Fluid Flow

Outline• Fundamentals and applications of rheology• Shear stress and shear rate• Viscosity and types of viscometers• Rheological classification of fluids• Apparent viscosity• Effect of temperature on viscosity• Reynolds number and types of flow• Flow in a pipe• Volumetric and mass flow rate• Friction factor (in straight pipe), friction coefficient (for fittings,

expansion, contraction), pressure drop, energy loss• Pumping requirements (overcoming friction, potential energy,

kinetic energy, pressure energy differences)2

Fundamentals of Rheology

• Rheology is the science of deformation and flow– The forces involved could be tensile, compressive, shear

or bulk (uniform external pressure)• Food rheology is the material science of food

– This can involve fluid or semi-solid foods• A rheometer is used to determine rheological

properties (how a material flows under different conditions)– Viscometers are a sub-set of rheometers

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Applications of Rheology• Process engineering calculations

– Pumping requirements, extrusion, mixing, heat transfer, homogenization, spray coating

• Determination of ingredient functionality– Consistency, stickiness etc.

• Quality control of ingredients or final product– By measurement of viscosity, compressive strength etc.

• Determination of shelf life– By determining changes in texture

• Correlations to sensory tests– Mouthfeel

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Stress and Strain• Stress: Force per unit area

(Units: N/m2 or Pa)• Strain: (Change in dimension)/(Original dimension)

(Units: None)• Strain rate: Rate of change of strain

(Units: s-1)• Normal stress: [Normal (perpendicular) force] / [Area]• Shear stress: [Shear (parallel) force] / [Area]

– Units: Pa

• Shear rate: Abbreviation for shear strain rate– It is the velocity gradient (du/dx) in many cases– Units: s-1

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Shear Forces

Max resistance Min resistance

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Shear Rate

• Max shear rate for fluid flow in small gaps– (Velocity difference)/(Gap thickness)

• Max shear rate for brushing/spreading a paste– (Velocity of brush)/(Thickness of paste)

• Max shear rate for fluid flow in large gaps– (Mid-point velocity – End-point velocity)/(Half height)– (Maximum velocity)/(Height)

• Max shear rate for pipe flow– 4V/R3

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Shear Stress

• In general,– Shear stress = (shear rate)– Shear stress = (velocity gradient)

• In a rotational viscometer,– Max shear stress

• Torque/(2R2L)

• For pipe flow,– Max shear stress

• 4V/R3 OR 4u/R.

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Shear Stress & Shear Rate for Pipe Flow

• Shear stress– Maximum at wall– Zero at center of pipe

• Velocity– Zero at wall (under ideal “no-slip” conditions)– Maximum at center of pipe

• Shear rate– Maximum at wall– Zero at center of pipe

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Shear Rate for Various ProcessesProcess Shear Rate (s-1)

Sedimentation 10-6 to 10-3

Leveling 10-2 to 10-1

Extrusion 100 to 103

Pumping 100 to 103

Chewing/Brushing 101 to 102

Stirring/Mixing 101 to 103

Curtain coating 102 to 103

Rubbing/Spraying 103 to 105

Lubrication (bearings) 103 - 107

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Viscosity () -- Newtonian Fluids• A measure of resistance to flow

– Ratio of shear stress () to shear rate ()• SI Units: Pa s

– cgs units: centipoise (cP); 1 cP = 0.001 Pa s• Viscosity () of water at 20 °C = 1 cP

– Viscosity of water dec. by ~3% for every 1 °C inc. in temp.• Measurement of viscosity

– Tube viscometer (Cannon-Fenske)– Rotational viscometer (Brookfield, Haake)– Empirical technique (Bostwick consistometer)

Where is viscosity used?NRe = (Density) (Avg. vel.) (Diameter) / (Viscosity)Reynolds number determines flow type: Laminar/Turbulent

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Flow Behavior for Time-Independent Fluids(Herschel-Bulkley Model for Shear Stress vs. Shear Rate)

