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Fluid Flow of Food Processing

Production Line for milk processing

Fluid Mechanics Overview

Fluid

Mechanics

Gas

Liquids

Statics

Dynamics

, Flows

Water, Oils, Alcohols, etc.

Stability

Air, He, Ar, N2, etc.

Buoyancy

Compressible/ Incompressible

Pressure

Laminar/ Turbulent

Surface Tension

Steady/Unsteady

Compressibility

Viscosity

Density

Vapor Pressure

Viscous/Inviscid

Fluid Dynamics

Introduction

Characteristics of Fluids

- Gas or liquid state
- Large molecular spacing relative to a solid
- Weak intermolecular cohesive forces
- Can not resist a shear stress in a stationary

state - Will take the shape of its container
- Generally considered a continuum
- Viscosity distinguishes different types of fluids

Measures of Fluid Mass and Weight Density

The density of a fluid is defined as mass per

unit volume.

m mass, and v volume.

- Different fluids can vary greatly in density
- Liquids densities do not vary much with pressure

and temperature - Gas densities can vary quite a bit with pressure

and temperature - Density of water at 4 C 1000 kg/m3
- Density of Air at 4 C 1.20 kg/m3

Alternatively, Specific Volume

Measures of Fluid Mass and Weight Specific

Weight

The specific weight of fluid is its weight per

unit volume.

g local acceleration of gravity, 9.807 m/s2

- Specific weight characterizes the weight of the

fluid system - Specific weight of water at 4 C 9.80 kN/m3
- Specific weight of air at 4 C 11.9 N/m3

Measures of Fluid Mass and Weight Specific

Gravity

The specific gravity of fluid is the ratio of the

density of the fluid to the density of water _at_ 4

C.

- Gases have low specific gravities
- A liquid such as Mercury has a high specific

gravity, 13.2 - The ratio is unitless.
- Density of water at 4 C 1000 kg/m3

Viscosity Kinematic Viscosity

- Kinematic viscosity is another way of

representing viscosity - Used in the flow equations
- The units are of L2/T or m2/s and ft2/s

Surface Tension

At the interface between a liquid and a gas or

two immiscible liquids, forces develop forming

an analogous skin or membrane stretched over

the fluid mass which can support weight. This

skin is due to an imbalance of cohesive forces.

The interior of the fluid is in balance as

molecules of the like fluid are attracting each

other while on the interface there is a net

inward pulling force. Surface tension is the

intensity of the molecular attraction per unit

length along any line in the surface. Surface

tension is a property of the liquid type, the

temperature, and the other fluid at the

interface. This membrane can be broken with a

surfactant which reduces the surface tension.

Surface Tension Liquid Drop

The pressure inside a drop of fluid can be

calculated using a free-body diagram

Real Fluid Drops

Mathematical Model

R is the radius of the droplet, s is the surface

tension, Dp is the pressure difference between

the inside and outside pressure.

The force developed around the edge due to

surface tension along the line

This force is balanced by the pressure difference

Dp

Surface Tension Liquid Drop

Now, equating the Surface Tension Force to the

Pressure Force, we can estimate Dp pi pe

This indicates that the internal pressure in the

droplet is greater that the external pressure

since the right hand side is entirely positive.

Surface Tension Capillary Action

Capillary action in small tubes which involve a

liquid-gas-solid interface is caused by surface

tension. The fluid is either drawn up the tube

or pushed down.

Wetted

Non-Wetted

Cohesion gt Adhesion

Adhesion gt Cohesion

h is the height, R is the radius of the tube, q

is the angle of contact.

The weight of the fluid is balanced with the

vertical force caused by surface tension.

Surface Tension Capillary Action

Free Body Diagram for Capillary Action for a

Wetted Surface

Equating the two and solving for h

For clean glass in contact with water, q ? 0,

and thus as R decreases, h increases, giving a

higher rise. For a clean glass in contact with

Mercury, q ? 130, and thus h is negative or

there is a push down of the fluid.

