Flow of Fluids

Abstract

Fluids are substances that flow without disintegration when pressure is applied. This definition of a fluid includes gases, liquids, and certain solids. A number of foods are fluids. In addition, gases such as compressed air and steam are also used in food processing, and they exhibit resistance to flow just like liquids. In this chapter, the subject of fluid flow will be discussed from two standpoints: the resistance to flow and its implications in the design of a fluid handling system and evaluation of rheological properties of fluid foods.

Problems

• 5.1. The following data were obtained when tomato catsup was passed through a tube having an inside diameter of 1.384 cm and a length of 1.22 m.

  Flow rate (cm3/s)	P (dynes/cm2)
  107.5	50.99 × 104
  67.83	42.03 × 104
  50.89	33.07 × 104
  40.31	29.62 × 104
  10.10	15.56 × 104
  8.80	14.49 × 104
  33.77	31.00 × 104
  53.36	35.14 × 104
  104.41	46.85 × 104

  Determine the fluid consistency index K and the flow behavior index n of this fluid.

• 5.2. Figure 5.30 shows a water tower and the piping system for a small manufacturing plant. If water flows through the system at 40 L/min, what would be the pressure at point B? The pipes are wrought iron pipes. The water has a density of 998 kg/m³ and a viscosity of 0.8 centipoise. The pipe is 1.5-in. (nominal) wrought iron pipe.

• 5.3. The catsup in Problem 1 (n = 0.45, K = 6.61 Pa · sⁿ) is to be heated in a shell and tube heat exchanger. The exchanger has a total of 20 tubes 7 m long arranged parallel inside a shell. Each tube is a 3/4-in. outside diameter, 18-gauge heat exchanger tube. (From a table of thickness of sheet metal and tubes, an 18-gauge wall is 0.049 in.) It is possible to arrange the fluid flow pattern by the appropriate selection of heads for the shell, such that number of passes and the number of tubes per pass can be varied. Calculate the pressure drop across the heat exchanger (the pressure at the heat exchanger inlet necessary to push the product through the heat exchanger) if a flow rate of 40 L of product per min (density = 1013 kg/m³) is going through the system for (a) two-pass (10 tubes/pass) and (b) five-pass system (four tubes per pass). Consider only the tube resistance, and neglect the resistance at the heat exchanger heads.

• 5.4. Figure 5.31 shows a deaerator operated at 381-mm-Hg vacuum. (Atmospheric pressure is 762 mm Hg.) It is desired to allow a positive suction flow into the pump. If the fluid has a density of 1040 kg/m³, the flow rate is 40 L/min, and the pipe is 1.5-in. sanitary pipe, calculate the height “h”; the bottom of the de-aerator must be set above the pump level in order that the pressure at the pump intake is at least 5 kPa above atmospheric pressure. The fluid is Newtonian and has a viscosity of 100 centipoises.

  [Note: Total length of straight pipe from first elbow below the pump to the entrance to the pump = 4 m. Distance from pump level to horizontal pipe = 0.5 m.]

• 5.5. Calculate the total equivalent length of 1-in. wrought iron pipe that would produce a pressure drop of 70 kPa due to fluid friction, for a fluid flowing at the rate of 50 L/min. The fluid has a density of 988 kg/m³ and a viscosity of two centipoises.

• 5.6. Calculate the horsepower required to pump a fluid having a density of 1040 kg/m³ at the rate of 40 L/min through the system shown in Fig. 5.32. ΔP_f/ρ calculated for the system is 120 J/kg. Atmospheric pressure is 101 kPa.

• 5.7. Calculate the average and maximum velocities of a fluid flowing at the rate of 20 L/min through a 1.5-in. sanitary pipe. The fluid has a density of 2030 kg/m³ and a viscosity of 50 centipoises. Is the flow laminar or turbulent?

