Measuring Filter and Cake Resistances Using Calcium Carbonate SlurryBrent Gregory

Group 12, Chemical Engineering LaboratorySubmitted to Prof. Kim H. Henthorn

Wednesday, May 02, 2012

Summary

The objectives of the project were to find the canvas filter resistance, the cake resistance, and cake compressibility when using a plate and frame type filter press and calcium carbonate and water slurry. Three trials were performed at each constant pressure differential of 30, 40, and 50 psid. Flow rate data over time was collected using DeltaV Operator System. The filter medium resistance was found to be 52x1010, 58x1010, and 88x1010 ft-1 respectively and the specific cake resistance was found to be 6x1010, 6x1010, and 6x1010 ft/lbm respectively. A positive correlation existed between the filter medium and change in pressure drop, which was expected because solid particles are more likely to get lodged in the filter medium at higher velocities. The specific cake resistance did not change with a change in pressure drop, which would normally indicate that the cake is incompressible. However, the uncertainties associated with the specific cake resistance and the compressibility factor are too large to yield conclusive results. Recommendations are to increase the length of the data collection period to decrease the random error of the results. Another option is increase the concentration of calcium carbonate in the slurry to shorten trial times.

Introduction

The main objectives of the filter press project were to calculate the canvas filter

resistance, the cake resistance, and cake compressibility when using a plate and frame type filter

press. Calcium carbonate was removed from water based slurry by maintaining constant pressure

drop across the filter press.

The filter resistance, cake resistance, and cake compressibility are important

considerations when designing, sizing, or scaling up a filter press. Plate and frame type filter

presses are used for separating solids from liquid and are utilized in a wide variety of

applications, which can range from high solids mineral processing, separating pulp from fruit

juices, to removing sludge from bug ponds. Filter presses are common in both the food and

pharmaceutical industries. Depending on the application, either the filtrate or the filter cake is

then used in further processing.

A filter press involves a slurry, which is mixture of solids suspended in water, that is

forced through a filter membrane. In this case, calcium carbonate slurry was forced through a

95.1 in2 square canvas filter. The calcium carbonate did not pass through the canvas membrane;

however, the water was forced through by a constant pressure differential. The resistance to the

flow of water that was attributed to the membrane is known as the filter resistance. During the

time after initializing a run, a cake of calcium carbonate builds up on the filter membrane, which

results in additional resistance to the flow of water passing through the filter. The extra resistance

due to the buildup of calcium carbonate is known as the specific cake resistance. The total

resistance is best represented by the total pressure drop across the filter press (∆ p) as shown in

Equation (1), where ∆ pc is the pressure drop over the cake, and ∆ pm is the pressure drop over

the filter medium. All nomenclature can be found in table A.1 of Appendix A.

∆ p=∆ pc+∆ pm (1)

Equation (2) is then used to relate the pressure drop over the cake to the specific cake

resistance (α).[1]

α=∆ pc gc A

μumc

(2)

The linear superficial velocity of filtrate, u, is defined as:[1]

u=( dVdt )A

(3)

where V is the volume of filtrate that is passed through the filter over the duration of the run.

The pressure drop over the filter medium can be related by the filter-medium resistance (

Rm) defined in equation (4)[1]. For the purposes of this experiment Rm is assumed to be constant,

although it may change with time due to degradation effects.

Rm=∆ pmgcμu

(4)

By rearranging Equations (2) and (4), Equation (1) can be adjusted as follows to find the

overall pressure drop.[1]

∆ p=μugc

(mc α

A+Rm) (5)

The mass of the dry filter cake can be found by relating it to the concentration of the

slurry and volume (V), which is shown as:

c=mc

V (6)

Inserting Equations (6) and (3) into Equation (5), results in:[1]

∆ p= dtdV

= μAgc∆ p (αcVA +Rm) (7)

Also since all trials used a constant total pressure drop,[1]

dtdV

=1q=K cV + 1

qo (8)

where 1qo

is the inverse initial volumetric flow rate of the slurry defined below in Equation (9), 1q

is the inverse volumetric flow rate of the slurry at time t , and K c is defined below in Equation

(10):[1]

1qo

=μ Rm

A∆ p gc(9)

K c=μcα

A2∆ pgc(10)

Integrating Equation (8) between the limits of (0,0) and (t,V) gives the following:[1]

tV

=K c

2V + 1

qo(11)

This equation shows that a plot of tV

vs V will be linear, with a slope of K c

2 and an

intercept of1qo

. Once these values are determined, 1qo

can be substituted into Equation (9) to find

the value of Rm, and K c can be substituted into Equation (10) to find the value for α .

