EX01

Fundamentals of Industrial Separations, SOLUTIONS TO PROBLEMS 10-1

Chapter 10: Filtration

Exercise 10.1

A slurry is filtered with a laboratory leaf filter with a filtering surface area of 0.05 m2 to determine

the specific cake and cloth resistance using a vacuum giving a pressure difference of 0.7 bar. The

volume of filtrated collected in the first 5 min was 250 cm3 and, after a further 5 min, an additional

150 cm3 was collected. The filtrate viscosity is 10

-3 Pa.s, the slurry contains 5 vol% of solids with

a density of 3000 kg m-3

.

Calculate the specific cake resistance α and the cloth resistance RM.

Exercise 10.2

The data given in the table below were obtained from the constant pressure period of a pilot scale

plate and frame filter press.

The mass of dry cake per unit volume of filtrate amounts to 125 kg/m3, filter area A = 2.72 m

2,

viscosity η = 10-3

Pa.s, pressure difference ∆P = 3 bar.

time: 92 160 232 327 418 472 538 (s)

filtrate volume V: 0.024 0.039 0.054 0.071 0.088 0.096 0.106 (m3)

Calculate the cake resistance α.

Exercise 10.3

Calculate the specific cake resistance α and the medium resistance RM when the same slurry

data apply from the following constant rate data obtained on the same pilot scale plate and frame

filter press:

V: 0.016 0.032 0.040 0.056 0.064 0.072 0.088 0.096 0.114 (m3)

∆P: 0.9 1.2 1.35 1.7 1.8 1.85 2.3 2.4 2.7 (bar)

Exercise 10.4

Laboratory filtrations conducted at constant pressure drop on a slurry of CaCO3 in water gave the

data shown in the following table:

Filtrate

Volume

Test I

(0.45 bar)

Test 2

(1.10 bar)

Test 3

(1.95 bar)

Test 4

(2.50 bar)

Test 5

(3.40 bar)

V (l) t (s) t (s) t (s) t (s) t (s)

0.5 17 7 6 5 4

1.0 41 19 14 12 9

1.5 72 35 24 20 16

2.0 108 53 37 30 25

2.5 152 76 52 43 35

3.0 202 102 69 57 46

3.5 131 89 73 59

4.0 163 110 91 74

4.5 134 111 89

5.0 160 133 107

5.5 157

6.0 183

10-2 FUNDAMENTALS OF INDUSTRIAL SEPARARTIONS

The filter area A = 440 cm2, the mass of solid per unit volume of filtrate was 23.5 g/L, viscos-

ity η = 10-3

Pa.s and the temperature 25°C. Evaluate the quantities α and RM as a function of

pressure drop ∆P, and fit an empirical equation to the results for α.

Exercise 10.5

Calculate the relationship between the average specific resistance and the filtration pressure from

the following data, obtained from a series of constant pressure filtration experiments:

Filtration pressure ∆P : 70 104 140 210 400 800 (kPa)

Specific resistance α: 1.4 1.8 2.1 2.7 4.0 5.6 (×1011

m/kg)

Exercise 10.6

A slurry, containing 0.1 kg of solid (solid density ρS = 2500 kg m-3

) per kilogram of water, is fed to

a rotary drum filter with length L = 0.6 m and diameter D = 0.6 m. The drum rotates at a speed of

one revolution in 6 min and 20 per cent of the filtering surface is in contact with the slurry at any

instant. Specific cake resistance α = 2.8*1010

m/kg and medium resistance RM = 3.0*109 m

-1.

Liquid density ρL = 1000 kg m-3

, liquid viscosity ηL = 0.001 kg m-1

s-1

.

a. Determine the filtrate and dry solids production rate when filtering with a pressure

difference of 65 kN/m2.

b. Calculate the thickness of the cake produced when it has a porosity ε = 0.5.

Exercise 10.7

A rotary drum filter with 30 percent submergence is to be used to filter an aqueous slurry of

CaCO3 containing 230 kg of solids per cubic meter of water. The pressure drop is to be 0.45 bar.

Liquid density and viscosity as in previous exercise, solid density ρS as in Exercise 10.4.

The specific cake resistance α = 1.1*1011

m/kg and medium resistance RM = 6.0*109 m

-1.

Calculate the filter area required to filter 40 ltr/min of slurry when the filter cycle time is 5 min.