Pressure

- Pressure is the force on an object that is spread

over a surface area. The - equation for pressure is the force divided by the

area where the force is applied. - Although this measurement is straightforward when

a solid is pushing on a - solid, the case of a solid pushing on a liquid or

gas requires that the fluid be - confined in a container. The force can also be

created by the weight of an - object.

Unit of pressure is Pa

Force Equilibrium of a Fluid Element

- Fluid static is a term that is referred to the

state of a fluid where its velocity is zero and

this condition is also called hydrostatic. - So, in fluid static, which is the state of fluid

in which the shear stress is zero throughout the

fluid volume. - In a stationary fluid, the most important

variable is pressure. - For any fluid, the pressure is the same

regardless its direction. As long as there is no

shear stress, the pressure is independent of

direction. This statement is known as Pascals

law

Force Equilibrium of a Fluid Element

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Standard Atmosphere

1 atm 101325 Pa

760 mmHg 760 torr

1 bar

h 76 cm

Mercury

Mercury barometer

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- A pressure is quoted in its gauge value, it

usually refers to a standard atmospheric pressure

p0. A standard atmosphere is an idealised

representation of mean conditions in the earths

atmosphere. - Pressure can be read in two different ways the

first is to quote the value in form of absolute

pressure, and the second to quote relative to the

local atmospheric pressure as reference. - The relationship between the absolute pressure

and the gauge pressure is illustrated in Figure

2.6.

- The pressure quoted by the latter approach

(relative to the local atmospheric pressure) is

called gauge pressure, which indicates the

sensible pressure since this is the amount of

pressure experienced by our senses or sensed by

many pressure transducers. - If the gauge pressure is negative, it usually

represent suction or partially vacuum. The

condition of absolute vacuum is reached when only

the pressure reduces to absolute zero.

Pressure Measurement

- Based on the principle of hydrostatic pressure

distribution, we can develop an apparatus that

can measure pressure through a column of fluid

(Fig. 2.7)

Pressure Measurement

We can calculate the pressure at the bottom

surface which has to withstand the weight of four

fluid columns as well as the atmospheric

pressure, or any additional pressure, at the free

surface. Thus, to find p5, Total fluid

columns (p2 p1) (p3 p2) (p4 p3) (p5

p4) p5 p1 ?og (h2 h1) ?wg (h3 h2)

?gg (h4 h3) ?mg (h5 h4) The p1 can be the

atmospheric pressure p0 if the free surface at z1

is exposed to atmosphere. Hence, for this case,

if we want the value in gauge pressure (taking

p1p00), the formula for p5 becomes p5 ?og

(h2 h1) ?wg (h3 h2) ?gg (h4 h3) ?mg

(h5 h4) The apparatus which can measure the

atmospheric pressure is called barometer (Fig

2.8).

- For mercury (or Hg the chemical symbol for

mercury), the height formed is 760 mm and for

water 10.3 m. - patm 760 mm Hg (abs) 10.3 m water (abs)
- By comparing point A and point B, the atmospheric

pressure in the SI unit, Pascal, - pB pA ?gh
- pacm pv ?gh
- 0.1586 13550 (9.807)(0.760)
- ? 101 kPa

- This concept can be extended to general pressure

measurement using an apparatus known as

manometer. Several common manometers are given

in Fig. 2.9. The simplest type of manometer is

the piezometer tube, which is also known as

open manometer as shown in Fig. 2.9(a). For

this apparatus, the pressure in bulb A can be

calculated as - pA p1 p0
- ?1gh1 p0

- Here, p0 is the atmospheric pressure. If a known

local atmospheric pressure value is used for p0,

the reading for pA is in absolute pressure. If

only the gauge pressure is required, then p0 can

be taken as zero.

Although this apparatus (Piezometer) is simple,

it has limitations, i.e. It cannot measure

suction pressure which is lower than the

atmospheric pressure, The pressure measured is

limited by available column height, It can only

deal with liquids, not gases. The restriction

possessed by the piezometer tube can be overcome

by the U-tube manometer, as shown in Fig. 2.9(b).