An empirical relationship between ∆ p vs α can be used to find the compressibility factor,

s, if multiple trials are run at varying values of ∆ p. One common relationship is defined as

follows:[1]

α=αo (∆ p ) s (12)

where α o is an empirical constant and s is the compressibility factor of the filter cake. Equation

(12) can be linearized taking the log of both sides, where the resulting equation has a slope equal

to s when plotted. The compressibility factor should have a value somewhere between 0.2 and

0.8.[1]

log (α )=log (αo )+s log (∆ p) (13)

Equipment, Materials, and Methods

First, the system needed to be prepared to operate in recycle mode. The calcium

carbonate that settled in the bottom of the feed tank (TK-100A) was broken up with a slim

wooden pole. All equipment is shown in Figure B.1 in Appendix B. Manual mixing was needed

to prevent large chucks of calcium carbonate from damaging the agitator (M-100), or pump (P-

100). The feed tank had an inner diameter of 24 inches and height of 30 inches at the walls and

33 inches at the center. It was filled to approximately three-quarters of its capacity with slurry.

The agitator, pump and the cooling water were turned on and valve MV100A was opened to

begin the recycle process. The goal of the recycle process was to achieve a uniform calcium

carbonate concentration. The variable speed agitator was set at the maximum speed. The pump

was a 1.5 hp pump that could deliver a maximum of approximately 59 gal/min or 109 head in

feet. After running in recycle mode for five minutes, a sample of the slurry was collected and

tested for solid concentration with the SMART Moisture Content Analyzer. The slurry needed to

be at a concentration between 10 to 12 wt%.[2]

The plate and canvas of the filter press (FP-100) was wetted down, weighed, and then

locked into place on the plate and frame type filter press. The weight was taken so that the

weight of the filter cake could be found after completing the trial. The plates were square with an

area of 95.1 in2. A metal pan was weighed and placed under the filter press to collect possible

runoff. DeltaV (complexities detailed in Figure B.1) was used to collect flow rate of the filtrate

from FIT-100A (Fig. 1A) and to find the pressure differential from PIT-100A&B. The flow

transmitter was a Proline Promag 50H Electromagnetic Flow Measuring System with

measurement up to 1250 gpm. The pressure transmitters were Cerabar M PMC41 transmitters

with a range of 600 psi. DeltaV was initiated and the pressure drop was set. Three runs were

performed at each of the pressure drops of 30, 40, and 50 psid.

The run was started by opening valve MV-100C and initiating the run in DeltaV

simultaneously. The trial was ended when the flow rate was less than 0.1 gal/min, which

indicated that the plate was filled with calcium carbonate filter cake. The filled plate and canvas

membrane were then weighed again along with the tray that collected possible runoff. Five

samples of the filter cake were then collected from each of the four corners as well as the center

of the plate and were analyzed to find solid concentration using the SMART Moisture Content

Analyzer. All raw data gathered are shown in Table C.1.

Results and Discussion

The filter medium resistance and specific cake resistance were determined for a calcium

carbonate slurry and a constant pressure differential. The three pressure differentials used were

30, 40, and 50 psid. From DeltaV, data on how the volume changed over time were used to plot

tV

versus V . A trend line was fitted through the linear portion of each plot (shown in Figures C.1

through C.9). Values for K c

2 and

1qo

were then obtained and used to find values for Rm and α

with the Equations (9), (10), and (11) outlined in the Introduction. The resulting average values

for Rm and α at each pressure drop are given in Table 1. Uncertainties are calculated as shown in

Appendix D.

Table 1: Average experimental values and corresponding uncertainties of the filter medium resistance (Rm) and specific

cake resistance (α ) for specified pressure differentials.

∆p (psi) 30 40 50

α*10-10 (ft/lbm) 6 ± 2 6 ± 2 5 ± 2

Rm*10-10 (ft-1) 52 ± 4 58 ± 3 88 ± 4

The filter medium resistance had a definite positive correlation with increasing pressure.

As seen in Table 1, the Rm at a pressure drop of 50 psi was 88x1010 ft-1, but was only 52x1010 ft-1

for a pressure drop of 30 psi. This result can be explained “since the higher velocity caused by a

large pressure drop may force additional particles of solid into the filter medium.”[1] However,

the correlation was not linear, and there was variation between trials at the same pressure drop as

shown in Figure 1. However, a positive trend still exists.

