PROBLEMS

14.2-1.

Constant-Pressure Filtration and Filtration Constants. Data for the filtration of CaCO3 slurry in water at 298.2 K (25°C) are reported as follows (R1, R2, M1) at a constant pressure (−Δp) of 46.2 kN/m2 (6.70 psia). The area of the plate-and-frame press was 0.0439 m2 (0.473 ft2) and the slurry concentration was 23.47 kg solid/m3 filtrate. Calculate the constants α and Rm. Data are given as t = time in s and V = volume of filtrate collected in m3.

V × 103tV × 103tV × 103t
0.517.31.572.02.5152.0
1.041.32.0108.33.0201.7

A1: Ans. α = 1.106 × 1011 m/kg (1.65 × 1011 ft/lbm), Rm = 6.40 × 1010 m1 (1.95 × 1010 ft1)
14.2-2.

Filtration Constants for Constants-Pressure Filtration. Data for constant-pressure filtration at 194.4 kN/m2 are reported for the same slurry and press as in Problem 14.2-1 as follows, where t is in s and V in m3:

V × 103tV × 103TV × 103t
0.56.32.551.74.5134.0
1.014.03.069.05.0160.0
1.524.23.588.8  
2.037.04.0110.0  

Q3:Calculate the constants α and Rm.
A3: Ans. α = 1.61 × 1011 m/kg
14.2-3.

Compressibility of Filter Cake. Use the data for specific cake resistance α from Example 14.2-1 and Problems 14.2-1 and 14.2-2 and determine the compressibility constant s in Eq. (14.2-11). Plot ln α versus ln(−Δp) and determine the slope s.

14.2-4.

Prediction of Filtration Time and Washing Time. The slurry used in Problem 14.2-1 is to be filtered in a plate-and-frame press having 30 frames and 0.873 m2 area per frame. The same pressure, 46.2 kN/m2, will be used in constant-pressure filtration. Assume the same filter-cake properties and filter cloth, and calculate the time to recover 2.26 m3 of filtrate. At the end, using through-washing and 0.283 m3 of wash water, calculate the time of washing and the total filter-cycle time if cleaning the press takes 30 min.

14.2-5.

Constants in Constant-Pressure Filtration. McMillen and Webber (M2), using a filter press with an area of 0.0929 m2, performed constant-pressure filtration at 34.5 kPa of a 13.9 wt % CaCO3 solids in water slurry at 300 K. The mass ratio of wet cake to dry cake was 1.59. The dry-cake density was 1017 kg/m3. The data obtained are as follows, where W = kg filtrate and t = time in s:

WtWtWt
0.91243.632446.35690
1.81714.543727.26888
2.721465.445248.161188

Calculate the values of α and Rm.
14.2-6.

Constant-Pressure Filtration and Washing in a Leaf Filter. An experimental filter press having an area of 0.0414 m2 (R1) is used to filter an aqueous BaCO3 slurry at a constant pressure of 267 kPa. The filtration equation obtained was


where t is in s and V in m3.
  1. If the same slurry and conditions are used in a leaf press having an area of 6.97 m2, how long will it take to obtain 1.00 m3 of filtrate?

  2. After filtration, the cake is to be washed with 0.100 m3 of water. Calculate the time of washing.

A7: Ans. (a) t = 381.8 s
14.2-7.

Constant-Rate Filtration of Incompressible Cake. The filtration equation for filtration at a constant pressure of 38.7 psia (266.8 kPa) is


where t is in s, −Δp in psia, and V in liters. The specific resistance of the cake is independent of pressure. If the filtration is run at a constant rate of 10 liters/s, how long will it take to reach 50 psia?
14.2-8.

Effect of Filter-Medium Resistance on Continuous Rotary-Drum Filter. Repeat Example 14.2-4 for the continuous rotary-drum vacuum filter but do not neglect the constant Rm, which is the filter-medium resistance to flow. Compare with results of Example 14.2-4.

A9: Ans. A = 7.78 m2
14.2-9.