• 5.8. Determine the inside diameter of a tube that could be used in a high-temperature, short-time heater-sterilizer such that orange juice with a viscosity of 3.75 centipoises and a density of 1005 kg/m³ would flow at a volumetric flow rate of 4 L/min and have a Reynolds number of 2000 while going through the tube.

• 5.9. Calculate the pressure generated at the discharge of a pump that delivers a pudding mix (ρ = 995 kg/m³; K = l.0 Pa · sⁿ, n = 0.6) at the rate of 50 L/min through 50 m of a l.5-in. straight, level l.5-in. stainless steel sanitary pipe. What would be the equivalent viscosity of a Newtonian fluid that would give the same pressure drop?

• 5.10. What pipe diameter will give a rate of flow of 4 ft./min for a fluid delivered at the rate of 2 gal/min?

• 5.11. Calculate the viscosity of a fluid that would allow a pressure drop of 35 kPa over a 5 m length of ³/₄-in. stainless steel sanitary pipe if the fluid is flowing at 2 L/min and has a density of 1010 kg/m³. Assume laminar flow.

• 5.12. A fluid is evaluated for its viscosity using a Brookfield viscometer. Collected data of rotational speed in rev/min and corresponding apparent viscosity in centipoises, respectively, are 20, 7230; 10, 12,060; 4, 25,200; and 2, 39,500. Is the fluid Newtonian or non-Newtonian? Calculate the flow behavior index, n, for this fluid.

• 5.13. A fluid having a viscosity of 0.05 lb_mass/ft. (s) requires 30 s to drain through a capillary viscometer. If this same viscometer is used to determine the viscosity of another fluid and it takes 20 s to drain, calculate the viscosity of this fluid. Assume the fluids have the same densities.

• 5.14. Figure 5.33 shows a storage tank for sugar syrup that has a viscosity of 15.2 centipoises at 25 °C and a density of 1008 kg/m³. Friction loss includes an entrance loss to the drain pipe that equals the kinetic energy gain of the fluid and resistance to flow through the short section of the drain pipe.

  • (a) Formulate an energy balance equation for the system.

  • (b) Formulate the continuity equation that represents the increment change in fluid level in the tank as a function of the fluid velocity through the drain pipe.

  • (c) Solve simultaneously equations formulated in (a) and (b) to calculate the time to drain the tank to a residual level 1 m from the bottom of the tank.

  • (d) Calculate the amount of residual fluid in the tank including the film adhering to the side of the tank.

• 5.15. A fluid tested on a tube viscometer 0.75 cm in diameter and 30 cm long exhibited a pressure drop of 1200 Pa when the flow rate was 50 cm³/s.

  • (a) Calculate the apparent viscosity and the apparent rate of shear under this condition of flow.

  • (b) If the same fluid flowing at the rate of 100 cm³/s through a viscometer tube 0.75 cm in diameter and 20 cm long exhibits a pressure drop of 1300 Pa, calculate the flow behavior and consistency indices. Assume wall effects are negligible.

• 5.16. Apparent viscosities in centipoises (cP) of 4000, 2500, 1250, and 850 were reported on a fluid at rotational speeds of 2, 4, 10, and 20 rev/min. This same fluid was reported to require a torque of 900 dyne cm to rotate a cylindrical spindle 1 cm in diameter and 5 cm high within the fluid at 20 rev/min.

  • (a) Calculate the flow behavior and consistency indices for this fluid.

  • (b) When exhibiting an apparent viscosity of 4000 cP, what would have been the shear rate under which the measurement was made?

• 5.17. The flow behavior and consistency indices of whey at a solids content of 24% have been reported to be 0.94 and 4.46 × 10⁻³ Pa · sⁿ, respectively. If a rotational viscometer having a full-scale torque of 673.7 dyne Acm is to be used for testing the flow behavior of this fluid, determine the diameter and height of a cylindrical spindle to be used such that at the slowest speed of 2 rev/min. The minimum torque will be 10% of the full-scale reading of the instrument. Assume a length to diameter ratio of 3 for the spindle. Would this same spindle induce a torque within the range of the instrument at 20 rev/min?