25 30 35 40 45 50 553.0E+11

4.0E+11

5.0E+11

6.0E+11

7.0E+11

8.0E+11

9.0E+11

1.0E+12

1.1E+12

f(x) = 18249646135.3202 x − 69298596720.7333R² = 0.553174971577128

∆p (psi)

Rm (ft

-1)

Figure 1: Plot of values of the filter medium resistance with constant pressure drops of 30 psid, 40 psid, and 50 psid. A linear line may not be line of best fit, but it’s used in this figure to illustrate the general positive trend of the filter medium resistance with increasing pressure.

Unlike the filter medium resistance, the specific cake resistance did not have an obvious

correlation with a change in pressure. If the specific cake resistance is found to be independent,

the cake is then considered to be incompressible.[1] The specific cake resistances also have large

uncertainties when compared to their mean values.

The compressibility factor (s) was found to be the slope of Equation (13) as stated in the

Introduction. The compressibility factor has a value of -0.3087 as shown on Figure 2; however,

the expected values for a compressible cake would be between 0.2 and 0.8.[1] It should be noted

that the uncertainties of ln(α) are nearly as large as the values themselves, thus the results are

inconclusive. The large uncertainties associated with the specific cake resistance were mainly

attributed to the elemental error associated with weighing the calcium carbonate cake. A more

precise scale would have significantly reduced the uncertainty in the specific cake resistance.

3.3 3.4 3.5 3.6 3.7 3.8 3.9 424

25

25

f(x) = − 0.308711727439545 x + 25.826868922085R² = 0.503413463574481

ln(∆p)

ln(α

)

Figure 2: Plot of the natural log of the specific cake resistance (α ) versus the natural log of the pressure differential (∆p)

for each pressure drop. The slope of trend line is the value of the compressibility factor (s). If the calcium carbonate

slurry was compressible, one would see increasing values α with increasing pressure, which would produce a positive

slope.

Conclusions and Recommendations

The objectives of the project were to find the canvas filter resistance, the cake resistance,

and cake compressibility when using a plate and frame type filter press. Constant pressure drops

of 30, 40, and 50 psid were used to determine the filter medium resistance to be 52x1010,

58x1010, and 88x1010 ft-1, respectively, and the specific cake resistance of 6x1010, 6x1010, and

6x1010 ft/lbm, respectively. A positive correlation exists between the filter medium and change in

pressure drop as expected. The specific cake resistance does not change with a change in

pressure drop, which would normally indicate that the cake is incompressible. However, the

uncertainties associated with the specific cake resistance and the compressibility factor are too

large to yield conclusive results.

A recommended modification would be to expand the number of trials for each pressure

drop and to increase the number set pressure differentials. Time restrictions in this project

prevented extending the data collection period. Expanding the data collection period would

likely reduce much the random error associated with the results. Increasing the calcium

carbonate concentration in the slurry could also reduce the amount of time for each trial, thus

allowing more data to be collected.

References

[1] McCabe, Warren L., Julian C. Smith and Harriott Peter. “Unit Operations of Chemical Engineering”, Fifth Edition, McGraw-Hill, Inc. New York, NY. (1993)

[2] Serbezov, Atanas. ChemicalEngineeringLaboratory:FilterPressSOP. Terre Haute: Rose-Hulman. (2010)

[3] Wheeler, Anthony J., Ahmad R. Ganji. “Introduction to Engineering Experimentation”, Third Edition, Pearson Higher Education. Upper Saddle River, NJ. (2010)

[4] The Engineering ToolBox website. http://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html. March, 28th 2012.

Appendix A: Nomenclature

Table A.1: Nomenclature used in the report.

