Fluidized Bed

Evaluation of Liquid Flow Consants Through Application of Equations for Bed Fuidization D.S. Corpuz, J.L. de Guzman and J.M. Golbin Department of Chemical Engineering, University of the Philippines-Diliman, Quezon City, Philippines

D.S. Corpuz, J.L. de Guzman and J.M. Golbin, 2008. The purpose of this experiment is to validate graphically the CarmanKozeny, Burke-Plummer, and Ergun Equations. This was done by passing water upward at varying flow rates into a bed of particles. The experiment gave values of 5x106 for the Kozeny-Carman constant and 36.07 for the Burke-Plummer constant, which is far from theoretical value. The experiment, however, was able to verify the direct relationship of bed height and bed porosity with the Reynolds number and relationship of pressure drop with the Reynolds number. Keywords: bed height, bed porosity, Burke-Plummer Equation, Carman-Kozeny Equation, Ergun Equation, friction factor, mean velocity, minimum fluidization velocity, porosity, pressure drop, Reynolds number Kozeny-Carman Equation

OBJECTIVES The main objective of this experiment is to graphically determine the experimental constants of the three equations namely: Carman-Kozeny, Burke-Plummer, and Ergun Equations. In the process, the experimental data could be compared to the theoretical data. Other important objectives were to run the experiment at various flow rates using the bucket method and calculate the corresponding pressure drops. The minimum fluidization values were also determined. THEORETICAL BACKGROUND Packed beds and fluidized beds play a major role in many chemical engineering processes. Packed-bed situations include processes such as filtration, wastewater treatment, and the flow of crude oil in a petroleum reservoir. Fluidized beds are used extensively in the chemical process industries, particularly for the cracking of high-molecular-weight petroleum fractions (http://www.eng.buffalo.edu). When a fluid is pumped upward through a bed of fine solid particles at a very low flow rate, the fluid penetrates through the void spaces without disturbing the bed. This process is known as the packed bed process. If the upward flow rate is steadily increased, the pressure drop increases. The pressure drop across the bed counterbalances the force of gravity on the particles. Increasing the velocity further makes the particles separated enough to move about the bed and become suspended in the fluid. This is the onset of fluidization and is termed as fluidized bed. A minimum velocity is needed to fluidize a bed. Consider a column packed with solid particles through which a fluid flows, if the velocity is too small the bed stays fixed and operates as a packed bed. The particles in the bed will remain in a packed bed as long as the gravitational forces holding the solid particles down are greater than the force exerted by the fluid passing up through the bed of particles. At the point where the two forces become equal the solid particles begin to move. The force balance given in Eq. 1 describes this condition known as incipient fluidization. Incipient or minimum fluidization velocity is the superficial fluid velocity at which the fluidization of the bed commences.

* equation 7.49 (McCabe, 2001)

After achieving incipient fluidization, increasing the fluid flow velocity does not result in any significant increase in the pressure drop as the bed expands to reduce the resistance to flow. Finally, at conditions of entrainment, the pressure drop decreases as the entrained particles offer little resistance to flow.

* equation 7.17 (McCabe, 2001)

Eq. 2 is applicable for flow through beds at Reynolds numbers (NRe) up to about 2100. No sharp transition to turbulent flow can be observed at this NRe, but at higher NRe, considerable kinetic energy losses results due to the frequent energy changes in the shape and direction of the channels in the bed. The theoretical constant is 150. Burke-Plummer equation

* equation 7.20 (McCabe, 2001)

Eq. 3 is used to determine the pressure drop in turbulent flow region where the kinetic energy losses significantly affect the system. NRe is greater than 4000. The theoretical constant is 1.75. Ergun Equation

* equation 7.50 (McCabe, 2001)

Eq. 4 covers the entire range of flow rates assuming that the kinetic and drag/viscous losses are additive. The theoretical constants are 150 and 1.75. The variables used in Eq. 2 to 4 are: ∆P = pressure drop in the fluidized bed L = height of fluidized bed = superficial or empty tower fluid velocity µ = absolute viscosity Φs = sphericity Dp = particle diameter ε = bed porosity PROCEDURE Flow meter Calibration The carbon tetrachloride (CCl4) and mercury (Hg) manometer were connected to the valve regulating water flow. Several sample points were gathered to measure the actual flow rate from no flow to maximum flow. Using the bucket method, flow rates (F) were calculated by measuring the time required to fill

the bucket with water and the corresponding weight of the water at different sample points. Concurrently, the manometer level or the height difference (∆h) of both fluid (Hg and CCl4) was read. These calculated F, together with the ∆h, determined the meter constant Cv by regression analysis.