= Shear stress (Pa)

0 = Yield stress (Pa)

= Shear rate (s-1)

K = Consistency coeff.(Pa sn)

n = Flow behavior index

Newtonian0 = 0, n = 1Then, K =

.n < 1

n > 1

n = 1

Yield stress

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Herschel-Bulkley Model: Power-law Model: 12

Examples of Types of Fluids(Herschel-Bulkley Model)

• Newtonian: Water, clear fruit juices, milk, honey, vegetable oil, corn syrup

• Shear thinning (Pseudoplastic): Applesauce, banana puree, orange juice concentrate, French mustard, dairy cream

• Dilatant: Some types of honey, 40% raw corn starch solution

• Bingham plastic: Tomato paste, toothpaste• Herschel-Bulkley: Minced fish paste, raisin paste

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Apparent Viscosity: Non-Newtonian Fluids(Power-Law Fluids)

• For Newtonian fluids, the ratio of shear stress to shear rate is independent of the magnitude of shear rate– This ratio of shear stress to shear rate is called viscosity ()

• Eg., The viscosity of water at 20 °C is 0.001 Pa s

• For power-law fluids (shear thinning, dilatant), the ratio of shear stress to shear rate is dependent on the magnitude of shear rate– This ratio of shear stress to shear rate is called the apparent

viscosity (app); app = / = K n/ = K n-1

– The magnitude of apparent viscosity MUST be accompanied with the magnitude of shear rate

• Eg., The apparent viscosity of fluid ‘A’ at 20 °C is 20 Pa∙s at a shear rate of 25 s-1

....

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Apparent Viscosity (contd.)For pseudoplastic and dilatant fluids,

For pseudoplastic fluids, app decreases with an increase in shear rateFor dilatant fluids, app increases with an increase in shear rateNote: For pseudoplastic & dilatant fluids, app & do NOT change with time

Single point apparent viscosity: Human perception of thickness of a fluid food is correlated to app at 60 s-1

. ..

(Pseudoplastic Fluid) (Dilatant Fluid)

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Time Dependent Fluids• Thixotropic fluids

– Exhibit a decrease in shear stress (and app) over time at constant shear rate

• Eg., starch-thickened baby foods, yogurt, condensed milk, mayonnaise, egg white

• Rheopectic fluids– Exhibit an increase in shear stress (and app) over

time at constant shear rate• Eg., Whipping cream, lubricants, printer’s inks

• Thixotropy and rheopecty may be reversible or irreversible

Time

Shea

r St

ress

Shea

r R

ate

Thixotropy

Constant shear rate

Shea

r St

ress

Shear Rate

Thixotropy

Rheopecty

Step inc. in shear rateInc. & dec. in shear rate

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Effect of Temperature on Viscosity

Arrhenius equation for Newtonian fluids:

Temperature

Vis

cosi

ty

: Viscosity at temperature, T (Pa s)1: Viscosity at temperature, T1 (Pa s)2: Viscosity at temperature, T2 (Pa s)BA: Arrhenius const. or frequency factor (Pa s)Ea: Activation energy for viscous flow (J/kg mol)R: Universal gas constant

(= 8.314 J/mol K = 8314 J/kg K)T: Temperature (must be in K; °C NOT okay)

Determining Ea & BA:ln () = ln (BA) + Ea/RTPlot ln () versus 1/TSlope = Ea/R; intercept = ln (BA)Ea = R * SlopeBA = eintercept

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Rotational Viscometer (Newtonian Fluid)• Principle

– Measure torque [a measure of shear stress (in Pa] versus rpm [a measure of shear rate (in s-1].