Term Definition UnitsA Filter area i n2

c Mass of cake per volume of filtrate lbmgal

gc Newton’s-law proportionality constant lbm ft

l b f s2

K c Constant of linear trend s

ga l2

mc Mass of dry filter cake kg∨l bm∆ p Overall pressure drop psid∆ pc Pressure drop over the cake psid∆ pm Pressure drop over the filter medium psid

1qo

Inverse initial volumetric flow rate of the slurry sgal

Rm Filter-medium resistance f t−1

s Cake compressibilty -u Linear superficial velocity of filtrate ft

sV Volume of filtrate collected galα Specific cake resistance ft

lbmα o Empirical constant of cake resistance ft

lbmμ Viscosity of filtrate lb f ∙ s

f t 2

Appendix B: Process Schematic

Figure B.1: Filter press unit operations process schematic

Legend

MV – Manual ValvePIT – Pressure Indicating TransmitterFE – Flow ElementFIT – Flow Indicating TransmitterTE – Temperature Element

Appendix C: Analyzed Data

0.3 0.8 1.3 1.8 2.3 2.8500

600

700

800

900

1000

1100

1200

f(x) = 249.680408463784 x + 432.360528416494R² = 0.997035231951381

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.1: Flow rate data from trial 1 were gathered using the DeltaV Operator System at a constant pressure drop of 30 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.200600

620

640

660

680

700

720

740

760

780

f(x) = 89.9178070821711 x + 481.227336602766R² = 0.976449002732211

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.2: Flow rate data from trial 2 were gathered using the DeltaV Operator System at a constant pressure drop of 50 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.200700

750

800

850

900

950

1000

f(x) = 150.642160196821 x + 492.211292520266R² = 0.973347314208022

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.3: Flow rate data from trial 3 were gathered using the DeltaV Operator System at a constant pressure drop of 40 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800950

1000

1050

1100

1150

1200

1250

1300

1350

f(x) = 242.846850605859 x + 636.896217886871R² = 0.972462339065488

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.4: Flow rate data from trial 4 were gathered using the DeltaV Operator System at a constant pressure drop of 30 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000700

720

740

760

780

800

820

840

860

880

900

f(x) = 105.503000786232 x + 537.297662379414R² = 0.960216525899458

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.5: Flow rate data from trial 5 were gathered using the DeltaV Operator System at a constant pressure drop of 50 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

0.500 1.000 1.500 2.000 2.500 3.000500

600

700

800

900

1000

1100

1200

f(x) = 294.010523835945 x + 319.64570287941R² = 0.983849008260391

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.6: Flow rate data from trial 6 were gathered using the DeltaV Operator System at a constant pressure drop of 30 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

1.000 1.500 2.000 2.500 3.000550

600

650

700

750

800

850

f(x) = 128.00698296542 x + 402.445173105532R² = 0.98248752632586

Volume (gal)

Tim

e/Vo

lum

e (s

/gal

)

Figure C.7: Flow rate data from trial 7 were gathered using the DeltaV Operator System at a constant pressure drop of 50 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

0.500 1.000 1.500 2.000 2.500 3.000 3.500400

500

600

700

800

900

1000

f(x) = 187.031946558249 x + 316.620800912701R² = 0.989019356683129

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.8: Flow rate data from trial 8 were gathered using the DeltaV Operator System at a constant pressure drop of 40 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

0.600 1.100 1.600 2.100 2.600 3.100500

600

700

800

900

1000

1100

f(x) = 191.7072025637 x + 362.11294543535R² = 0.98910022247153

Volume (gal)

Tim

e/V

olum

e (s

/gal

)

Figure C.9: Flow rate data from trial 9 were gathered using the DeltaV Operator System at a constant pressure drop of 40 psid across the filter press. Collected data was cropped and a trendline was made to fit the linear portion, where time/volume of the slurry was graphed versus the volume of filtrate.

25 30 35 40 45 50 5537.4

37.6

37.8

38

38.2

38.4

38.6

38.8

39

Pressure Differential

Perc

ent M

oist

ure

Figure C.10: Percent moisture data from all the trials were gathered using the SMART Moisture Content Analyzer for the percent moisture of different spots in the cake. Each point is the result of the average of data taken from five different areas in the filter cake and also the average of three different trials. The figure does include error bars, but they are too small to be visible.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.586.5

87

87.5

88

88.5

89

89.5

Week

Perc

ent M

oist

ure

Figure C.11: Percent moisture data from all the trials were gathered using the SMART Moisture Content Analyzer for the value of the chalk slurry at the start of each day. There are no error bars because only one test was done for each day.

Table C.1: Raw data gathered during each trial with their corresponding uncertainty.