The U-tube manometer is also an open manometer

and the pressure pA can be calculated as

followed p2 p3 pA ?1gh1 ?2gh2

p0 ? pA ?2gh2 - ?1gh1 p0

If fluid 1 is gas, further simplification can be

made since it can be assumed that ?1 ? ? 2, thus

the term ?1gh1 is relatively very small compared

to ?2gh2 and can be omitted with negligible

error. Hence, the gas pressure is

pA ? p2 ?2gh2 - p0 There is also a closed

type of manometer as shown in Fig. 2.9(c), which

can measure pressure difference between two

points, A and B. This apparatus is known as the

differential U-tube manometer. For this case, the

formula for pressure difference can be derived as

followed p2 p3 pA ?1gh1

pB ?3gh3 ?2gh2 ? pA - pB ?3gh3

?2gh2 - ?1gh1

Piezometer tube

Open

h

U-tube manometer

Pressure is defined as a force per unit area -

and the most accurate way to measure low air

pressure is to balance a column of liquid of

known weight against it and measure the height of

the liquid column so balanced. The units of

measure commonly used are inches of mercury (in.

Hg), using mercury as the fluid and inches of

water (in. w.c.), using water or oil as the fluid

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Example

- An underground gasoline tank is accidentally

opened during raining causing the water to seep

in and occupying the bottom part of the tank as

shown in Fig. E2.1. If the specific gravity for

gasoline 0.68, calculate the gauge pressure at

the interface of the gasoline and water and at

the bottom of the tank. Express the pressure in

Pascal and as a pressure head in metres of water.

Use ?water 998 kg/m3 and g 9.81 m/s2.

- For gasoline
- ?g 0.68(998) 678.64kg/m3
- At the free surface, take the atmospheric

pressure to be zero, or p0 0 (gauge pressure). - p1 p0 pgghg 0 (678.64)(9.81)(5.5)
- 36616.02 N/m2 36.6 kPa
- The pressure head in metres of water is
- h1 p1 p0 36616.02 - 0
- pwg (998)(9.81)
- 3.74 m of water
- At the bottom of the tank, the pressure
- p2 p1 pgghg 36616.02 (998)(9.81)(1)
- 46406.4 N/m2 46.6 kPa
- And, the pressure head in meters of water is
- h2 p1 p0 46406.4 - 0
- pwg (998)(9.81)
- 4.74 m of water

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Strain and Strain (Shear) Rate

- Strain
- a dimensionless quantity representing the

relative deformation of a material - Normal Strain Shear Strain

Shear Stress is the intensity of force per unit

area, acting tang

Simple Shear Flow

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Solids Elastic and Shear Moduli

- When a solid material is exposed to a stress, it

experiences an amount of deformation or strain

proportional to the magnitude of the stress - Stress (?) ? Strain (? or ?)
- Stress Modulus ? Strain
- Normal stress elastic modulus (E)
- Shear stress shear modulus (G)

Fluid Viscosity

- Newtonian fluids
- viscosity is constant (Newtonian viscosity, ?)
- Non-Newtonian fluids
- shear-dependent viscosity (apparent viscosity, ?

or ?a)

Viscosity Introduction

The viscosity is measure of the fluidity of the

fluid which is not captured simply by density or

specific weight. A fluid can not resist a shear

and under shear begins to flow. The shearing

stress and shearing strain can be related with a

relationship of the following form for common

fluids such as water, air, oil, and gasoline

- is the absolute viscosity or dynamics viscosity

of the fluid, u is the velocity of the fluid and

y is the vertical coordinate as shown in the

schematic below

No Slip Condition

Viscosity

Viscosity is a property of fluids that indicates

resistance to flow. When a force is applied to a

volume of material then a displacement

(deformation) occurs. If two plates (area, A),

separated by fluid distance apart, are moved (at

velocity V by a force, F) relative to each

other,

Newton's law states that the shear stress (the

force divided by area parallel to the force, F/A)

is proportional to the shear strain rate . The

proportionality constant is known as the

(dynamic) viscosity

Shear stress

The unit of viscosity in the SI system of units

is pascal-second (Pa s)