Kozeny-Carman Equation ∆K O2/ P 1 - .N g  MN N L c QR .S

11

* from equation 22.76 (Foust, 1980)

k2 is the Kozeny-Carman experimental constant obtained using: Pressure drop Measurement The flow rate was varied, and the ∆h between the CCl4 and Hg manometers was recorded. RESULT AND ANALYSIS

x‐axis 

ft F   average velocity  V s ρH2O Ap

y‐axis 

ε3

ΔP L

(

, lbm

lbm

ft 2 s 2

(ft 2s 2+

gY,

∆K [O2/N 1 - . g c  MZ L QR . S

+

k4 is the experimental constant obtained using:

x‐axis 

y‐axis  7

*from equation 7.14 (McCabe, 2001) )

bed porosity = ε = ( + 1 - ./ 

,

lb

(ft2ms 2+ (

lbm

ft 2 s 2

+

∆K QRN .S ]^_ gc 2  MN \ MZ 1 - . L PO/ 1 - .N

13

The corresponding equations for the axes of the Ergun equation are NRe x‐axis  1‐ε ∆K QRN .S y‐axis  ` L a PO/ 1 - .N

*from equation 7.2 (McCabe, 2001)

pressure drop  ∆hρA ‐ρB 

Dp ε3 ΔP g , L Y

* from equation 22.85 (Foust, 1980)

8

µ

02 1‐ε ρV

Ergun Equation

)*

222 V0 ·ρ·Dp

12

* from equation 22.82 (Foust, 1980)

6

ft 0    ε · average velocity super icial velocity  V s

9

The flow meter was first calibrated, and the value of the flow meter constant was determined. The calibration equation was given as <  => ?∆@AABC (10) where F is the flow rate in lb/s, and ∆h is in inches. Determination of Cv 1.4

During fluidization, several regions are expected to be observed which include the laminar, transition, and turbulent regimes. For the laminar flow, the Kozeny-Carman equation is used. Turbulent flow obeys the Burke-Plummer equation. Meanwhile, the Ergun equation can be applied for the entire range of flow rates, and it is assumed that the kinetic energy and viscous losses are additive. From the calculated flow rates, the Reynolds number for each data point as then calculated using Eq. 10 and Eq. 5 to 8. Table 1 shows the calculated values for a single trial.

1.2 Flow rate, lb/s

D2p

Burke-Plummer equation

The data recorded in this experiment included the time to fill the bucket with a certain mass of water in seconds (t), the mass of water in pounds (mw), bed height in cm (L), and manometer level (∆h). These variables will be used to calculate for the following terms: mass water lb 5 volumetric low rate  F    time s

NRe 

0µ 1‐ε2 V

1 0.8

y = 1.137x - 0.091 R² = 0.966

0.6

Table. 1 Sample NRe calculation ?∆@bc , √de

0.4 0.2 0 0

0.5 1 sqrt(H CCl4), sqrt(in)

1.5

Fig.1 Determination of Cv It was determined from the slope of Fig.1 that the flow meter DE⁄F constant is 1.137 , with an experimental error of -0.091. This √IJ error is probably due to the inaccurate bucket method, and that the meter just got repaired. The Cv was used for subsequent determinations of flow rates. From the flow rates and manometer readings, and using some necessary laboratory constants, such as bed porosity ε and pipe diameter, the experimental data were plotted (using the following equations) and compared to the theoretical constants given by Kozeny-Carman, Burke-Plummer, and Ergun equations.

F, lb/s

mean v, ft/s superficial v, ft/s

Nre

0.70710

0.80398 0.620968704

0.602774321

887.6871286

0.836660

0.95128 0.734740079

0.717517771

1056.666265

0.894427

1.016963 0.785470184

0.770741047

1135.046539

0.948683

1.07865 0.83311694

0.818095201

1204.78354

In the experiment, all data points fell under the laminar flow regime, with NRe < 2100. None had an NRe > 4000. The Kozeny-Carman equation is applicable to laminar flow. Fig. 2 shows the determination of the experimental Kozeny-Carman constant.

