VISCOSITY :Shear 1. (a) A yield stress material is found to be best described by the "Bingham-Plastic" model. Explain the terms which are required to represent the material in this model and suggest an alternative model for materials which display the following characteristics: shear-thinning, shear-thickening. Illustrate your flow models using diagrams of shear stress against shear rate and apparent viscosity against shear rate. What are the limitations of the model when applied to shear-thinning fluids and can it be modified to account for a yield stress ? Your answer should refer to the following terms and define each of them: shear stress ; shear rate ; apparent viscosity ; zero-shear viscosity; consistency. Suggest a flow model for (i) shear-thinning, (ii) shear-thickening fluids and (iii) Newtonian fluids and explain your answer using appropriate diagrams of shear stress against shear rate and apparent viscosity against shear rate. 1. (b) A pressure drop of 105 N/m2 is developed when a fluid flows through a straight circular pipe 20 m in length and 25 mm in diameter. The fluid is Newtonian with a shear viscosity of 1 Pa s. A chemical is added to the fluid and its flow properties are changed. A graph of shear stress against shear rate and apparent viscosity against shear rate shows (i) that the changed fluid is shear-thinning, with a shear-thinning power-law index = 1/3 ; and (ii) that its apparent viscosity is the same as that of the original fluid at a shear rate of 1000 s-1. Calculate the volumetric flowrate for the shear-thinning fluid at the original value of the pressure drop. 1. (c) A power-law fluid has a density of 1075 kg/m3. It is pumped at a rate of 2500 kg/hour through a pipe of internal diameter 25 mm.The flow is laminar and the power law constants are K2 = 3 Pa.sn and n = 0.5. Estimate the pressure drop over a 10 m straight length of pipe and the centre-line velocity for these conditions. 2. (a) You are provided with information from a viscometer for 3 different fluids. The information consists of the results of measurements of shear stress for different values of shear rate. Using appropriate diagrams, explain how you would use this information to characterise the fluids in terms of their non-Newtonian flow characteristics according to (i) the Bingham-Plastic; (ii) the shear-thinning fluid and (iii) the shear-thickening fluid models, respectively. Suggest a suitable model for the shear-thinning fluid and discuss any limitations it may have. 2.(b) A material flows in a pipe of 0.15m diameter at a velocity of 0.5 ms-1. The relation between shear stress σ and shear rate ϒ is σ = 2.5 ϒ 0.2 (SI Units) As a result of temperature change the Power Law index becomes 0.25 but the apparent viscosity is unchanged when the shear rate is 100 s-1. Calculate the percentage increase in the mean velocity of the material at the same pressure drop and at the new temperature. (b) A horizontal pipe of circular cross-section and 600 mm diameter carries water under a head of 30 m with a velocity of flow of 3 m s-1. If the pipe turns through a 75 degree bend, calculate the magnitude and direction of the resultant force on the bend. NEWTON"S SECOND LAW:Bernoulli 1. (a) Explain how the Bernoulli Energy Equation can be obtained from considerations of the forces acting on a streamtube of fluid. Ensure that your answer explains the significance of the terms in the Equation and its limitations. What principle could be used to create a differential head flowmeter based on this Equation ? 1. (b) Water flows through a pipe of inside diameter 200 mm at a rate of 100 m3h-1. The flow abruptly enters a section which reduces the pipe diameter to 150 mm, for which the head loss is equivalent to 0.2 velocity heads based on the smaller pipe. If the gauge pressure is 80 kNm-2 upstream of the reducing section, find the force needed to hold the section in position. 2. (a) You are required to create a differential head flowmeter based on a convergence in a section of pipe. Starting with an expression for the forces acting on an ideal fluid, show how you would estimate the volumetric flowrate based on measurements of differential pressure. How would you modify your answer to account for the fact that a real fluid will have viscosity and how would you ensure that energy degradation is minimised? 2.(b) A manometer uses a manometric fluid of density 1075 kg/m3 to measure the pressure drop across an orifice plate with a throat diameter of 75 mm. The orifice plate is placed inside a vertical pipe with a diameter of 225 mm and oil with a density of 860 kg/m3 is flowing upwards inside the pipe. The deflection of the manometer fluid is 0.5 m and the discharge coefficient of the orifice is 0.659. What is the flowrate of the oil? 3. (a) Beginning with Newton's Second Law calculate the force required to stabilise a 90o horizontal pipe bend against movement due to hydrodynamic reaction forces. State any assumptions you would make and explain how you would calculate the direction of the force. 3. (b) A jet of water of 22.5 cm diameter impinges normally on a flat plate moving at 0.6 m s-1 in the same direction as the jet. If the discharge is 0.14 m3 s-1 find the force and the work done per second on the plate. 4. (a) Explain how you would create a differential head flowmeter based on convergence at an orifice plate placed in a section of pipe. Starting with an expression for the forces acting on an ideal fluid, show how you would estimate the volumetric flowrate based on measurements of differential pressure. 