In cgs unit , the unit of viscosity is expressed

as poise

Shear rate

1 poise 0.1 Pa s

1 cP 1 m Pa s

Example Shear stress in soybean oil

- The distance between the two parallel plates is

0.00914 m and the lower plate is being pulled at

a relative velocity of 0.366 m/s greater than the

top plate. The fluid used is soybean oil with

viscosity of 0.004 Pa.s at 303 K - Calculate the shear stress and the shear rate
- If water having a viscosity of 880x10-6 Pa.s is

used instead of soybean oil, what relative

velocity in m/s needed using the same distance

between plates so that the same shear stress is

obtained? Also, what is the new shear rate?

Kinematics Viscosity

Reynolds Number

Experiment for find Re

Laminar flow

Turbulent flow

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Example At what velocity does water flow

convert from laminar to transition flow in a 5

cm diameter pipe at 20 C

Properties of water _at_ 20 C

Milk is flowing at 0.12 m min in a 2.5-cm

diameter pipe. If the temperature of the milk is

21C, is the flow turbulent or streamline

3

-1

Given

Density of milk 1030

kg /m 3 Viscosity

2.12 cP

Velocity Profile in a Liquid Flowing under Fully

Developed Flow Condition

The velocity profile for a larminar , fully

develop flow in horizontal pipe is

Measurement of Viscosity

Viscosity of a liquid can be measurement

Viscosity Measurements

A Capillary Tube Viscosimeter is one method of

measuring the viscosity of the fluid. Viscosity

Varies from Fluid to Fluid and is dependent on

temperature, thus temperature is measured as

well. Units of Viscosity are Ns/m2 or lbs/ft2

Movie Example using a Viscosimeter

Capillary Tube Viscometer

Rotational Viscometer

Influence of Temperature on Viscosity

There is considerable evidence that the influence

of temperature on viscosity for liquid food may

be described by an Arrhenius type relationship

Energy Balance

In addition to the mass balance, the other

important quantity we must consider in the

analysis of fluid flow, is the energy balance.

Referring again to this picture , we shall

consider the changes in the total energy of unit

mass of fluid, one kilogram, between Section 1

and Section 2

There may be energy interchange with the

surroundings including (4) Energy lost to

the surroundings due to friction. (5)

Mechanical energy added by pumps. (6) Heat

energy in heating or cooling the fluid.

Firstly, there are the changes in the intrinsic

energy of the fluid itself which include changes

in (1) Potential energy. (2) Kinetic

energy. (3) Pressure energy.

Bernoulli Equation

Bernoullis equation is a consequence of

Conservation of Energy applied to an ideal fluid

Assumes the fluid is incompressible and

non-viscous, and flows in a non-turbulent,

steady-state manner

States that the sum of the pressure, kinetic

energy per unit volume, and the potential energy

per unit volume has the same value at all points

along a streamline

A 3 m diameter stainless steel tank contains

wine. In the tank , the wine is filled to 5 cm

depth. A discharge port, 10 cm diameter , is

opened to drain the wine. Calculate the discharge

velocity of wine ,assuming the flow is steady and

frictionless and the time required in emptying it

5 m

Location 2

Then volumetric flow rate from the discharge port

using

Water flows at the rate of 0.4 m3 / min in a 7.5

cm diameter pipe at a pressure of 70 kPa. If the

pipe reduces to 5 cm diameter calculate the new

pressure in the pipe

P 70 kPa D 7.5 cm A 4.42 x 10 -3 m 2

P ? kPa D 5 cm A 1.90 x 10 -3 m 2

P2 65.3 k Pa.