• 5.18. Egg whites having a consistency index of 2.2 Pa · sⁿ and a flow behavior index of 0.62 must be pumped through a 1.5-in. sanitary pipe at a flow rate, which would induce a shear rate at the wall of 150/s. Calculate the rate of flow in L/min and the pressure drop due to fluid flow resistance under these conditions.

• 5.19. The system shown in Fig. 5.34 is used to control the viscosity of a batter formulation used on a breading machine. A float indicates the fluid level in the reservoir, and by attaching the float to an appropriate transducer, addition of dry ingredients and water into the mixing tank may be regulated to maintain the batter consistency. The appropriate fluid level in the reservoir may be maintained when a different consistency of the batter is required, by changing the length of the pipe draining the reservoir. If the fluid is Newtonian with a viscosity of 100 cP and if fluid density is 1004 kg/m³, calculate the feed rate that must be metered into the reservoir to maintain the level shown. Assume entrance loss is negligible compared to fluid resistance through the drain pipe.

• 5.20. The system shown in Fig. 5.35 has been reported to be used for disintegrating wood chips in the pulp industry, after digestion. It is desired to test the feasibility of using the same system on a starchy root crop such as cassava or sweet potatoes, to disrupt starch granules for easier hydrolysis with enzymes to produce sugars for alcoholic fermentation. In simulating the system on a small scale, two parameters are of importance: the shear rate of the slurry as it passes through the discharge pipe and the impact force of the fluid against the plate. Assume that entrance loss from the tank to the discharge pipe is negligible. Under the conditions shown, calculate the shear rate through the drain pipe and the impact force against the plate at the time the drain pipe is first opened. The slurry has flow behavior and consistency indices of 0.7 and 0.8 Pa · sⁿ and a density of 1042 kg/m³.

• 5.21. In a falling film direct-contact steam heater for sterilization, milk is pumped into a header that distributes the liquid to several vertical pipes, and the liquid flows as a film in laminar flow down the pipe. If the fluid has a density of 998 kg/m³ and a viscosity of 1.5 cP, calculate the flow rate down the outside surface of each of 3.7-cm outside diameter pipes in order that the fluid film will flow at a Reynolds number of 500. Calculate the fluid film thickness when flow develops at this Reynolds number .

• 5.22. A sauce product is being formulated to match a reference product (Product A) that has a consistency index of 12 Pa sⁿ and a flow behavior index of 0.55. Rheological measurements of the formulated product (Product B) on a wide-gap rotational viscometer using a cylindrical spindle 1 cm in diameter and 5 cm long are as follows, with speed in rev/min and torque in % of full scale, respectively: 2, 11; 4, 18; 10, 34; and 20, 56. The viscometer constant is 7187 dyne cm. Calculate the apparent viscosity of Product A and Product B at 0.5 rev/min. At this rotational speed, did the apparent viscosity of Product B match that of Product A?

• 5.23. An FMC deaerator 2.97 m high and 96.5 cm in diameter is rated to deaerate from 4.2 to 8.4 kg/s of product. If the product has a density of 1008 kg/m³ and has a flow behavior index of 0.44 and a consistency index of 8.1 Pa s ⁿ, calculate the film thickness and film velocity to achieve the mid-range (6.3 kg/s) of the specified capacity. If half the deaerator height is to be covered by the fluid film, calculate the time available for any gas bubbles to leave the film into the vapor space in the chamber.

Fig. 5.30

Water tower and piping system for Problem 2

Fig. 5.31

Diagram of Problem 4

Fig. 5.32

Diagram for Problem 6

Fig. 5.33

Diagram for Problem 14

Fig. 5.34

Diagram for Problem 19

Fig. 5.35

Diagram for Problem 20