T: Torque (N∙m)N: Revolutions per second (s-1)L: Spindle length (m)Ri, Ro: Radius of spindle, cup resp. (m)

Plot “T” on y-axis versus “N” on x-axis. The slope of this graph is “82L/[1/Ri

2 – 1/Ro2]”. Obtain from this.

nist.gov18

Tube Viscometer

• Principle– Measure pressure drop (P) versus volumetric

flow rate (V) across a straight section of tube of length, L and radius, R

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Plot P on y-axis vs. V on x-axis

Slope = (8 L) / ( R4)

Obtain from slope

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.UnitsR, L: mP: PaV: m3/s: Pa∙s

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Tube Viscometer (contd.)• Capillary tube

– When gravitational force provides the pressure in a (capillary) tube viscometer, ΔP = ρgL

‘t’ is the time taken for a certain mass ‘m’ of the fluid to traverse distance, ‘L’ of the tube

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Bostwick Consistometer

Compartment: 5 x 5 x 3.8 cmInclined trough: Graduated (5 cm x 24 cm)Spring loaded gateHow far does the product travel in 30 s?Good for Quality control

foodqa.byu.edu

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Usefulness of Viscometric Data

• How can information from rotating a spindle in a beaker of fluid translate to practical situations?

• First step would be to determine for a Newtonian fluid or K & n (and app at shear rate of interest) for a non-Newtonian fluid

• Then, you can determine pumping power required to pump a fluid in a pipeline at a given flow rate

• You can also determine the uniformity of processing based on the velocity profile during pipe flow

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Factors Affecting Type of Flow in a Pipe

• What factors affect if flow in a pipe is going to be steady/streamlined OR erratic/random?– System parameter

• Diameter of pipe

– Process parameter• Mass flow rate of product

– Product property• Viscosity

The above 3 parameters are grouped into ONE dimensionless (unitless) quantity called Reynolds Number (NRe) and the magnitude of this number can be used to determine if the flow will be steady or erratic.

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Reynolds Number (for Newtonian Fluids)

dh = Hydraulic diameter = 4 (Across-section)/(Wetted perimeter)= Inside diameter of pipe (D) for flow in a pipe

= dio – doi for flow in an annulus (dio is the inside diameter of the outer pipe and doi is the outside diameter of the inner pipe)

For flow inside a pipe of diameter, D:

: Density of fluid (kg/m3)u: Average velocity of fluid (m/s): Viscosity of fluid (Pa s)

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Reynolds Number and Types of Flow

• Laminar flow: NRe < 2,100– Steady, streamlined flow

• Transitional flow: 2,100 < NRe < 4,000– Neither steady nor completely erratic or random

• Turbulent flow: NRe > 4,000– Erratic, random flow

dh = Hydraulic diameter = 4 (Across-section)/(Wetted perimeter)= Inside diameter of pipe for flow in a pipe= dio – doi for flow in an annulus (dio is the inside diameter of the outer pipe and doi is the outside diameter of the inner pipe)

: Density of fluid (kg/m3)u: Average velocity of fluid (m/s): Viscosity of fluid (Pa s)

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Significance of Reynolds Number

• It is the ratio of inertial forces and viscous forces• The magnitude gives us an indication of which

forces dominate– High NRe => Inertial forces dominate over viscous forces

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Reynolds Number (for Power-Law Fluids)

• NGRe: Generalized Reynolds number• K: Consistency coefficient (Pa∙sn)• n: Flow behavior index• : Density of fluid (kg/m3)• u: Average velocity of fluid (m/s)• dh: Hydraulic diameter (m)

The critical Reynolds number [NRe(critical)], beyond which flow is no longer laminar, is given by:

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Poiseuille Flow(Pressure Driven Flow in a Cylindrical Pipe)

• For laminar flow of Newtonian fluids in a circular conduit of radius, R:

• For laminar flow of power-law (shear thinning or shear thickening) fluids in a circular conduit of radius, R:

r: Radial distance from center (m)u: Velocity at radial distance, r (m/s)u: Average velocity (m/s)

r: Radial distance from center (m)u: Velocity at radial distance, r (m/s)n: Flow behavior indexu: Average velocity (m/s)

This equation translates to a parabolic velocity profile

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Velocity Profiles and their ImplicationNewtonianLaminar(n = 1)

DilatantLaminar(n > 1)

PseudoplasticLaminar(n < 1)

umax > 2 u

umax < 2 u

umax = 2 u

NewtonianTurbulent

(n = 1)umax = 1.2 u

Example: Heating of a fluid food product in an indirect contact heat exchanger with the fluid food flowing in the inside tube and hot water flowing through the outside tube.