Exercise 10.8

Calculate the dry solids production from a 10 m2 rotating vacuum filter operating at 68 kPa

vacuum and the following conditions:

cake resistance α = 1×1010

m/kg, medium resistance RM = 1×1010

1/m, drum speed = 1 rpm,

fraction submerged f = 0.3, solids concentration xS = 0.1 kg solids/kg slurry, cake moisture 3.5 kg

wet cake/kg dry cake, liquid density ρL = 1000 kg/m3, liquid viscosity ηL = 0.001 Pa.s

Exercise 10.9

Calculate the filtration time, required area, operational speed and cake thickness required to

produce 5 m3/h of filtrate with a 1 m wide horizontal belt filter operating at a pressure difference

∆P = 60 kPa. The slurry has the following properties: specific cake resistance α = 5×109 m/kg,

slurry solids concentration cS = 350 kg/m3, solids density ρS = 2000 kg/m

3, liquid density ρL =

1000 kg/m3 and viscosity ηL = 0.001 Pa.s. The cake porosity ε = 0.43.

Exercise 10.10

It is proposed to use an existing horizontal belt filter to separate phosphoric acid from a slurry

containing gypsum at 30% w/w. Cake formation at a constant pressure difference of ∆P = 50 kPa

is to be followed by displacement washing and deliquoring. Of the total 9 m belt length 1.5 m is

available for the filtration stage. Calculate the solids production rate (kg/s).


Filter data: width 2 m, linear velocity 0.1 m/s, medium resistance RM = 2·109 1/m

Cake properties: αav = 7.1·108∆P

0.46 m/kg, εav = 0.84∆P

-0.054, solid density ρS = 2350 kg/m

3

Filtrate: density ρL = 1390 kg/m3, viscosity ηL = 0.001 Pa.s

Exercise 10.11

A tank filter is operated at a constant rate of 25 L/min from the start of the run until the pressure

drop ∆P = 3.5 bar, and then at a constant pressure drop of 3.5 bar until a total of 5 m3 of filtrate is

obtained.

Given: specific cake resistance α = 1.8·1011

m/kg, medium resistance RM = 1.0·1010

m-1

,

solids concentration cS = 150 kg/m3 filtrate, viscosity ηL = 0.001 Pa.s.

Calculate the total filtration time required.

Exercise 10.12

Calculate the washing rate of a filter cake 0.025 m thick deposited on a centrifuge basket (0.635

m diameter, 0.254 m height) rotating at 20 rps. The cake porosity ε = 0.53, its specific resistance

α = 6×109 m/kg. A medium with a resistance of RM = 1·10

8 m

-1 is used to line the perforate basket.

Solid density ρS = 2000 kg/m3, liquid density ρL = 1000 kg/m

3 and liquid viscosity ηL = 0.001 Pa.s.

Assume the cake is incompressible.

a. What time is required for the passage of two void volumes of wash if there is almost no

supernatant liquid layer over the cake.

b. How does the washing rate change when a 5 cm thick supernatant liquid layer is present?


SOLUTIONS


In the graph at the following page a plot at each pressure difference is given. The table

summerizes the calculated values of αav and RM.


Test

slope

x 1011

kg/m7s

αααα

x 1010

m/kg

intercept

x 109 kg/m·s

RM

x 1010

1/m

1 6.05 4.98 1.26 5.55

2 8.37 6.89 1.26 5.53

3 8.88 7.32 1.88 8.29

4 9.46 7.79 2.00 8.80

5 15.5 12.8 3.29 14.5

0 0.002 0.004 0.006 0.008

0

2 109

4 109

6 109

8 109

1 1010

Test 1

Test 2

Test 3

Test 4

Test 5

∆P ⋅ t

V

kg

m4s

V /m3


A plot of ln(α) vs ln(∆P) is presented in the following figure:

Closer inspection of Eqs. 10.14 and 10.15 suggests plotting ∆P ⋅

t

Vvs V ⋅ ∆P

n to get a unified plot

10 11 12 13

24

25

26

lnα

ln∆P

0 0.2 0.4 0.6 0.8

0

2 109

4 109

6 109

8 109

.

Test 1

Test 2

Test 3

Test 4

Test 5

∆P ⋅ t

V

kg

m4s

V ⋅∆Pn

m3įPa

n







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