Forces due to Friction

When a fluid moves through a pipe or through

fittings, it encounters frictional resistance and

energy can only come from energy contained in the

fluid and so frictional losses provide a drain on

the energy resources of the fluid. The actual

magnitude of the losses depends upon the nature

of the flow and of the system through which the

flow takes place. In the system, let the energy

lost by 1 kg fluid between section 1 and section

2, due to friction, be equal to Eƒ (J).

Common parameter used in laminar and turbulent

flow is Fanning friction factor (ƒ) ƒ is defined

as drag force per wetted surface unit area

product of density

times velocity head

? ½?v2

Major loss

f

Type of pipe Roughness (?) Roughness (?)

Ft m

Riveted steel 0.003-0.03 0.0009-0.009

Concrete 0.001-0.01 0.0003-0.003

Wood stave 0.0006-0.003 0.0002-0.0009

Cast iron 0.00085 0.00026

Galvanized iron 0.0005 0.00015

Asphalted cast iron 0.0004 0.0001

Commercial steel or wrought iron 0.00015 0.000046

Drawn brass or copper tubing 0.000005 0.0000015

Glass and plastic smooth smooth

Moody Diagram for the Fanning friction factor

Moody Diagram for the Moody friction factor

Laminar Zone

Turbulent Zone

Laminar

Lamina Zone

Turbulent Zone

Transition

Water at 30 C is being pumped through a 30 m

section of 2.5 cm. diameter steel pipe at a mass

flow rate 2.5 kg/s. Compute the pressure loss due

to friction in pipe section

Re

240.08 kPa

A liquid is flowing through a horizontal straight

commercial steel pipe at 4.57 m/s. The inside

diameter of the pipe is 2.067 in. The viscosity

of the liquid is 4.46 cP and the density 801

kg/m3. Calculate the mechanical-energy friction

loss in J/kg for a 36.6 m section of pipe. Given

roughness of commercial steel pipe is 4.6 X

10-5 m.

For Non-circular pipe

4(Cross-sectional area)

4 Hydraulic radius

Wetted Perimeter

0.875 m

Water at 30 C is being pumped through a 30 m

section of annular area of steel pipe at a mass

flow rate 2 kg/s. Compute the pressure loss due

to friction in pipe section

2.5 cm

5 cm

Determine Equivalent Diameter

0.025 m

Determine Reynolds Number and

Determine Friction factor

240.08 kPa

Energy Equation for Steady Flow of Fluids

Frictional Energy Loss

The frictional energy loss for a liquid flowing

in pipe is composed of major and minor loses

The minor losses are due to various components

used in pipeline system such as values , tees

and elbow and contraction of fluid

The major losses are due to the flow of viscous

liquid in the straight portions of a pipe

General Equation for Minor Losses

hLm minor loss K minor loss coefficient Le

equivalent length

Expansion and Contraction

Transition Loss Equations

- Contraction
- Expansion

Pipe Bends

Friction loss of turbulent flow through valves

and fittings

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Example Friction losses and mechanical-energy

balance

- An elevated storage tank contains water at

82.2ºC as shown in figure. It is desired to have

a discharge rate at point 2 of 6.315 X 10-3 m3/s.

What must be the height H in m of the surface of

the water in the tank relative to the discharge

point? The pipe used is commercial steel pipe.

Pump

- Power and work
- Using the total mechanical-energy-balance

equation on pump and piping system, the actual or

theoretical mechanical energy (Ws) added to fluid

by the pump can be calculated. - If ? fractional efficiency and Wp the shaft

work delivered to the pump - Wp Ws/ ?
- The mechanical energy Ws in J/kg added to fluid

often expressed as the developed head (H) of pump

in m of fluid - Wp H.g.mº/ ?

