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) 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
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 Stress and Strain Stress: Force per unit area (Units: N/m 2 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 Shear Forces Max resistance Min resistance
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/ R 3 Shear Stress In general, Shear stress = (shear rate) Shear stress = (velocity gradient) In a rotational viscometer, Max shear stress Torque/(2 R 2 L) For pipe flow, Max shear stress 4V/ R 3 OR 4u/R 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
Shear Rate for Various Processes Process Shear Rate (s -1 ) Sedimentation 10-6 to 10-3 Leveling 10-2 to 10-1 Extrusion 10 0 to 10 3 Pumping 10 0 to 10 3 Chewing/Brushing 10 1 to 10 2 Stirring/Mixing 10 1 to 10 3 Curtain coating 10 2 to 10 3 Rubbing/Spraying 10 3 to 10 5 Lubrication (bearings) 10 3-10 7 Viscosity ( ) -- Newtonian Fluids A measure of resistance to flow Ratio of shear stress ( ) to shear rate ( ) Units: Pa s or centipoise (cp) 1 cp = 0001 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? N Re = (Density) (Avg vel) (Diameter) / (Viscosity) Reynolds number determines flow type: Laminar/Turbulent Flow Behavior for Time-Independent Fluids (Herschel-Bulkley Model for Shear Stress vs Shear Rate) Yield stress n < 1 n = 1 n > 1 = Shear stress (Pa) 0 = Yield stress (Pa) = Shear rate (s -1 ) K = Consistency coeff (Pa s n ) n = Flow behavior index Newtonian 0 = 0, n = 1 Then, K = Herschel-Bulkley Model: Power-law Model:
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 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 0001 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 Apparent Viscosity (contd) For pseudoplastic and dilatant fluids, For pseudoplastic fluids, app decreases with an increase in shear rate For dilatant fluids, app increases with an increase in shear rate Note: For pseudoplastic & dilatant fluids, app & do NOT change with time (Pseudoplastic Fluid) (Dilatant Fluid) Single point apparent viscosity: Human perception of thickness of a fluid food is correlated to app at 60 s -1
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 Shear Stress Thixotropy Rheopecty Shear Stress Shear Rate Constant shear rate Thixotropy Shear Rate Time Effect of Temperature on Viscosity Arrhenius equation for Newtonian fluids: Viscosity Temperature 1 : Viscosity at temperature, T 1 (Pa s) 2 : Viscosity at temperature, T 2 (Pa s) B A : Arrhenius const or frequency factor (Pa s) E a : Activation energy for viscous flow (J/kg mol) R: Universal gas constant (= 8314 J/mol K) T: Temperature (K; C NOT okay) Determining E a & B A : ln ( ) = ln (B A ) + E a /RT Plot ln ( ) versus 1/T Slope = E a /R; intercept = ln (B A ) E a = R * Slope B A = e intercept 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) R i, R o : Radius of spindle, cup resp (m) nistgov Plot T on y-axis versus N on x-axis The slope of this graph is 8 2 L /[1/R i2 1/R o2 ] Obtain from this
Tube Viscometer Principle Measure pressure drop ( P) versus volumetric flow rate (V) across a straight section of tube of length, L and radius, R Units R, L: m P: Pa V: m 3 /s : Pa s Plot P on y-axis vs V on x-axis Slope = (8 L ) / ( R 4 ) Obtain from slope 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 Bostwick Consistometer Step 1 Step 2 Step 3 Step 4 Step 5 foodqabyuedu Compartment: 5 x 5 x 38 cm Inclined trough: Graduated (5 cm x 24 cm) Spring loaded gate How far does the product travel in 30 s? Good for Quality control
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 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 clubbed into ONE dimensionless (unitless) quantity called Reynolds Number (N Re ) and the magnitude of this number can be used to determine if the flow will be steady or erratic Reynolds Number (for Newtonian Fluids) : Density of fluid (kg/m 3 ) u: Average velocity of fluid (m/s) : Viscosity of fluid (Pa s) d h = Hydraulic diameter = 4 (A cross-section )/(Wetted perimeter) = Inside diameter of pipe (D) for flow in a pipe = d io d oi for flow in an annulus (d io is the inside diameter of the outer pipe and d oi is the outside diameter of the inner pipe) For flow inside a pipe of diameter, D:
