Fluid Mechanics: Applications

Abstract

In this chapter, we will focus on some selected applications of fluid mechanics. Food processing and manufacturing operations involve handling various types of fluids, including liquids, gases, steam, and fluidized materials. They follow many of the same laws and exhibit similar resistance to flow and are thus grouped together for analysis and process design purposes. Based on their unifying theory, engineering operations are often categorized into unit operations such as heating, cooling, drying, sheeting, and homogenization. The fluid flow properties and design of the handling and transport system in each unit operation and how fluids are moved from one operation to another control the resulting quality, safety, and consistency of the final products. Thus, a good understanding of the design considerations is needed. Although specific requirements depend on the product characteristics and the processes involved, elimination of dead spots where microorganisms could grow, ease of clean-in-place (CIP) and overall operational efficiency apply to all food handling systems. In Chap. 5, we discussed the basics of the flow of fluids. In this chapter, we will learn how those principles are utilized in selecting and setting up equipment and pipeline networks that are used to process and transport fluids such as liquid foods and related materials.

Problems

• 6.1 A centrifugal pump is operating at 2000 rpm against a 28 m head. If the speed of the pump is halved, calculate the new velocity and the developed head.

• 6.2 You are designing a pipeline system using a centrifugal pump with an impeller diameter of 1500 mm to transport milk at a mass flow rate of 6 kg/s and the energy requirement to pump is 15 kJ/kg. If the pump efficiency is 90%, compute the brake power needed. However, the supplier can only provide a centrifugal pump with an impeller diameter of 2500 mm. If you decide to use the 2500 mm impeller diameter pump, calculate the power the new pump will require.

• 6.3 Compute the impeller diameter needed to develop a head of 20 m by a centrifugal pump running at 2500 rpm.

• 6.4 A storage tank is located 30 m above a body of water. If the pipe has an inside diameter of 5 cm, what power would have to be supplied to the fluid to pump the water into the tank at a rate of 0.03 m³/s? If the pump has an efficiency of 70%, what will be the power input required by the pump? Assume no energy loss due to friction in the connecting pipe.

• 6.5 Water flows through a pipe with 1” I. D. at a mass flow rate of 10 kg/s. The total length of the pipe is 20 m. There are two 45° elbows, one gate valve half open, and one swing check valve between the inlet and outlet. Determine the equivalent length of the straight pipe due to the friction loss caused by the fittings and valves. Assume a Fanning friction factor of 0.005.

• 6.6 Corn oil (S. G = 0.8) is being pumped at 10.22 m³/h through a pipe system of 30.48 mm I.D. to a height of 6.1 m above the initial level in the tank. The level in the tank drops at a rate of 61.0 mm/s. The total friction loss in the pipe system is 74.7 J/kg. If the flow is assumed to be laminar, what is the horsepower of the pump required for this process?

• 6.7 Vegetable oil (Newtonian fluid) is pumped from a feed tank at a rate of 57.5 m³/h to a storage tank located 30 m away and at a level 5 m higher than the feed tank. A pipe of 3” i.d. is used, and the piping system includes four standard 90° elbows and two gate valves (one fully opened and the other half opened). The viscosity of the oil is 60 mPa·s, and the density is 920 kg/m³. Determine the pumping power needed. Assume the pipe is smooth.

• 6.8 To pump milk (ρ = 1150 kg/m³; η = 2 × 10⁻³ kg/m·s) from a storage tank located at ground level to a process vessel situated 10.0 m above ground level, a 200 m long, 75 mm I.D. pipeline is used. The friction loss for the associated valves, fitting and entrance effects equals 10 times the velocity heads (based on the flow in the 75 mm I.D. pipe).

  A centrifugal pump with the following characteristics is available for this job:

  Flow rate (m3/h)	10.0	15.1	19.5	25.2	30.0
  Head (m)	26.1	25.1	23.9	21.0	16.1

  If this equipment is to be used for the above process, develop a system’s curve of dynamic head versus flow rate using the flow rates given in the table above and specify the optimal flow rate for using this and the power required if the pump is 85% efficient.

• 6.9 Orange juice (S.G. = 1.2) in a tank is being pumped at 20.0 L/s through a uniform pipe system (10.0-cm I.D.) to 5.0 m above the liquid surface. The total friction loss in the pipe system is 36 J/kg. If the flow is assumed to be turbulent, calculate (a) the velocity of the flow, (b) the total dynamic head of the pumping system, and (c) if the efficiency of the pump is 75%, calculate the fluid power and the break power requirements.

• 6.10 Chocolate milk at 24 °C (ρ = 1025 kg/m³) in a large tank at atmospheric pressure is pumped through the system at a rate of 3.465 kg/s. The gauge pressure at the end of the discharge line is 217 kPa. The discharge height is 3.66 m, and the pump suction is 0.91 m above the level in the reservoir. The discharge line has an exit velocity of 2.1 m/s. The pressure drop by friction in the suction line is known to be 4.8 kPa and that in the discharge line is 33.1 kPa. If the vapor pressure of chocolate milk at 24 °C is 2.1 kPa and it is flowing turbulently, calculate (a) the differential head between the tank and the pump in kJ/kg and (b) if the pump manufacturer specifies an NPSHR of 3.048 m, what is the NPSHA, and will the pump be suitable for this service?