Throughput in Continuous Rotary-Drum Filter. A rotary-drum filter having an area of 2.20 m2 is to be used to filter the CaCO3 slurry given in Example 14.2-4. The drum has a 28% submergence and the filter-cycle time is 300 s. A pressure drop of 62.0 kN/m2 is to be used. Calculate the slurry feed rate in kg slurry/s for the following cases:

  1. Neglect the filter-medium resistance.

  2. Do not neglect the value of B.

14.3-1.

Settling Velocity of a Coffee-Extract Particle. Solid spherical particles of coffee extract (F1) from a dryer having a diameter of 400 μm are falling through air at a temperature of 422 K. The density of the particles is 1030 kg/m3. Calculate the terminal settling velocity and the distance of fall in 5 s. The pressure is 101.32 kPa.

A11: Ans. νt = 1.49 m/s, 7.45 m fall
14.3-2.

Terminal Settling Velocity of Dust Particles. Calculate the terminal settling velocity of dust particles having a diameter of 60 μm in air at 294.3 K and 101.32 kPa. The dust particles can be considered spherical, with a density of 1280 kg/m3.

A12: Ans. νt = 0.1372 m/s
14.3-3.

Settling Velocity of Liquid Particles. Oil droplets having a diameter of 200 μm are settling from still air at 294.3 K and 101.32 kPa. The density of the oil is 900 kg/m3. A settling chamber is 0.457 m high. Calculate the terminal settling velocity. How long will it take the particles to settle? (Note: If the Reynolds number is above about 100, the equations and form-drag correlation for rigid spheres cannot be used.)

14.3-4.

Settling Velocity of Quartz Particles in Water. Solid quartz particles having a diameter of 1000 μm are settling from water at 294.3 K. The density of the spherical particles is 2650 kg/m3. Calculate the terminal settling velocity of these particles.

14.3-5.

Hindered Settling of Solid Particles. Solid spherical particles having a diameter of 0.090 mm and a solid density of 2002 kg/m3 are settling in a solution of water at 26.7°C. The volume fraction of the solids in the water is 0.45. Calculate the settling velocity and the Reynolds number.

14.3-6.

Settling of Quartz Particles in Hindered Settling. Particles of quartz having a diameter of 0.127 mm and a specific gravity of 2.65 are settling in water at 293.2 K. The volume fraction of the particles in the slurry mixture of quartz and water is 0.25. Calculate the hindered settling velocity and the Reynolds number.

14.3-7.

Density Effect on Settling Velocity and Diameter. Calculate the terminal settling velocity of a glass sphere 0.080 mm in diameter having a density of 2469 kg/m3 in air at 300 K and 101.32 kPa. Also calculate the diameter of a sphalerite sphere having a specific gravity of 4.00 with the same terminal settling velocity.

14.3-8.

Differential Settling of Particles. Repeat Example 14.3-3 for particles having a size range of 1.27 × 102 mm to 5.08 × 102 mm. Calculate the size range of the various fractions obtained using free settling conditions. Also calculate the value of the largest Reynolds number occurring.

14.3-9.

Separation by Settling. A mixture of galena and silica particles has a size range of 0.075–0.65 mm and is to be separated by a rising stream of water at 293.2 K. Use specific gravities from Example 14.3-3.

  1. To obtain an uncontaminated product of galena, what velocity of water flow is needed and what is the size range of the pure product?

  2. If another liquid, such as benzene, having a specific gravity of 0.85 and a viscosity of 6.50 × 104 Pa · s is used, what velocity is needed and what is the size range of the pure product?

14.3-10.

Separation by Sink-and-Float Method. Quartz having a specific gravity of 2.65 and hematite having a specific gravity of 5.1 are present in a mixture of particles. It is desired to separate them by a sink-and-oat method using a suspension of fine particles of ferrosilicon having a specific gravity of 6.7 in water. At what consistency in vol % ferrosilicon solids in water should the medium be maintained for the separation?

14.3-11.

Batch Settling and Sedimentation Velocities. A batch settling test on a slurry gave the following results, where the height z in meters between the clear liquid and the suspended solids is given at time t hours:

t (h)z (m)t (h)z (m)t (h)z (m)
00.3601.750.15012.00.102
0.500.2853.000.12520.00.090
1.000.2115.000.113  

The original slurry concentration is 250 kg/m3 of slurry. Determine the velocities of settling and concentrations and make a plot of velocity versus concentration.
14.4-1.