Table 2. Comparison of Experimental and Book Values

Determination of the Kozeny-Carman constant 14000

KC

BP

12000

Trial 1

6.00E+06

22.24

10000

Trial 2

4.00E+06

51.25

8000

Trial 3

5.00E+06

34.65

average

5.00E+06

36.04667

theoretical

1.50E+02

1.75

% error

3.33E+06

1959.81

y = 6E+06x + 2643. R² = 0.971

6000 4000 2000 0 0

0.0005

0.001

0.0015

0.002

Fig. 2 Experimental Kozeny-Carman constant The slope of Fig. 2 gives the experimental Kozeny-Carman constant, and shown to be is 6x106 with an error of 2643. This error accounts for the inaccuracy of the manometer. Some of the error may have also been due to the calculated Cv. Based on the calculated NRe, the flow of the liquid was never turbulent since NRe < 4000. Thus, the Burke-Plummer constant cannot be proven because the values obtained experimentally would not permit it. Fig. 3 shows the experimental results using the Burke-Plummer equation. Determination of the Burke-Plummer constant 14000 12000 10000

In the experiment, the behavior of the bed from being fixed to an expanded one was observed. Given that the bed operates at zero velocity (or even at a very low velocity), the bed was considered to be operating as a packed bed. From the given value in the laboratory, the porosity of bed at no flow, ε0, is 0.414. Minimum fluidization velocity is required in order to observe the start of expansion of bed.

8000 y = -189.0x + 19839 R² = 0.667

6000

The experimental constants were determined for three trials, and were averaged to complete the table. The very high percentage error may be attributed to the inaccuracy of the manometer and flow rate readings, and to the fact that the equipments were old. In fact, the CCl4 manometers just got broken and repaired prior to this experiment. Another source of error was that the initial ∆h of the manometers were never zero, despite fully closing the flow regulating valve. This could have affected the flow meter constant. Ideally, the ∆h should have been set to zero at no flow rate.

4000 2000 0 0

20

40

60

80

Fig. 3 Experimental Burke-Plummer constant Fig. 3 shows a negative slope, which is not possible according to Eq. 12. This is due to the fact the flow was not turbulent, yet the data points were forced into the Burke-Plummer equation. However, the Ergun equation provides a method for determining the experimental Burke-Plummer constant. Since the Ergun equation is applicable to all types of flow, Fig. 4 shows the plot of Eq. 13. Ergun Equation 1.20E+07 1.00E+07 8.00E+06 6.00E+06

Source: http://www.colorado.edu/che/TeamWeimer/images/bed_expansion

From Geankoplis (1993), the pressure drop across the bed is equivalent to the weight of the bed per unit area of cross section. This can be expressed by ∆f  `1 - .g[R - [hL At this point, minimum porosity, εM, is observed to be present. Applying these concepts, Ergun equation can be expressed as

y = 22.24x + 7E+06 R² = 0.621

N

1.75 QRNgjkl h [ N 150g1 - .kl hQR jkl [ QRS [g[R - [h` \ \ 0 S S PN mn .kl PN mn N .kl P

4.00E+06 2.00E+06 0.00E+00 0.00E+00

Fig. 5 Fixed Bed vs. Fluidized Bed

5.00E+04

1.00E+05

1.50E+05

Fig. 4 Experimental Illustration of the Ergun Equation According to Eq. 15, the slope of Fig. 3 gives the experimental Burke-Plummer constant, which is shown to be 22.24. The intercept, meanwhile, verifies the experimental Kozeny-Carman constant earlier determined by Fig. 2. The Kozeny-Carman constant from Fig. 4 is 7x106, which is consistent with the value obtained from Fig. 2. The experimental constants were compared to book values, as shown in Table 2.