Fluid streams close to the center of the pipe flow the fastest and also heat up the slowest due to their distance from hot water. Thus, they receive minimal heat treatment.

Fluid streams close to the wall of the pipe flow slowest and also heat up the fastest due to their proximity to hot water. Thus, they receive maximum heat treatment.

Greater the velocity difference between fluid streams at the center and wall, greater the quality difference in the food. Thus, flatter the velocity profile, more uniform the quality.

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Entrance and Exit Effects• As a fluid enters a pipe from a reservoir, the velocity profile is flat

(plug flow)• It then develops to a parabolic profile (for laminar flow of a

Newtonian fluid) after a certain distance– This distance is called the entrance length (Le)– A similar effect exists towards the discharge end of the pipe

• The flow in this region is not representative of flow in the rest (majority) of the system

• Thus, pressure gauges, flow meters etc are not placed in close proximity to entrances, exits, bends, valves etc.

Le / D = 0.06 NRe for Laminar flowLe / D = 4.40 NRe

1/6 for Turbulent flow

D: Pipe diameter30

Flow RatesVolumetric flow rate

Mass flow rate

V: Volumetric flow rate of product (m3/s)m: Mass flow rate of product (kg/s): Density of fluid (kg/m3)u: Average velocity of fluid (m/s)A: Cross-sectional area of pipe (m2)[A = R2 or D2/4 for circular pipesR: Radius of pipe (m), D: Diameter of pipe (m)]

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Effect of Change in Pipe Diameter on Flow Rate and Velocity

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Based on law of conservation of mass, the mass flow rate (and volumetric flow rate) of the fluid must be the same in both pipes

Thus, This simplifies to:

Hence, Since

D1 D2

Flow rate:Same in both pipes

Average velocity:Lower in pipe #2

Flow

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Friction in Pipes

• As a fluid flows in a straight pipe, it experiences friction due to the static wall

• As a fluid encounters fittings (such as valves, elbows, tees etc.), it experiences friction

• As a fluid encounters a change in area, it experiences friction

• The magnitude of frictional resistance and the associated pressure drop in the fluid depends on various system parameters and properties of the fluid

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Fanning Friction Factor (f)Laminar flowf = 16/NRe

Transitional flowf is determined from the Moody diagram (graph of f, NRe, /D)

Turbulent flow

is the average height of the roughness in a pipe (m)D is the diameter of the pipe (m)

For turbulent flow, f can also be determined from the Moody diagram (graph of f, NRe, /D)

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Roughness of Pipe ()

View under microscope

View under naked eye

is the average height of the roughness of the pipe

is greater for rough pipes such as wrought iron pipes and lesser for smooth pipes such as copper pipes

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Moody Diagram

= 259 x 10-6 m for cast iron; 1.5235 x 10-6 m for drawn tubing;152 x 10-6 m for galvanized iron; 45.7 x 10-6 m for steel or wrought iron

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Friction Coefficient (Cff) for Fittings• Regular 90° elbow (flanged) 0.3• Regular 90° elbow (threaded) 1.5• Branch flow tee (flanged) 1.0• Line flow tee (flanged) 0.2• Ball valve (1/3 closed) 5.5• Ball valve (2/3 closed) 210• Ball valve (open) 0.05• Diaphragm valve (open) 2.3• Diaphragm valve (1/2 closed) 4.3• Gate valve (1/2 closed) 2.1• Gate valve (open) 0.15• Globe valve (open) 10• Swing check valve (forward flow) 2.0

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Friction Coefficient for Expansion (Cfe)