Reynolds Number and Types of Flow : Density of fluid (kg/m 3 ) u: Average velocity of fluid (m/s) : Viscosity of fluid (Pa s) d h = Hydraulic diameter = 4 (A cross-section )/(Wetted perimeter) = Inside diameter of pipe for flow in a pipe = d io d oi for flow in an annulus (d io is the inside diameter of the outer pipe and d oi is the outside diameter of the inner pipe) Laminar flow: N Re < 2,100 Steady, streamlined flow Transitional flow: 2,100 < N Re < 4,000 Neither steady nor completely erratic or random Turbulent flow: N Re > 4,000 Erratic, random flow 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 N Re => Inertial forces dominate over viscous forces Reynolds Number (for Power-Law Fluids) N GRe : Generalized Reynolds number K: Consistency coefficient (Pa s n ) n: Flow behavior index : Density of fluid (kg/m 3 ) u: Average velocity of fluid (m/s) d h : Hydraulic diameter (m) The critical Reynolds number [N Re(critical) ], beyond which flow is no longer laminar, is given by:
Poiseuille Flow (Pressure Driven Flow in a Cylindrical Pipe) For laminar flow of Newtonian 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) This equation translates to a parabolic velocity profile 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) n: Flow behavior index u: Average velocity (m/s) Velocity Profiles and their Implication Newtonian Laminar (n = 1) u max = 2 u Newtonian Turbulent (n = 1) u max = 12 u Dilatant Laminar (n > 1) u max > 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 Pseudoplastic Laminar (n < 1) u max < 2 u Greater the velocity difference between fluid streams at the center and wall, greater quality difference in the food Thus, flatter the velocity profile, more uniform the quality 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 (L e ) A similar effect exists towards the discharge end of the pipe These effects are the reasons why pressure gauges, flow meters etc are not placed in close proximity to entrances, exits, bends, valves etc L e / D = 006 N Re for Laminar flow L e / D = 440 N Re 1/6 for Turbulent flow
Flow Rates Volumetric flow rate Mass flow rate V: Volumetric flow rate of product (m 3 /s) m: Mass flow rate of product (kg/s) : Density of fluid (kg/m 3 ) u: Average velocity of fluid (m/s) A: Cross-sectional area of pipe (m 2 ) [A = R 2 or D 2 /4 for circular pipes R: Radius of pipe (m), D: Diameter of pipe (m)] Effect of Change in Pipe Diameter on Flow Rate and Velocity 1 D 1 D 2 2 Flow rate: Same in both pipes Average velocity: Lower in pipe #2 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 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
Fanning Friction Factor (f) Laminar flow f = 16/N Re Transitional flow f is determined from the Moody diagram (graph of f, N Re, /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, N Re, /D) Moody Diagram Friction Factor (f) Relative Roughness ( /D) : Average height of non-uniformity in pipe Reynolds Number (N Re ) = 259 x 10-6 m for cast iron; 15235 x 10-6 m for drawn tubing; 152 x 10-6 m for galvanized iron; 457 x 10-6 m for steel or wrought iron Roughness of Pipe ( ) View under naked eye View under microscope 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
Friction Coefficient (C ff ) for Fittings Regular 90 elbow (flanged) 03 Regular 90 elbow (threaded) 15 Branch flow tee (flanged) 10 Line flow tee (flanged) 02 Ball valve (1/3 closed) 55 Ball valve (2/3 closed) 210 Ball valve (open) 005 Diaphragm valve (open) 23 Diaphragm valve (1/2 closed) 43 Gate valve (1/2 closed) 21 Gate valve (open) 015 Globe valve (open) 10 Swing check valve (forward flow) 20 Friction Coefficient for Expansion (C fe ) A 1 = Cross-sectional area of smaller pipe = R 12 (for a pipe of circular cross-section) A 2 = Cross-sectional area of larger pipe = R 22 (for a pipe of circular cross-section) Friction Coefficient for Contraction (C fc ) A 1 = Cross-sectional area of larger pipe = R 12 (for a pipe of circular cross-section) A 2 = Cross-sectional area of smaller pipe = R 22 (for a pipe of circular cross-section)