If the values for Φs and εmf are not provided, the equation below can be used. 1 - .kl o 11 S mnN .kl Substituting this to the Ergun equation provides a new equation for computing NRe. ]^_  p33.7N \ 0.0408

s

QRS g[R - [h` N r - 33.7 PN

Once NRe is calculated, the values for Vmf may be obtained by using QR jkl [ ]^_,kl  P Assuming the sphericity is unity (for spheres), and if the NRe falls between the range of 0.001 to 4000, the equation can be used. Since the value of pressure drop at minimum velocity is

also to be computed, the bed height at that instance, Lmf, must be calculated. 1 - .kl L/  Lkl 1 - ./ L, cm

ε0 is the porosity at no flow while L0 is the initial height. Obtaining Lmf, the pressure drop can be calculated.

Bed Height vs. Nre

∆fkl  Lkl g1 - .kl hg[R - [h` The data below is a summary of the values for the variables at minimum fluidization for trial 1.

20 18 16 14 12 10 8 6 4 2 0

y = 0.012x - 2.915 R² = 0.959

0

500

1000 Nre

1500

2000

Fig. 7 Bed Height vs. Reynolds number

trial 1 NRe

51.03558906

Vmf

0.034655163

εmf

1.61764E-08

Lmf

0.007690288

dP

4.492281741

Note, however, that the porosity at incipient fluidization is less than the porosity at no flow, which is impossible. This error can be traced to the use of inappropriate constants. An alternative method of determining the values at minimum fluidization may be read from the plot of log(∆P) vs. log(NRe) (Foust, 1980). Determination of the Minimum Fluidization Point 2.025 2.02

80 70

y = 0.053x - 15.30 R² = 0.959

60 50 40 30 20

2.015 log dP

Fig. 8 shows the plot of the bed expansion as a function of Reynolds number. McCabe (2001) provided a method of determining the bed height as a function of velocity. In Fig. 8, the abscissa was manipulated to make the bed height a function of Reynolds number: u  ]vw .S x 1 - . Bed Expansion vs Nre Bed Expansion

Table 3. Calculated Results of Minimum Fluidization Properties

As expected, the bed height is directly proportional to the Reynolds number. From the equations above, the bed height affects the calculation of the velocity and thus the Reynolds number. The non-linearity of the plot is due to the inaccuracy of the recorded manometer readings.

0

2.01

500

1000

1500

2000

Nre

2.005

Fig. 8 Bed Expansion as a function of Reynolds number

2 1.995 1.99 2.9

2.95

3

3.05 3.1 log Nre

3.15

3.2

3.25

Fig. 6 Determining the Minimum Fluidization Point The linear part of the graph in Fig. 6 illustrates the behavior of a fixed bed. The peak of the graph is where the fluidization starts, and is known as the point of fluidization. Beyond the peak, the bed is completely fluidized. Increasing further the flow rate would only result to the same pressure drop, shown by the horizontal part. Only the Reynolds number will be affected. The minimum NRe and the minimum pressure drop can be read directly from the graph, and the minimum fluidization velocity may be computed from NRe. The porosity at the fluidization point may be obtained from the modified Ergun equation. The values at incipient fluidization are shown in Table 4. Table 4. Graphical Results of Minimum Fluidization Properties Trial 1

Trial 2

Trial 3

Average

∆P

105.1961874

109.6478196

105.9253725

106.9231265

NRe

1096.478196

1202.264435

1083.926914

1127.556515

Vmf

0.744551614

0.816384611

0.736028802

0.765655009

The values obtained from Fig. 6 are much more realistic than the values in Table 3. The difference in the values of Table 3 and 4 may be because of the wrong values of constants used in Table 3. The relationship of the Reynolds number with the other relevant fluidization variables is shown in the following graphs. Fig. 7 shows the relationship of bed height L with NRe.

Notice that the plot is almost linear. This means that the bed becomes more fluidized as the flow rate (thus Reynolds number) is increased. Ideally, the intercept should be zero; however, the experimental results give a value of -15.30. This discrepancy can be due to the inaccuracy of the bed height and manometer readings, and in the inaccuracy of the experimental flow meter constant. Foust (1980) provided a method of determining the modified friction factor based on the Reynolds number. Fig. 9 shows the plot that calculated using ]vw u 1 - . -∆K`a QR .S x N L [Onk 1 - . Modified Friction Factor 250 200 150 100

y = 2E-08x2 - 0.004x + 347.2 R² = 0.992

50 0 0

50000

100000

150000

Nre/(1-e)

Fig. 9 Friction Factor vs. NRe The experimental plot displays a pattern similar to literature. The initial part of the graph represents the period of laminar

flow and the Kozeny-Carman is applicable. The later part represents the period of turbulent flow and the Burke-Plummer is applicable. All throughout, the Ergun equation is applicable.