A1 = Cross-sectional area of smaller pipe= R1

2 (for a pipe of circular cross-section)A2 = Cross-sectional area of larger pipe

= R22 (for a pipe of circular cross-section)

Flow

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Friction Coefficient for Contraction (Cfc)

A1 = Cross-sectional area of larger pipe= R1

2 (for a pipe of circular cross-section)A2 = Cross-sectional area of smaller pipe

= R22 (for a pipe of circular cross-section)

Flow

12

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Pressure Drop and Energy Loss due to Friction

• As a fluid flows in a pipe, it encounters friction due to the straight section of pipe, fittings (such as valves, elbows, and tees), and changes in area of pipe

• This friction manifests itself in the form of pressure drop and an associated loss in energy within the fluid in the pipe

• This loss in energy must be overcome by a pump in order to move the fluid through the pipe

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Ways of Expressing Pressure

• Gauge pressure– The pressure that a gauge displays

• Absolute pressure– Gauge pressure + atmospheric pressure

• Units of pressure– atm, bar, Pa, mm Hg, psi– 1 atm = 1.01325 bar = 101.325 kPa = 760 mm Hg =

10.36 m or 33.9 ft H2O = 14.696 psi

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Pressure Drop (P) and Energy Loss (Ef) due to Straight Pipe

Ef: Energy loss due to friction (J/kg or m2/s2)P: Pressure drop (Pa): Density of fluid (kg/m3)f: Friction factoru: Average velocity of fluid (m/s)L: Length of pipe (m)D: Diameter of pipe (m)

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Pressure Drop (P) and Energy Loss (Ef) due to Fittings

Ef: Energy loss due to friction (J/kg or m2/s2)P: Pressure drop (Pa): Density of fluid (kg/m3)Cff: Friction coefficientu: Average velocity of fluid (m/s)

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Pressure Drop (P) and Energy Loss (Ef) due to Expansion

Ef: Energy loss due to friction (J/kg or m2/s2)P: Pressure drop (Pa): Density of fluid (kg/m3)Cfe: Friction coefficient for expansionu1: Average velocity of fluid in smaller pipe (m/s)A1 = Cross-sectional area of smaller pipe (m2)

= R12 (for a pipe of circular cross-section)

A2 = Cross-sectional area of larger pipe (m2)= R2

2 (for a pipe of circular cross-section)

Flow

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Pressure Drop (P) and Energy Loss (Ef) due to Contraction

Ef: Energy loss due to friction (J/kg or m2/s2)P: Pressure drop (Pa): Density of fluid (kg/m3)Cfc: Friction coefficient for contractionu2: Average velocity of fluid in smaller pipe (m/s)A1 = Cross-sectional area of larger pipe (m2)

= R12 (for a pipe of circular cross-section)

A2 = Cross-sectional area of smaller pipe (m2)= R2

2 (for a pipe of circular cross-section)

Flow

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Pr. Drop (P) and Energy Loss (Ef) due to Straight Pipe, Fittings, and Change in Area

For a straight pipe

For fittings such as elbows, valves, tees

For an expansion in pipe diameter

For a contraction in pipe diameter

P: Pressure drop due to friction (Pa)Ef: Energy loss due to friction (J/kg or m2/s2)f: Fanning friction factor for straight pipeCff: Friction coefficient for fittingsCfe: Friction coefficient for expansion in pipeCfc: Friction coefficient for contraction in pipe

u: Average velocity in pipe (m/s)u1: Average velocity in pipe #1 (m/s)u2: Average velocity in pipe #2 (m/s)L: Length of pipe (m)D: Diameter of pipe (m)

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Pressures and Temperatures in a Heat-Hold-Cool System

