#
Etiket Arşivleri: Fluid mechanics

# CP502

Advanced Fluid Mechanics

What do we mean by ‘Fluid’?

Recalling vector operations

Continuity equation for incompressible (constant density) flow

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid

Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid

Steady, incompressible flow of Newtonian fluid in an infinite channel with stationery plates

– fully developed plane Poiseuille flow

Continuity and Navier-Stokes equations

for incompressible flow of Newtonian fluid

Steady, incompressible flow of Newtonian fluid in a pipe

– fully developed pipe Poisuille flow

Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder

– fully developed pipe Couette flow

…

# FE 222 FLUID MECHANICS

Transport Processes and Separation Process Principles (Includes Unit Operations) (4th Edition) by Christie John Geankoplis, published by Pearson Education,Inc

(Used to be Transport Processes and Unit Operations, 3rd ed…..)

Unit Operations of Chemical Engineering (7th edition)(McGraw Hill Chemical Engineering Series) by Warren McCabe, Julian Smith, and Peter Harriott

2 exams 30 % each

final 40 %

# Fluids essential to life

Human body 65% water

Earth’s surface is 2/3 water

Atmosphere extends 17km above the earth’s surface

History shaped by fluid mechanics

Geomorphology

Human migration and civilization

Modern scientific and mathematical theories and methods

Warfare

Affects every part of our lives

History

Weather & Climate

Vehicles

Environment

Physiology and Medicine

Sports & Recreation

SPORTS

The main unit operations usually present in a typical food-processing line, including:

1. Flow of fluid – when a fluid is moved from one point to another by pumping, gravity, etc.

2. Heat transfer – in which heat is either removed or added (heating; cooling; refrigeration and freezing).

3. Mass transfer – whether or not this requires a change in state. Processes that use mass transfer include drying, distillation, evaporation, crystallization, and membrane processes.

4. Other operations requiring energy, such as mechanical separation (filtration, centrifugation, sedimentation, and sieving); size adjustment by size reductions (slicing, dicing, cutting, grinding) or size increase (aggregation, agglomeration, gelation); and mixing, which may include solubilizing solids, preparing emulsions or foams, and dry blending of dry powders (flour, sugar, etc.).

All theoretical equations in mechanics (and in other physical sciences) are dimensionally homogeneous; i.e., each additive term in the equation has the same dimensions.

Example is the equation from physics for a body falling with negligible air resistance:

S =S0+ V0t + ½ gt2

where S0 is initial position, V0 is initial velocity, and g is the acceleration of gravity.

Each term in this relation has dimensions of length {L}.

However, many empirical formulas in the engineering literature, arising primarily from correlations of data, are dimensionally inconsistent. Referred as dimensional equations. Defined units for each term must be used with it.

Fluids: Statics vs Dynamics

Continium mechanics : Branch of engineering science that studies behavior of solids and fluids

Fluid mechanics: Branch of engineering science that studies behavior of fluids

Fluid statics: Deals with fluids in the equilibrium state of no shear stress (study of fluids at rest)

Fluid dynamics: Deals with the fluids when portions of the fluid are in motion relative to other parts. (study of fluids in motion)

Density

Pressure

Pressure field

Pressure is a scalar field:

p = p(x; y; z; t)

The value of p varies in space, but p is not associated with a direction.

The pressure at any point in a stationary fluid is independent of direction.

A pressure sensor will not detect different values of pressure when the orientation of the sensor is changed at a fixed measurement point.

Atmospheric Pressure

Equality of pressure at the same level in a static fluid

Variation of pressure with elevation

General variation of pressure in a static fluid due to gravity

Variation of pressure in an incompressible fluid (liquid)

Variation of pressure in an compressible fluid (gas)

Pressure distribution for a fluid at rest

Incompressible fluid

Liquids are incompressible (density is constant):

Some Pressure Levels

10 Pa – The pressure at a depth of 1 mm of water

10 kPa – The pressure at a depth of 1 m of water, or the drop in air pressure when going from sea level to 1000 m elevation

10 MPa – A “high pressure” washer forces the water out of the nozzles at this pressure

10 GPa – This pressure forms diamonds

Some Alternative Units of Pressure

1 bar – 100,000 Pa

1 millibar – 100 Pa

1 atmosphere – 101,325 Pa

1 mm Hg – 133 Pa

1 inch Hg – 3,386 Pa

The bar is common in the industry. One bar is 100,000 Pa, and for most practical purposes can be approximated to one atmosphere even if 1 Bar = 0.9869 atm

Bourdon Gauge:

Pressure scales

Absolute pressure: pabs is measured relative to an absolute vacuum; it is always positive. (The pressure of a fluid is expressed relative to that of vacuum (=0)

Gauge pressure: pgauge is measured relative to the current pressure of the atmosphere; it can be negative or positive. (Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere.)

Usual pressure gauges record gauge pressure.

Measuring Pressure

Barometers

Piezometer

Rough edges or burrs on or near the edges of the

piezometer holes deflect the water into or away from the piezometer , causing erroneous indications. The case as in W shows the tube pushed into the flow,

causing the flow to curve under the tip which pulls the water level down.

Errors caused by faulty piezometer tap installation increase with velocity.

By determining the height to which liquid rises and using the relation P = ρgh, gauge pressure of the liquid can be determined.

To avoid capillary effects, a piezometer’s tube should be about 1/2 inch or greater.

Measuring Pressure with Manometers

Manometers

A somewhat more complicated device for measuring fluid pressure consists of a bent tube containing one or more liquid of different specific gravities. Such a device is known as manometer.

In using a manometer, generally a known pressure (which may be atmospheric) is applied to one end of the manometer tube and the unknown pressure to be determined is applied to the other end.

In some cases, however, the difference between pressure at ends of the manometer tube is desired rather than the actual pressure at the either end. A manometer to determine this differential pressure is known as differential pressure manometer.

The manometer in its various forms is an extremely useful type of pressure measuring instrument, but suffers from a number of limitations.

While it can be adapted to measure very small pressure differences, it can not be used conveniently for large pressure differences – although it is possible to connect a number of manometers in series and to use mercury as the manometric fluid to improve the range. (limitation)

A manometer does not have to be calibrated against any standard; the pressure difference can be calculated from first principles. ( Advantage)

Some liquids are unsuitable for use because they do not form well-defined menisci. Surface tension can also cause errors due to capillary rise; this can be avoided if the diameters of the tubes are sufficiently large – preferably not less than 15 mm diameter. (limitation)

A major disadvantage of the manometer is its slow response, which makes it unsuitable for measuring fluctuating pressures.(limitation)

It is essential that the pipes connecting the manometer to the pipe or vessel containing the liquid under pressure should be filled with this liquid and there should be no air bubbles in the liquid.(important point to be kept in mind)

Manometers – measure DP

Manometers – Various forms

Simple U – tube Manometer

Inverted U – tube Manometer

U – tube with one leg enlarged

Two fluid U – tube Manometer

Inclined U – tube Manometer

Simple U – tube Manometer

The maximum value of P1 – P2 is limited by

the height of the manometer.

To measure larger pressure differences

we can choose a manometer with higher density, and to measure smaller pressure differences with accuracy we can choose a manometer fluid which is having a density closer to the fluid density

Simple U – tube manometer

U – tube with one leg enlarged

Two fluid U-tube Manometer

Small differences in pressure

in gases are often measured

with a manometer of the

form shown in the figure.

Inclined Manometer

To measure small pressure differences need to magnify Rm some way.

…