure and temperature varies

The density ρ at constant pressure is varied by the following expression

𝛼 = −1

𝜌𝑜

𝜕𝜌

𝜕𝑇𝑝

If the velocity is high (about 400fps or more for air), the pressure change associated with

the generation of this velocity would be large enough to affect the density, and thus flow

becomes compressible. In such cases not only temperature but also pressure influences the

gas density.

Mach Number

A property that is used usually in compressible fluid dynamics is the Mach Number.

Mach # = Ma = 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑓𝑙𝑜𝑤𝑠 (𝑢)

𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝑆𝑜𝑢𝑛𝑑 (𝐶)

Where “C” = 346m/sec at room temperature and sea level

Ma > 1 = Supersonic velocity

Ma = 1 Sonic velocity

Ma < 1 Subsonic Velocity

Ma >> 1 Hypersonic velocity

Derivation of General Energy Equation

General energy equation can be written in the following form

[Rate of change of energy of a system] = [Net rate of transfer of energy to the system]

Energy may be transferred to the system by two different processes namely heat and work

Heat transfer (Q); Type of energy transfer across a system boundary because oftemperature difference between the system and surrounding

Work (W); it is energy transfer by the action of force through a distance or any otherenergy transfer that could replaced by the action of force through a distance*.

*The flow of electric current into a system may be considered work, because a mechanism could be devised ( a frictionless electric motor) that completely converts the current flow to the action of force through a distance ( e.g. using motor to raise the weight). In the fluid mechanics, we are almost always concerned with work that actually is the action of force through a distance.

Intrinsic Energy It is the energy stored in the system (i.e. energy contained in the system’s mass). There are five types of intrinsic energies;

Kinetic Energy: The energy of motion

Potential energy: the energy owing to position in the force field (usually a gravitationalfield but sometimes an electric or magnetic field).

Internal Energy: The energy of molecular structure and motion.

Chemical Energy: The energy associated with the arrangement of atoms into molecules(released only in chemical reaction)

Nuclear Energy: The energy associated with the internal structure of atoms (releasedonly by nuclear fission and fusion).

In Fluid Mechanics/ Dynamics chemical and nuclear energies can be neglected.

Specific Energy is usually reported in the Fluid Mechanics/Dynamics which is energy perunit mass.

Classification of EnergiesEnergy can be classified as either Thermal energy and/or Mechanical energy

Thermal Energy: it is associated with temperature, molecular structure and heat transfer.

Mechanical Energy: It is associated with force and motion.

Based on previous discussion, we amplify the energy balance for a system to read as follows;

[Rate of heat transfer to the system] + [Rate of work done on the system] = Rate of increase of Intrinsic (kinetic + Potential + Internal ) energy of the system]

Above equation makes no distinction between the thermal and mechanical energies of the system; anenergy transfer may result in an increase of the system’s mechanical energy, thermal energy or both.

Mathematically it can be written as follows;𝑑𝑄 − 𝑑𝑊 = 𝑑𝐸𝑠𝑦𝑠

𝑑𝑄 − 𝑑𝑊 = 𝑑(𝑈 + 𝐾. 𝐸. +𝑃. 𝐸. )

Mechanical Energies Thermal Energies

Work

Kinetic Energy

Potential Energy

Heat

Internal Energy

Analysis of Work Term• If we neglect electrical and other forms of work, three types of work might be done on or by the fluid

inside the control volume as given under

Shaft work (Wshaft): It is transmitted by a rotating shaft, such as a pump drive shaft or a turbine output

shaft that is cut by control surface. It is sometimes called “pump work”(Wp) or “turbine work” (Wt) if

these devices are present.

Shear work (Wshear): It is done by shear stresses in the fluid acting on boundaries of a control volume.

Pressure work (Wpressure): It is done by fluid pressure acting on the boundaries of control volume.

Concept of “Friction Loss”

Let’s start with 2- point form of energy balance

−𝑊𝑠 − 𝑢2 − 𝑢1 + 1

2

𝑃𝑑𝑣 − 𝑞 − 1

2𝑑𝑃

𝜌= 𝑣22

2− 𝑣12

2+ 𝑔𝑍2 − 𝑔𝑍1

Besides grouping these terms, we have used following

𝑃2𝜌2−𝑃1𝜌1=

1

2

𝑑(𝑃

𝜌) =

1

2𝑑𝑃

𝜌+

1

2

𝑃 𝑑1

𝜌=

1

2𝑑𝑃

𝜌+

1

2

𝑃𝑑𝑣

So the bracketed term

ℎ𝐿 = 𝑢2 − 𝑢1 − 𝑞 + 1

2

𝑃𝑑𝑣

ℎ𝐿 is energy loss

The term ℎ𝐿 has following characteristics

ℎ𝐿 is always positive or zero ( -ve value violates the second law of thermodynamics)

ℎ𝐿 represents the loss of potential to perform useful work

ℎ𝐿 is zero only in an ideal process

We now write the energy equation for steady flow along a streamline as

𝑊𝑠 − 12 𝑑𝑃

𝜌−ℎ𝐿 =

𝑣22

2+ 𝑔𝑍2 − (

𝑣12

2+ 𝑔𝑍1)

