F 24NReh

where the Reynolds number, NRe,h, is based upon the hydraulic diameter of the channel. Arrange your solution for the power law fluid in dimensionless form, and solve for the friction factor, f.

(c) Set your result from (b) equal to 24/NRe;h and determine an equivalent expression for the power law Reynolds number for slit flow.

45. You are drinking a milk shake through a straw that is 8 in. long and 0.3 in. in diameter. The milk shake has the properties of a Bingham plastic, with a yield stress of 300 dyn/cm2, a limiting viscosity of 150 cP, and a density of 0.8 g/cm3.

(a) If the straw is inserted 5 in. below the surface of the milk shake, how hard must you suck to get the shake flowing through the entire straw (e.g., how much vacuum must you pull, in psi)?

(b) If you pull a vacuum of 1 psi, how fast will the shake flow (in cm3/s)?

46. Water is to be transferred at a rate of 500 gpm from a cooling lake through a 6 in. diameter sch 40 pipeline to an open tank in a plant that is 30 mi from the lake.

(a) If the transfer pump is 70% efficient, what horsepower motor is required to drive the pump?

(b) An injection station is installed at the lake that injects a high polymer into the pipeline, to give a solution of 50 ppm concentration with the following properties: a low shear limiting viscosity of 80 cP, a flow index of 0.5, and a transition point from low shear Newtonian to shear thinning behavior at a shear rate of 10 s—1. What horsepower is now required to drive the same pump, to achieve the same flow rate?

47. You measure the viscosity of a sludge in the lab and conclude that it can be described as a power law fluid with a flow index of 0.45, a viscosity of 7 poise at a shear rate of 1 s—1, and a density of 1.2 g/cm3.

(a) What horsepower would be required to pump the sludge through a 3 in. sch 40 pipeline, 1000 ft long, at a rate of 100 gpm?

(b) The viscosity data show that the sludge could also be described by the Bingham plastic model, with a viscosity of 7 poise at a shear rate of 1 s—1 and a viscosity of 0.354 poise at a shear rate of 100 s—1. Using this model, what required horsepower would you predict for the above pipeline?

(c) Which answer do you think would be the most reliable, (a) or (b), and why?

48. An open drum, 3 ft in diameter, contains a mud that is known to be described by the Bingham plastic model, with a yield stress of 120 dyn/cm2, a limiting viscosity of 85 cP, and a density of 98 lbm/ft3. A 1 in. ID hose, 10 ft long, is attached to a hole in the bottom of the drum to drain the mud out. How far below the surface of the mud should the end of the hose be lowered in order to drain the mud at a rate of 5 gpm?

49. You would like to determine the pressure drop-flow rate relation for a slurry in a pipeline. To do this, you must determine the rheological properties of the slurry, so you test it in the lab by pumping it through a 1/8 in. ID pipe that is 10 ft long. You find that it takes 5 psi pressure drop in the pipe to produce a flow rate of 100 cm3/s and that 10 psi results in a flow rate of 300 cm3/s.

(a) What can you deduce about the rheological characteristics of the slurry from these data?

(b) If it is assumed that the slurry can be adequately described by the power law model, what are the values of the fluid properties, as deduced from the data?

(c) If the Bingham plastic model is used instead of the power law model to describe the slurry, what are its properties?

50. A pipeline is installed to transport a red mud slurry from an open tank in an alumina plant to a disposal pond. The line is 5 in. sch 80 commercial steel, 12,000 ft long, and is designed to transport the slurry at a rate of 300 gpm. The slurry properties can be described by the Bingham plastic model, with a yield stress of 15 dyn/cm2, a limiting viscosity of 20 cP, and an SG of 1.3. You may neglect any fittings in this pipeline.

(a) What delivered pump head and hydraulic horsepower would be required to pump this mud?

(b) What would be the required pump head and horsepower to pump water at the same rate through the same pipeline?

(c) If 100 ppm of fresh Separan AP-30 polyacrylamide polymer were added to the water in case (b), what would the required pump head and horsepower be?

51. Determine the power required to pump water at a rate of 300 gpm through a 3 in. ID pipeline, 50 mi long, if:

(a) The pipe is new commercial steel

(b) The pipe wall is hydraulically smooth

(c) The pipe wall is smooth, and "degraded" Separan AP-30 polyacrylamide is added to the water at a concentration of 100 wppm.

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