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 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 = 101325 bar = 101325 kpa = 760 mm Hg = 1036 m or 339 ft H 2 O = 14696 psi Pressure Drop due to Straight Pipe E f : Energy loss due to friction (J/kg or m 2 /s 2 ) P: Pressure drop (Pa) : Density of fluid (kg/m 3 ) f: Friction factor u: Average velocity of fluid (m/s) L: Length of pipe (m) D: Diameter of pipe (m)
Pressure Drop due to Fittings E f : Energy loss due to friction (J/kg or m 2 /s 2 ) P: Pressure drop (Pa) : Density of fluid (kg/m 3 ) C ff : Friction coefficient u: Average velocity of fluid (m/s) Pressure Drop due to Expansion E f : Energy loss due to friction (J/kg or m 2 /s 2 ) P: Pressure drop (Pa) : Density of fluid (kg/m 3 ) C fe : Friction coefficient for expansion u 1 : Average velocity of fluid in smaller pipe (m/s) A 1 = Cross-sectional area of smaller pipe (m 2 ) = R 12 (for a pipe of circular cross-section) A 2 = Cross-sectional area of larger pipe (m 2 ) = R 22 (for a pipe of circular cross-section) Pressure Drop due to Contraction E f : Energy loss due to friction (J/kg or m 2 /s 2 ) P: Pressure drop (Pa) : Density of fluid (kg/m 3 ) C fc : Friction coefficient for contraction u 2 : Average velocity of fluid in smaller pipe (m/s) A 1 = Cross-sectional area of larger pipe (m 2 ) = R 12 (for a pipe of circular cross-section) A 2 = Cross-sectional area of smaller pipe (m 2 ) = R 22 (for a pipe of circular cross-section)
Pr Drop due to Pipes, Area Change & Fittings For a straight pipe For fittings such as elbows, valves, tees 1 For an expansion in pipe diameter 2 For a contraction in pipe diameter E f : Energy loss due to friction f: Fanning friction factor for straight pipe C ff : Friction coefficient for fittings C fe : Friction coefficient for expansion in pipe C fc : Friction coefficient for contraction in pipe Pressures and Temperatures in a Heat-Hold-Cool System Indirect Heating Hot water Product 20 C 60 psi 130 C 40 psi Holding 125 C 30 psi Indirect Cooling Cold water 35 C 15 psi 1209 C P gauge = 15 psi P abs = 29696 psi = 20475 kpa = Max P sat From steam tables, T sat = 1209 C 30 C 0 psi Filling at atmospheric pressure T > T sat in the dotted box Product flashes in this region! Increase back pressure to prevent this 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 packaging should 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 P sat = 10135 kpa, T sat = 100 C When P sat = 16906 kpa, T sat = 115 C Thus, to prevent flashing when T prod at end of hold tube = 115 C, we need to ensure a P reqd of at least 16906 10135 kpa (plus a factor of safety) in above equation If the P calculated using the above equation < P reqd, additional pressure must be added to prevent product boiling (flashing)
Bernoulli s Equation As 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/m 3 ) P 2 P 1 = P u: Velocity (m/s) g (z 2 z 1 ): PE ½ (u z: Height (m) 22 u 12 ): KE g: Acceleration due to gravity (m/s 2 ) PE: Potential Energy Assumptions: KE: Kinetic energy Fluid is incompressible (density is constant) Flow is inviscid Flow is steady No shaft work is done by or on the fluid No heat transfer takes place between the fluid and surroundings Pumping Capacity When a pump is used to pump a fluid from point 1 to 2, the energy to be supplied (E p ) by the pump is given by: E p = E f + PE + KE + P/ This can be expanded as follows: E f : Energy loss due to friction (J/kg or m 2 /s 2 ) u: Average velocity (m/s) = 05 for laminar flow; = 10 for turbulent flow P: External pressure difference between points 1 and 2 : Density of fluid (kg/m 3 ) The power (P) to be supplied by the pump is given by: P = m E p 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)
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 Summary (contd) Friction is associated with straight pipes (f), fittings (C ff ), expansion in pipe (C fe ), and contraction in pipe (C fc ) Friction factor Laminar flow: f = 16/N Re Transition & turbulent flow: Moody diagram Friction coefficient (C ff ) 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 (E f ) 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