∆p (lbf/in^2) 30 30 30 40 40 40 50 50 50Trial 1 4 6 3 8 9 2 5 7 unc

mc (kg) 9.0 9.0 9.0 9.0 8.9 9.0 9.0 6.7 9.0 ±1.5

V (gal) 2.909 2.698 2.886 3.190 3.388 3.275 3.149 3.027 3.286 ±0.003

c (lbm/gal) 7 7 7 6 6 6 6 5 6 ±1

Kc (s/gal^2) 499 ± 3 486 ± 10 588 ± 9 301 ± 7 374 ± 5 383 ± 5 179 ± 5 211 ± 7 256 ± 5

1/qo (s/gal) 432 ± 3 640 ± 10 319 ± 9 492 ± 4 316 ± 5 362 ± 6 481 ± 5 537 ± 8 402 ± 6

Appendix D: Sample Calculations

Data were obtained using the DeltaV Operating System, and the relationship of

time/volume vs time, as can be seen in Figures C.1-C.9 in Appendix C. The data used for the

following sample calculations are represented in Figure B.1, which represents Trial 1 that was

performed at a constant pressure drop of 30 psid. For this trail, the following data were obtained

using regression statistics in Excel: K c=499±2s

ga l2 and

1qo

=432±2sgal . The following data

were also known for Trial 1: L=9.75±0.0625∈¿, mc=9.0±1.5kg, V=2.909±0.003 gal,

μ=1.908 ∙10−5 l b f ∙ s

f t 2 ,[5] and ∆ p=30.0±1.2 psid. When calculating uncertainties, it was found

that all of the equations were in a series form, so an equivalent method of calculation could be

used instead of taking derivatives.[4]

Before using the values of K c and 1qo

, the area of the filter medium, A, and the value of

the mass of the cake per the volume of the filtrate, c.

A=L2=¿¿ (D.1)

ωA=± A√∑ ( aix iωi)2

=± (95.0625 i n2 )√¿¿¿ (D.2)

A=95.1±1.3 i n2 (D.3)

c=mc

V=

( 9.0kg )(2.205lbmkg )

2.909 gal=6.814

l bmgal

(D.4)

ωc=±c√∑( aix iωi)2

=±(6.814lbmgal )√( 1

9.0kg(1.5kg ))

2

+( −12.909 gal

(0.003gal ))2

(D.5)

ωc=±1.1lbmgal

(D.6)

c=7±1l bmgal

(D.7)

Since K c and 1qo

are known and their theoretical equations are known, then these

equations can be rearranged to find the specific cake resistance, α , and the filter medium

resistance, Rm. The following calculations were performed to find these resistances.

Note: In the following equations, gc has been taken care of by the value of μ obtained

from EngineeringToolbox, so its value is not written in the equations that follow though it is still

represented.

α=K c A

2∆ p gcμc

=(499

s

ga l2 )( 95.1i n2 )2 (30 psid )(7.48

gal

f t 3 )(1.908 ∙10−5 l bf ∙ s

f t 2 )(7 l bmgal )(144i n2

f t2 )=5.409 ∙1010 ft

lbm (D.8)

ωα

α=±√∑ ( a ix iωi)

2

(D.9)

¿√( 1

499s

gal2(2

s

gal2 ))2

+( 2

95.1 i n2(1.1i n2 ))

2

+( 130 psid

(1.2 psid ))2

+( −1

7l bmgal

(1 lbmgal ))2

(D.10)

ωα=±(5.409 ∙1010 ftlbm ) (0.172 )=±9.29 ∙109 ft

l bm(D.11)

α=5.4 ∙1010±0.9 ∙1010 ftl bm

(D.12)

Rm=A ∆ p gc ( 1

qo )μ

=

(95.1 in2 ) (30 psid )(432sgal )(7.48

galf t3 )

(1.908 ∙10−5 l bf s

f t2 )=4.834 ∙1011 f t−1

(D.13)

ωRm

Rm

=±√∑ ( a ix iωi)2

(D.14)

¿±√( 195.1 in2 (1.1 i n2 ))

2

+( 130 psid

(1.2 psid ))2

+( 1

432sgal

(2 sgal ))

2

(D.15)

ωRm=( 4.834 ∙1011 f t−1 ) (±0.0515 )=±1.93 ∙1010 f t−1 (D.16)

Rm=48 ∙1010±2∙1010 f t−1 (D.17)

After finding the resistances, the specific resistances of the cake were plotted on a log-log

plot to find the compressibility. Since triplicates were taken at each pressure drop, the standard

error of the compressibility was used as the sole factor to determine the uncertainty associated

with the compressibility. As can be seen from Figure 2, the compressibility factor was found to

be s=−0.3±0.3.