A.1 Determination of Porosity, ε Trial 1 ∆hHg,0 = 0.4 in

CONCLUSIONS AND RECOMMENDATIONS Evidently, the data gathered from the experiment did not fully satisfy the primary objective of the activity as stated in the manual. This may be due to the repeated bursting of the pipes in the set-up. The valve which regulated the flow of water may also add to the inaccuracy of the readings in the gages. It was difficult to control the fluid flow. Experimental errors can be traced all the way back to the determination of the flow meter constant. It is recommended that a 1000 ml graduated cylinder be used instead of a normal bucket. That way, the volume readings, and thus the flow rate, could be more accurate. Also, it would be better if the graduations of the manometers were clearer. It was hard to read accurately the height difference in the manometers. The use of more sensitive manometers would also help. A more accurate flow regulating valve would also result in better experimental results. REFERENCES Foust, A.S. (1980). Principles of Unit Operations. Singapore: John Wiley & Sons (Asia) Pte Ltd. Geankoplis, C.J. (1993). Transport Processes and Unit Operations. Singapore: Prentice Hall. McCabe, W.L. (2001). Unit Operations of Chemical Engineering. Singapore: McGraw-Hill Book Co. (n.d.).Retrieved March 8, 2008 from http://www.eng.buffalo.edu/Courses/ce427/Beds.PDF (n.d.). Retieved March 10, 2008 from http://chemical.uakron.edu (n.d.). Retieved March 10, 2008 from http://www.colorado.edu/che/TeamWeimer/images/bed_expansion

APPENDICES Constants used: Bed height at no flow, Lo = 0.4 cm Bed porosity at no flow, ε0 = 0.414

∆hCCl4,0 = 0.3 in

∆hHg, in

∆hCCl4, in

L, cm

L, in

ε

1.5

0.5

8

3.149606299

0.9707

1.6

0.7

10

3.937007874

0.97656

1.6

0.8

12.5

4.921259843

0.981248

1.6

0.9

13

5.118110236

0.981969231

1.6

1.1

13.5

5.31496063

0.982637037

1.6

1.3

15

5.905511811

0.984373333

1.6

1.35

15

5.905511811

0.984373333

1.6

1.4

16

6.299212598

0.98535

1.6

1.5

17.5

6.88976378

0.986605714

1.6

1.55

17.5

6.88976378

0.986605714

Trial 2 ∆hHg,0 = 0.3 in

∆hCCl4,0 = 1 in

∆hHg, in

∆hCCl4, in

L, cm

L, in

ε

0.9

0.8

8

3.149606299

0.9707

1.6

0.9

8.3

3.267716535

0.971759036

1.6

1

9

3.543307087

0.973955556

1.6

1.05

11.5

4.527559055

0.979617391

1.6

1.1

13

5.118110236

0.981969231

1.6

1.3

13

5.118110236

0.981969231

1.6

1.35

13

5.118110236

0.981969231

1.6

1.55

15

5.905511811

0.984373333

1.6

1.6

15

5.905511811

0.984373333

1.6

1.7

17

6.692913386

0.986211765

Trial 3 ∆hHg,0 = 0.2 in

∆hCCl4,0 = 1 in

∆hHg, in

∆hCCl4, in

L, cm

L, in

ε

1.3

0.6

8

3.149606299

0.9707

1.6

0.7

8.5

3.346456693

0.972423529

1.6

0.8

11

4.330708661

0.978690909

1.6

0.9

12

4.724409449

0.980466667

1.6

1.1

13

5.118110236

0.981969231

1.6

1.3

14

5.511811024

0.983257143

1.6

1.4

15

5.905511811

0.984373333

1.6

1.5

15.5

6.102362205

0.984877419

Water density, ρH2O = 62.42796058 lbm/ft3 Water viscosity, µH2O = 0.00060054 lbm/ft-s Mercury density, ρHg = 849 lbm/ft3