• Mathematics on the White board

Summary and Comparison of various forms of energy

equation

General Energy Equation

𝑢1 +𝑝1𝜌1+𝑉12

2+ 𝑔𝑧1 + 𝑞 − 𝑤𝑠 = 𝑢2 +

𝑝2𝜌2+𝑉22

2+ 𝑔𝑧2

Mechanical Energy Equation

𝑝1𝜌1+𝑉12

2+ 𝑔𝑧1 −𝑤𝑠 =

𝑝2𝜌2+𝑉22

2+ 𝑔𝑧2 + ℎ𝐿

Bernoulli’s Equation

𝑝1𝜌1+𝑉12

2+ 𝑔𝑧1 =

𝑝2𝜌2+𝑉22

2+ 𝑔𝑧2

Important terms

Head

It is the elevation that would produce potential energy equal to the particular energy in question.

“Velocity head is the elevation that would yield the same specific potential energy as actual kinetic energy.”

head is the specific energy divided by gravitational acceleration.

Pressure form of Bernoulli’s equation is given below

𝑝 +𝜌𝑣2

2+ 𝑔𝜌𝑧 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Static Pressure (p)

It is true fluid pressure, because it is the pressure that would be measured by an instrument that is static with respect tofluid. of course if the instrument were static with respect to fluid, it would have to move along with fluid.

Dynamic Pressure (𝝆𝒗𝟐

𝟐)

This term is called dynamic pressure per unit volume. If the fluid is brought to rest ( V = 0) without shear stress or shaftwork, the pressure of the particle rises by an amount equal to 1/2ρV2 . The kinetic energy will have been converted intopressure.

Hydrostatic pressure (𝒈𝝆𝒛)

This actually potential energy per unit volume. This the pressure when the fluid is brought to zero elevation.

Stagnation Pressure (po)

This is the total pressure that could be attained by fluid particle when it is brought to rest and zero elevation so thatkinetic pressure and potential pressure could be converted true pressure called as stagnation pressure.

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Assumptions

Assumptions in the analysis of Compressible Fluid Flow

Flow is steady and one dimensional

Velocity gradient within the cross-section is neglected. So, α = β = 1 and Ū = U

Shaft work is zero

Gravitational effects are negligible

And mechanical Potential energy is neglected

Fluid is an ideal gas with constant specific heat

Friction is restricted to wall shear.

Stagnation Properties

• All the state properties are called static properties. The properties

measured by instruments are called static properties.

• Stagnation properties are the properties of the fluid that it would

obtain if it were brought to a condition of zero velocity and

elevation in a reversible process with no heat transfer and no work.

Stagnation Property Relation

• On the Board

Speed of Sound & Mach Number

Consider the air around you.

Air composed of different molecules that are moving about in a random motion with

different instantaneous velocities and energies at different times. However, the average

molecular velocity and energy can be defined and for a perfect gas these are the function

of temperature only.

Assume a small firecracker detonates nearby.

Energy released by fire cracker will be absorbed by surrounding air molecules, which

result in an increase in their mean velocity.

Faster molecules colloids with their neighbors, transfer some of their newly acquired

energy.

In turn, these neighbors eventually colloids with others resulting in net transfer or

propagation of firecracker energy through space.

This wave of energy travels through air at a velocity that must be related to mean

molecular velocity because molecular collisions are propagating the wave.

As the wave passes by you, this small pressure variation is picked up by your eardrum,

and is transmitted to your brain as a sense of sound.

Such a weak wave is defined as a Sound Wave.

And calculations are made to find out how fast it is propagated through air ????

Valve in the Pipeline

Consider a case of suddenly closing a valve at the end of long pipe.

At instant the valve is closed, the fluid at valve is brought to rest.

A wave propagates from closed valve back into the fluid, informing the fluidabout closed valve.

Wave causes the pressure change/increase as the kinetic energy of the fluid israpidly changed.

The wave is called compression wave.

The speed at which such a wave travels in a particular fluid is a measure of fluidscompressibility.

More compressible the fluid, the slower is the wave.

Because the existence and speed of such disturbance waves are intimatelyconnected with fluid compressibility, analysis of compressible flow often involvesconsideration of wave speed.

Derivation (Mathematics on the Board)

Consider the control volume encloses the wave front and moves with it

If the observer is riding on the compression wave then the fluid on its right will appear tobe moving toward the wave front with finite speed “C”.

Fluid on the left seems to be moving away from the wave front with the speed of “c-dv”.

One-dimensional Compressible Flow

When compressible fluid flows through the passage, the fluid motion depends on the followings;

Change in Cross-sectional Area

Fluid friction

Heat transfer

Work

Primary affect of Area change is the acceleration or De-acceleration.

If there is no heat transfer or work, flow in variable area passage isiso-energetic.

If there is no friction/shocks the flow is isentropic.

Nozzles and Diffusers

Nozzles and Diffusers in Daily life

Mathematical Treatment

• On board

Summary of Results