4. (b) A horizontal venturi meter with a discharge coefficient of 0.96 is to be used to measure the flowrate of water up to 0.025m3s-1 in a pipe of internal diameter 100 mm. The meter is connected to a differential manometer containing mercury (Specific Gravity,SG = 13.6). If the maximum allowable difference in mercury levels is 80 cm, what is the diameter of the throat ? 5. (a) Using Newton's Second Law as a starting point, explain how you would create a flowmeter based on a converging section of pipe for a real (non-ideal) fluid. Your answer must explain how the degradation of energy is minimised and how you would estimate the volumetric flowrate based on measurements of differential pressure. 5. (b) Obtain an expression for the force exerted by a jet of liquid which leaves a nozzle and strikes a stationary flat plate normally with a velocity v. How would this expression be modified if the plate were to be moving in the same direction as the jet with a velocity u ? Explain any assumptions which you make. 5. (c) Two pressure gauges are located at tapping points 50 cm apart on a vertical Venturi tube which has an inlet diameter of 150 mm, a throat diameter of 70 mm and a discharge coefficient of 0.98. If a liquid of density 1,000 kgm-3 flows upward through the Venturi tube at a rate of 0.075 m3s-1 what is the difference in reading of the two pressure gauges ? 5. (d) Droplets of oil (density = 960 kg/m3) are dispersed as an emulsion in a solution with a density and shear viscosity the same as water. Calculate how long an 80 µm spherical droplet will take to rise from the bottom of a tank to the surface 1.4 m above in still liquid. Neglect acceleration effects and state any assumptions you make. RUSHTON TURBINE 1. (a) Explain what factors influence the amount of power input required for fluid agitation and mixing in a standard Rushton turbine. Use this mixer configuration to explain why scale-up under conditions of 'same torque per unit volume' is equivalent to performing scale-up at constant tip speed in the fully turbulent region of mixing. Explain why the 'same mixing time criterion' can be prohibitively expensive. [ 1. (b). You are required to agitate water with a standard 'Rushton' turbine. The tank diameter is 2 m and you are required to work to a tip speed design criterion of 3 m s-1. Assuming a Power Number of 6, calculate: (i) the power required per unit volume of fluid (ii) the speed that the impeller should be driven at in a geometrically similar 4 m diameter tank on the basis of scale-up at equal power per unit volume State any assumptions you make 2. (a) Describe the configuration of a standard Rushton turbine and use this mixer configuration to explain (i) why the 'same torque per unit volume' scale-up criterion is equivalent to performing scale-up at constant tip speed in the fully turbulent region of mixing; and (ii) why the conditions under which scale-up based on the 'same mixing time criterion' could prove very expensive. (b). A horizontal pipe with an inside diameter of 200 mm has a 180o U-bend and carries a fluid of density 900 kgm-3 at a rate of 150 m3h-1. Find the force exerted by the liquid on the bend if the gauge pressures upstream and downstream of the bend are 100 kPa and 80 kPa, respectively. 3. (a) What are the main parameters influencing power input for fluid agitation and mixing. Explain why the 'same mixing time' scale-up criterion may be prohibitively expensive. Refer to the fully turbulent region of mixing and the standard 'Rushton-Turbine' configuration in your answer. 3(b). Tests on a small scale tank 0.3 m in diameter show that a blending process between two miscible liquids (both aqueous with properties the same as water) is complete after 1 minute when using an impeller speed of 250 revolutions per minute. It is decided to scale up the process to a tank of 2.5 m diameter using the criterion of constant tip speed. (i) What speed should be chosen for the larger impeller ? (ii) What power will be required ? (iii) What will be the blend time in the large tank ? State any assumptions you make. Assume a standard Rushton-turbine configuration, and a Power Number of 6. LAMINAR/TURBULENT FLOW: Reynold's Number 1. (a) Explain the meaning of the terms 'friction factor', 'Reynold's Number' and 'relative roughness' and how they are used in the construction of the Moody plot. Verify the relationship between friction factor and Reynold's Number given in the Moody plot for the laminar flow region. 1. (b) A foam consists of droplets of oil (density 960 kg/m3) which are dispersed as an emulsion in a solution whose density and viscosity is the same as water. Calculate how long an 80 micron diameter ( 1 µm = 1 x 10-6 m ) spherical drop will take to rise from the bottom of a tank to the surface 1.4 m above in a still liquid (neglect acceleration effects). 2. (a) The flow of a viscous fluid past a spherical particle is characterised by the Reynold's Number, Re'. Explain the form of the curve obtained when Re' is plotted against the Drag Coefficient. Ensure that your answer includes: (i) an explanation of the terms involved in Re' and the Drag Coefficient, (ii) an explanation of the term 'separation', (iii) an account of the changing drag force, F, on the particle. 2. (b) A water softener consists of a vertical cylindrical pipe 0.5 m long and 50 mm internal diameter packed with an ion-exchange resin consisting of spherical particles whose diameter is 1 mm. The bed porosity ε is 0.33. The column runs full of water, under a head of 0.2 m, but water trickles out slowly from the bottom of the column which is supported by a perforated plate. Calculate the flowrate and verify any assumptions which you make.