A.2 Determination of Reynolds Number Trial 1

Total particle mass = 0.986 kg Particle diameter, Dp = 0.014166667 ft

?Δh,yyDZ

F, lb/s

mean v, ft/s

superficial velocity, ft/s

Nre

0.70710678

0.803980

0.6209687

0.602774321

887.6871286

0.83666002

0.951282

0.7347400

0.717517771

1056.666265

0.89442719

1.0169637

0.7854701

0.770741047

1135.046539

0.94868329

1.0786529

0.833116

0.818095201

1204.78354

1.04880884

1.192495

0.9210454

0.905053355

1332.844127

1.14017542

1.2963794

1.0012819

0.985635249

1451.514596

1.16189500

1.3210746

1.0203557

1.004410941

1479.164977

1.18321595

1.3453165

1.0390793

1.023856871

1507.802398

1.22474487

1.3925349

1.0755493

1.06114313

1562.712719

1.24498996

1.4155535

1.0933282

1.078683874

1588.544431

Pipe cross-sectional area = 0.02073942 ft

Trial 2

Trial 3

?Δh,yyDZ

F, lb/s

mean v, ft/s

0.89442719

1.01696371

0.785470184

0.94868329

1.07865291

0.83311694

superficial velocity, ft/s

BP x, lbm/ft2-s2

0.762455907

1122.84527

0.26246

0.10833

85.2119

10445.5

0.0018545

61.54811

0.809588915

1192.25659

0.27887

0.133333

104.876

12099.8

0.0017680

67.46245

0.36089

0.133333

104.876

9349.84

0.001114

59.19571

0.39370

0.133333

104.876

8570.69

0.0009894

60.93501

0.42650

0.133333

104.876

7911.41

0.0009292

68.64200

0.45931

0.133333

104.876

7346.30

0.0008687

75.22924

0.49212

0.133333

104.876

6856.55

0.0007835

75.52929

0.50853

0.133333

104.876

6635.37

0.0007587

78.27370

1

1.137

0.878182363

0.855310591

1259.58949

1.16507830

0.899869143

0.881527463

1298.19827

1.04880884

1.19249566

0.921045432

0.904438275

1331.93831

1.14017542

1.29637945

1.001281949

0.983228065

1447.96960

1.16189500

1.32107461

1.0203557

1.001957902

1475.55246

1.24498996

1.41555358

1.093328224

1.076243149

1584.95005

1.26491106

1.43820388

1.110822587

1.093464133

1610.31086

1.30384048

1.48246662

1.145009714

1.129222051

1662.97044

dP

KC x, lb m/ft2-s2

L, ft

1.02469507

∆hHg, ft

y, lb m/ft2-s2

Nre

A.4 Determination of Experimental Ergun Constants Trial 3 Trial 1

?Δh,yyDZ

F, lb/s

mean v, ft/s

superficial velocity, ft/s

Nre

0.77459666

0.880716413

0.680237133

0.660306185

972.412528

0.83666002

0.95128245

0.734740079

0.714478541

1052.19048

0.89442719

1.016963716

0.785470184

0.768732528

1132.08865

0.328083

0.133333333

104.87627

8119698.198

45079.61882

0.94868329

1.07865291

0.83311694

0.816843389

1202.94003

0.4101049

0.133333333

104.87627

9585474.866

60529.35897

0.4265091

0.133333333

104.87627

9412584.784

66818.19972

L, ft

∆hHg, ft

dP

y

Er x

0.2624671

0.125

98.321504

7119301.156

30296.48903

1.04880884

1.19249566

0.921045432

0.904438275

1331.93831

1.14017542

1.296379458

1.001281949

0.984517628

1449.86870

1.18321595

1.345316543

1.039079384

1.022842037

1506.30788

0.4429133

0.133333333

104.87627

8853494.639

76763.63362

1.22474487

1.392534919

1.075549345

1.059284263

1559.97522

0.4921259

0.133333333

104.87627

9080929.569

92887.02617

0.4921259

0.133333333

104.87627

8911177.599

94656.46185

0.5249343

0.133333333

104.87627

9352507.19

102921.6654

0.5741469

0.133333333

104.87627

9907651.371

116670.1049

0.5741469

0.133333333

104.87627

9746540.611

118598.667

A.3 Determination of Kozeny-Carman and Burke-Plummer Constants Trial 1 L, ft

