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 3 1 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 4 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 -1 – Units: s 5 Shear Forces Max resistance Min resistance 6 2 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 7 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 8 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 9 3 Shear Rate for Various Processes Process 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 10 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 11 Flow Behavior for Time-Independent Fluids (Herschel-Bulkley Model for Shear Stress vs. Shear Rate) Yield stress = Shear stress (Pa) n < 1 0 = Yield stress (Pa) . = Shear rate (s-1) n = 1 K = Consistency coeff. (Pa sn) n > 1 n = Flow behavior index Newtonian = 0, n = 1 . 0 Then, K = Herschel-Bulkley Model: Power-law Model: 12 4 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 13 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 . n . n-1 viscosity (app); app = / = K / = K – 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 14 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 -1 food is correlated to app at 60 s 15 5 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 Constant shear rate time at constant shear rate • Eg., Whipping cream, lubricants, printer’s inks • Thixotropy and rheopecty may be reversible or irreversible ShearRate Inc. & dec. in shear rate Step inc. in shear rate Thixotropy Thixotropy ShearStress ShearStress Rheopecty Time Shear Rate 16 Effect of Temperature on Viscosity Arrhenius equation for Newtonian fluids: : 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) Viscosity Determining Ea & BA: ln () = ln (BA) + Ea/RT Plot ln () versus 1/T Slope = Ea/R; intercept = ln (BA) Temperature Ea = R * Slope intercept BA = e 17 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 2 2 2 graph is “8 L/[1/Ri –1/Ro ]”. Obtain from this. nist.gov 18 6 Tube Viscometer • Principle – Measure pressure. drop (P) versus volumetric flow rate (V) across a straight section of tube of length, L and radius, R Plot. P on y-axis vs. V on x-axis . Slope = (8 L) / ( R4) Units R, L: m .P: Pa Obtain from slope V: m3/s : Pa∙s 19 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 20 Bostwick Consistometer foodqa.byu.edu Compartment: 5 x 5 x 3.8 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 21 7 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 22 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. 23 Reynolds Number (for Newtonian Fluids) : Density of fluid (kg/m3) u: Average velocity of fluid (m/s) : Viscosity of fluid (Pa s) 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: 24 8 Reynolds Number and Types of Flow : Density of fluid (kg/m3) u: Average velocity of fluid (m/s) : Viscosity of fluid (Pa s) 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) • 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 25 Significance of Reynolds