Indirect Heating Indirect Cooling

Product

Hot water Cold water

Filling at atmospheric

pressure

140 °C15 psi

20 °C20 psi

30 °C0 psi

135 °C10 psi

35 °C5 psi

Pgauge = 5 psiPabs = 34.475 psi

= 135.8 kPa= Max. Psat

From steam tables,Tsat = 108.3 °C

Holding

108.3 °C

T > Tsat in the dotted boxProduct could “flash” or boil in this region!Increase back pressure to prevent thisFor 140 °C, Pabs ≥ 316.3 kPa or 45.9 psi For 140 °C, Pgauge ≥ 215 kPa or 31.2 psi 47

External Back Pressure to be Applied?• In a heat-hold-cool-fill system, the pressure drop from the

end of the holding section to the discharge end at packagingshould be sufficient to prevent product boiling (flashing)– The existing pressure drop in the system can be calculated using:

Straight pipes Fittings Expansion Contraction

• From saturated steam tables,• When Tsat = 100 °C, Psat = 101.35 kPa• When Tsat = 140 °C, Psat = 316.3 kPa• Thus, to prevent flashing when Tprod at end of hold tube = 140 °C, we

need to ensure a Preqd of at least “316.3 – 101.35” kPa (plus a factor of safety) in above equation

• If the P calculated using the above equation < Preqd, additional pressure must be added to prevent product boiling (flashing)

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Bernoulli’s EquationAs a fluid flows in a pipe, Bernoulli’s equation for any two points (“1” and “2”) along a stream line is given by:

P: Pressure (Pa): Density (kg/m3)u: Velocity (m/s)z: Height (m)g: Acceleration due to gravity (m/s2)Assumptions:Fluid is incompressible (density is constant)Flow is inviscidFlow is steadyNo shaft work is done by or on the fluidNo heat transfer takes place between the fluid and surroundings

P2 – P1 = Pg (z2 – z1): PE½ (u2

2 – u12): KE

PE: Potential EnergyKE: Kinetic energy

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Pumping CapacityWhen a pump is used to pump a fluid from point “1” to “2”, the energy to be supplied (Ep) by the pump is given by:

Ep = Ef + PE + KE + P/This can be expanded as follows:

Ef: Energy loss due to friction (J/kg or m2/s2)u: Average velocity (m/s) = 0.5 for laminar flow; = 1.0 for turbulent flowP: External pressure difference between points “1” and “2”: Density of fluid (kg/m3)The power (P) to be supplied by the pump is given by: P = m Ep

If is the efficiency of the pump, then reqd. pump rating = (m Ep)/

..

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Summary• Rheological properties are important in determining

process outcome– How much power is needed for pumping a fluid?

• Rheological characterization of materials begin with a shear stress versus shear rate graph

• Majority of fluid foods can be described by the Herschel-Bulkley model– Thixotropy/rheopexy may be important for some fluid

foods and processes• The viscosity of most fluids decays exponentially

with an increase in temperature (Arrhenius model)51

Summary (contd.)• Ways to quantify flow behavior of fluids

– Rotational viscometer (Torque versus rpm)– Tube viscometer (Pressure drop versus volumetric flow rate)– Bostwick consistometer (Good for quality control)

• Apparent viscosity– Ratio of shear stress and shear rate (for non-Newtonian fluids)

• Determination of Reynolds number in various sections of a system is important

– Flow may be laminar in one section & turbulent in another section• The curvature of the velocity profile (flatter or more curved than

the parabolic profile for laminar Newtonian flow) depends on the magnitude of the flow behavior index

• Entrance and exit effects need to be factored while introducing flow meters, pressure gauges etc in a pipeline 52

Summary (contd.)• Friction is associated with straight pipes (f), fittings (Cff),

expansion in pipe (Cfe), and contraction in pipe (Cfc)• Friction factor

– Laminar flow: f = 16/NRe

– Transition & turbulent flow: Moody diagram• Friction coefficient (Cff) for fittings

– Determined from tables• Friction coefficient for expansion & contraction

– Determined from empirical correlations• Friction manifests itself in the form of pressure drop (P) and

an associated loss in energy (Ef)• These losses are additive and need to be overcome (along with

any PE, KE, and external pressure energy differences) to pump a fluid from one point in a system to another

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