Suction lift and cavitation

- Power can be calculated from difference in

pressure between discharge and suction. - Practically, lower limit of suction pressure is

fixed by vapor pressure (corresponding to

temperature at suction). If suction pressure is

equal to vapor pressure, liquid flashes into

vapor. This process call cavitation that can

occur in suction line and no liquid can be drawn

into pump. This process can cause severe erosion

and mechanical damage to pump. - Therefore, it should have net positive suction

head (NPSH)

Net positive suction head (NPSH)

- NPSH positive difference between pressure at

pump inlet and vapor pressure of liquid being

pumped to prevent cavitation. - NPSH
- required NPSH function of impeller design (its

value is provided by manufacturer) - available NPSH function of suction system.
- NPSHa ha hvp hs hf
- Where ha absolute pressure
- hvp vapor pressure
- hs static head of liquid above center line

of pump - hf friction loss

Centrifugal pump

- Pressure developed by rotating impeller.
- Impeller impact a centrifugal force on liquid

entering the center of impeller. - Affinity laws govern performance of

centrifugal pumps at various impeller speeds - Vº2 Vº1(N2 /N1)
- h2 h1(N2 /N1)2
- P2 P1(N2 /N1)3
- Where Vº volumetric flow rate
- h total head
- P power

Positive displacement

- Constant discharge pressure at difference flow

rates. - Regulation of flow rate is done by changing

displacement or capacity of intake chamber

Characteristic curve

- plot head, power consumption and efficiency with

respect to volumetric flow rate (capacity) - use for rating pumps

Pipe flow problems

- Basically, there are 3 types of problems
- Direct solution finding h or ?P for given L, D,

v, f - Finding v or Q for given L, D, h or ?P, ?/D
- Trial and error type solution
- Use Re.f1/2 constant (not depend on v or Q)
- Finding D for given L, Q, h or ?P, ?/D
- Trial and error type solution
- Use Re.f1/5 constant (not depend on D)

Assumption V

Trial and error type solution For finding v

Determine Re

Determine f

Determine V

YES

NO

Example Trial and error type solution For

finding v

2

Constant level

5 m

1

12 m

1 Equation 2 Variable cannot determine

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Whole milk flows through a horizontal stainless

steel pipe ( ? 4.2 x 10-6 m) , 30 m long and

having inside diameter D 2.54 cm . Determine

the flow rate of milk when a 2 HP motor with 75

efficiency is used.

Given

Density of whole milk 1030 kg /m

Viscosity of whole milk 0.00202 Pa s

1

2

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Flow Measurement

Flow measurement is essential in many industries

such as the oil, power, chemical, food, water,

and waste treatment industries. These industries

require the determination of the quantity of a

fluid, either gas, liquid, or steam, that passes

through a check point, either a closed conduit or

an open channel, in their daily processing or

operating. The quantity to be determined may be

volume flow rate, mass flow rate, flow velocity,

or other quantities related to the previous

three. These methods include (a) Pitot tube, (b)

Orifice plate and (c) Venturi tube are the

measurement involves pressure difference.

Differential pressure flow meters employ the

Bernoulli equation that describes the

relationship between pressure and velocity of a

flow. These devices guide the flow into a section

with difference cross section areas (different

pipe diameters) that causes variations in flow

velocity and pressure. By measuring the changes

in pressure, the flow velocity can then be

calculated. Many types of differential pressure

flow meters are used in the industry.

The Pitot Tube

The Pitot tube is a widely used sensor to measure

velocity of fluid

Total pressure

The principle is based on the Bernoulli Equation

where each term can be interpreted as a form of

pressure

Air

Static Pressure

Stagnation point (v 0)

Water

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The Orifice Meter

A flat plate with an opening is inserted into

the pipe and placed perpendicular to the flow

stream. As the flowing fluid passes through the

orifice plate, the restricted cross section area

causes an increase in velocity and decrease in

pressure. The pressure difference before and

after orifice plate is used to calculate the flow

velocity.

D1

D2

PA - PB

The Venturi Meter

To reduce energy loss due to friction created by

the sudden contraction in flow in an orifice

meter

Variable-Area Meter

Variable area flow meter 's cross section area

available to the flow varies with the flow rate.

Under a (nearly) constant pressure drop, the

higher the volume flow rate, the higher the flow

path area.