∆hHg, ft

dP

y, lbm/ft2-s2

KC x, lb m/ft2-s2

BP x, lb m/ft2-s2

0.2624671

0.125

98.32150

12052.5391

0.001692

51.29009

0.328083

0.13333

104.8762

10284.8334

0.001266

57.10019

0.4101049

0.13333

104.8762

8227.86671

0.000858

51.95647

0.4265091

0.13333

104.8762

7911.41030

0.000840

56.16163

0.4429133

0.13333

104.8762

7618.39511

0.000860

66.05478

0.4921259

0.13333

104.8762

6856.55559

0.000755

70.13434

0.4921259

0.13333

104.8762

6856.55559

0.000769

72.83182

0.5249343

0.13333

104.8762

6428.02087

0.000687

70.73853

0.5741469

0.13333

104.8762

5877.04765

0.000593

69.20669

0.5741469

0.1333

104.8762

5877.04765

0.000602

71.51358

∆hHg, ft

dP

y, lb m/ft2-s2

KC x, lb m/ft2-s2

BP x, lb m/ft2-s2

0.262467

0.075

58.99290

7231.523

0.00214141

82.06414

0.272309

0.133333

104.8762

12391.36

0.00210549

88.88824

0.295275

0.133333

104.8762

11427.59

0.00187907

90.87760

0.377296

0.133333

104.8762

8943.333

0.00116571

74.24607

0.426509

0.133333

104.8762

7911.410

0.00092922

68.64200

0.426509

0.133333

104.8762

7911.410

0.00101017

81.12236

0.426509

0.133333

104.8762

7911.410

0.00102941

84.24245

0.492125

0.133333

104.8762

6856.555

0.00082446

83.62172

0.492125

0.133333

104.8762

6856.555

0.00083765

86.31919

0.557742

0.133333

104.8762

6049.901

0.00066971

80.77339

Trial 2

Trial 2 L, ft

L, ft

∆hHg, ft

dP

y

Er x

0.2624671

0.075

58.99290296

3376981.05

38322.36418

0.2723097

0.1333333

104.8762719

5885251.913

42217.27703

0.2952755

0.1333333

104.8762719

6081515.696

48363.0779

0.3772965

0.1333333

104.8762719

7671975.516

63691.46817

0.4265091

0.1333333

104.8762719

8514003.282

73870.29922

0.4265091

0.1333333

104.8762719

7831743.94

80305.48175

0.4265091

0.1333333

104.8762719

7685343.294

81835.24743

0.4921259

0.1333333

104.8762719

8316414.642

101425.9845

0.4921259

0.1333333

104.8762719

8185439.297

103048.9033

0.5577427

0.1333333

104.8762719

9033496.129

120607.9247

L, ft

∆hHg, ft

dP

y

Er x

0.2624671

0.1083333

85.21197094

5632469.32

33188.14091

0.2788713

0.1333333

104.8762719

6843399.085

38155.37164

0.3608923

0.1333333

104.8762719

8391311.521

53127.02738

0.3937007

0.1333333

104.8762719

8661970.537

61583.96085

0.4265091

0.1333333

104.8762719

8514003.282

73870.29922

0.4593175

0.1333333

104.8762719

8456324.18

86596.25396

0.4921259

0.1333333

104.8762719

8750602.689

96393.42254

0.5085301

0.1333333

104.8762719

8744630.704

103155.3582

Trial 3

A.5 Determination of Point of Fluidization Trial 1 log Nre

log dP

2.948259923

1.992648517

3.023937842

2.020677241

3.055013669

2.020677241

3.080909025

2.020677241

3.124779363

2.020677241

3.161821407

2.020677241

3.170016615

2.020677241

3.17834443

2.020677241

3.193879147

2.020677241

3.200999366

2.020677241

Trial 2 log Nre

log dP

3.050319914

1.770799768

3.076369733

2.020677241

3.10022903

2.020677241

3.113341027

2.020677241

3.124484113

2.020677241

3.160759447

2.020677241

3.168954655

2.020677241

3.20001558

2.020677241

3.206909722

2.020677241

3.220884531

2.020677241

Trial 3 log Nre

log dP

2.987850546

1.930500611

3.02209437

2.020677241

3.053880439

2.020677241

3.080243979

2.020677241

3.124484113

2.020677241

3.161328677

2.020677241

3.177913749

2.020677241

3.193117701

2.020677241