Comparison of Forces in Centrifuges. Two centrifuges rotate at the same peripheral velocity of 53.34 m/s. The first bowl has a radius of r1 = 76.2 mm and the second r2 = 305 mm. Calculate the rev/min and the centrifugal forces developed in each bowl.

A22: Ans. N1 = 6684 rev/min, N2 = 1670 rev/min, 3806 g's in bowl 1951 g's in bowl 2
14.4-2.

Forces in a Centrifuge. A centrifuge bowl is spinning at a constant 2000 rev/min. What radius bowl is needed for the following?

  1. A force of 455 g's.

  2. A force four times that in part (a).

A23: Ans. (a) r = 0.1017 m
14.4-3.

Effect of Varying Centrifuge Dimensions and Speed. Repeat Example 14.4-2 but with the following changes:

  1. Reduce the rev/min to 10 000 and double the outer-bowl radius r2 to 0.0445 m, keeping r1 = 0.00716 m.

  2. Keep all variables as in Example 14.4-2 but double the throughput.

A24: Ans. (b) Dp = 1.747 × 106 m
14.4-4.

Centrifuging to Remove Food Particles. A dilute slurry contains small solid food particles having a diameter of 5 × 102 mm which are to be removed by centrifuging. The particle density is 1050 kg/m3 and the solution density is 1000 kg/m3. The viscosity of the liquid is 1.2 × 103 Pa · s. A centrifuge at 3000 rev/min is to be used. The bowl dimensions are b = 100.1 mm, r1 = 5.00 mm, and r2 = 30.0 mm. Calculate the expected flow rate in m3/s just to remove these particles.

14.4-5.

Effect of Oil Density on Interface Location. Repeat Example 14.4-3, but for the case where the vegetable-oil density has been decreased to 914.7 kg/m3.

14.4-6.

Interface in Cream Separator. A cream-separator centrifuge has an outlet discharge radius r1 = 50.8 mm and outlet radius r4 = 76.2 mm. The density of the skim milk is 1032 kg/m3 and that of the cream is 865 kg/m3 (E1). Calculate the radius of the interface neutral zone.

A27: Ans. r2 = 150 mm
14.4-7.

Scale-Up and Σ Values of Centrifuges. For the conditions given in Example 14.4-2, do as follows:

  1. Calculate the Σ value.

  2. A new centrifuge having the following dimensions is to be used: r2 = 0.0445 m, r1 = 0.01432 m, b = 0.394 m, and N = 26 000 rev/min. Calculate the new Σ value and scale up the flow rate using the same solution.

A28: Ans. (a) Σ = 196.3 m2
14.4-8.

Centrifugal Filtration Process. A batch centrifugal filter similar to Fig. 14.4-5 has a bowl height b = 0.457 m and r2 = 0.381 m and operates at 33.33 rev/s at 25.0°C. The filtrate is essentially water. At a given time in the cycle, the slurry and cake formed have the following properties: cS = 60.0 kg solids/m3 filtrate, ε = 0.82, ρp = 2002 kg solids/m3, cake thickness = 0.152 m, α = 6.38 × 1010 m/kg, Rm = 8.53 × 1010 m1, r1 = 0.2032 m. Calculate the rate of filtrate flow.

A29: Ans. q = 6.11 × 104 m3/s
14.5-1.

Change in Power Requirements in Crushing. In crushing a certain ore, the feed is such that 80% is less than 50.8 mm in size, and the product size is such that 80% is less than 6.35 mm. The power required is 89.5 kW. What will be the power required using the same feed so that 80% is less than 3.18 mm? Use the Bond equation. (Hint: The work index Ei is unknown, but it can be determined using the original experimental data in terms of T. In the equation for the new size, the same unknowns appear. Dividing one equation by the other will eliminate these unknowns.)

A30: Ans. 146.7 kW
14.5-2.

Crushing of Phosphate Rock. It is desired to crush 100 ton/h of phosphate rock from a feed size where 80% is less than 4 in. to a product where 80% is less than in. The work index is 10.13 (P1).

  1. Calculate the power required.

  2. Calculate the power required to crush the product further to where 80% is less than 1000 μm.

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