• 6.11 A Venturi meter is used to measure the velocity of water flowing through a pipe (I.D. = 75 cm). A manometer with mercury (S.G. = 13.5) is employed between the inlet and outlet of the nozzle, and the height difference in the manometer fluid level is 126 cm. If the coefficient of discharge of the Venturi meter is 0.9975 and the nozzle outlet diameter is 25 cm, calculate (a) the average velocity (in m/s) and (b) mass flow rate (in kg/s) of the water in the pipe.

• 6.12 Two identical particles (D_p = 1 mm) settle at terminal velocity in the Strokes flow regime, one in water (η_w = 10⁻² poise) and the other in glycerol (η_gly = 800 η_w). Neglecting any buoyancy difference, how much slower or faster will the particle in glycerol fall?

• 6.13 Calculate the terminal velocity of snowflakes of 3 mm diameter falling through air at 0 °C. (C_D = 2.5, ρ_air (0 °C) = 1.3 kg/m³, μ_air (0 °C) = 1.7 × 10⁻⁵ Pa·s, snow is on average 7% water by volume).

• 6.14 Corn starch has an average granular size of 10 μm and a particle density of 1500 kg/m³. If corn starch is suspended in water, calculate (a) the velocity of gravitational sedimentation (in cm/h), (b) the Reynolds number of starch settling in the water, and c) if the starch suspension is centrifuged at 8000 rpm in a centrifuge with a 10 cm diameter of rotation, calculate the relative centrifugal force (RCF) and the velocity of separation.

• 6.15 Spherical particles (ρ_p = 1500 kg/m³ and d_p = 10 μm) dispersed in water are harvested using a centrifuge consisting of several 20-cm long cylindrical tubes that are 85% full. The centrifuge operates at 1000 RPM, and the tubes spin perpendicular to the axis of rotation. During centrifugation, the distance between the top of the tube and the axis of rotation is 60 cm. How long would it take for all the particles to be completely separated?

• 6.16 A winery uses centrifugation to obtain clarified wine by separating impurities from wine of viscosity 2 cP and density 1.06 g/cm³. The impure moieties can be assumed to be spheres with a diameter of 7.0 μm and density of 1.12 g/cm³. The bowl of the separator has an outer diameter of 30 cm. If the centrifuge is operated at 600 rpm and the time it takes 90 min to separate the impurities, determine the distance between the surface of the liquid and the axis of rotation.

• 6.17 Milk (ρ = 1025 kg/m³, η = 3 cP) flows at a rate of 10 ft³/h through a 0.3 m diameter column packed with immobilized lactase-containing beads (diameter = 1 cm). If the bed porosity is 0.40, estimate (a) the superficial velocity, (b) the average interstitial velocity through the packed bed, and (c) the hydraulic radius of the beads.

• 6.18 An impeller stirrer (tip diameter = 0.1 m) is used for milk (ρ = 1030 kg/m³, η = 0.0012 Pa s) agitation during heating to prevent burning. Calculate the speed at the impeller tip and the Reynolds number for agitation when the impeller is operating at 10 rpm.

• 6.19 To mix a liquid drink (ρ = 1050 kg/m³, η = 0.002 Pa-s), an impeller stirrer is used to mix it during heating. The impeller diameter is 50 cm. You are told that to ensure adequate mixing and heat transfer within the fluid, the flow must be turbulent at 10% radial distance from the impeller center. Determine the minimum impeller speed in rpm and the Reynolds number at the tip.

• 6.20 A viscosity test is performed on a liquid food product with a density of 1200 kg/m³ by dropping a steel ball with a density of 7000 kg/m³ and a radius of 8 mm into a container filled with the food product. The ball attains a terminal velocity of 0.15 m/s. Compute the viscosity of the food. Verify whether Stroke’s law is valid in this case.

• 6.21 The kinematic viscosity of canola oil (ρ = 850 kg/m³) was measured at 25 °C using a capillary viscometer with a 40-mm long and 1 mm diameter capillary and reported to be 78 mm²/s. The flow rate through the capillary was 5 mm³/s. Determine the (a) dynamic viscosity of canola oil, (b) flow velocity of oil, (c) pressure drop across the capillary, and (d) head in mm required to produce the flow rate.

• 6.22 (A) You designed a new capillary viscometer and used a standard fluid of 1 cSt (centistokes) kinematic viscosity to calibrate it. It took 200 s for this fluid to drain from the start mark to the stop mark at 30 °C. Using this viscometer, you want to measure the viscosity of a new beverage (ρ_m = 1030 kg/m³) that drains in 250 s at the same temperature. Determine the viscosity of the new beverage.

  (B) To confirm your results, you use a falling-ball viscometer with a spherical bead (ρ_s = 1.05 g/cm³) of 2 mm radius. The ball travels a distance of 10 cm between the start and stop marks in 750 ms. Compute the viscosity of the new beverage and indicate which of the two methods seems more reliable to you and why?