Fluid Flow through Packed Beds: Experimental Data vs. Ergun’s Equation

Team 6 Jennifer Sandidge

David Shin Sebastian Vega-Fuentes

LaToya Williams

March 31, 2005

2

Abstract The goal of this experiment was to evaluate the validity of the Ergun Equation for packed beds.

In order to obtain experimental data, two packing materials, three rotameters, and two columns

of different diameters were used. From the experimental data, plots of pressure drop vs. flow-

rates were generated. This experimental data was compared to the theoretical data obtained from

Ergun’s Equation under the same experimental conditions. The conclusion of this comparison

showed that although Ergun’s Equation did not match the experimental data, the former can be

used as an estimation tool.

3

Table of Contents

Introduction..................................................................................................................................... 1

Theory ............................................................................................................................................. 1

Experimental Equipment ................................................................................................................ 3

Experimental Procedure.................................................................................................................. 4

Results............................................................................................................................................. 5

Discussion....................................................................................................................................... 8

Conclusion .................................................................................................................................. 100

Nomenclature and Abbreviations ............................................................................................... 111

References..................................................................................................................................... 12

Appendix....................................................................................................................................... 13

A1. Calibration Data ................................................................................................................. 13 A2. Rotameter Pressure Drop Graphs Pea Gravel .................................................................... 14 A3. Rotameter Pressure Drop Graphs Black Marbles .............................................................. 17 A4. Void Fraction Calculations ................................................................................................ 18 A5. Particle diameters............................................................................................................. 199 A6. Pressure Drop versus Volumetric Flow rate with initial void fraction .............................. 19 A7. Pressure Drop versus Volumetric Flow rate with modified void fraction ......................... 21 A8. Friction Factor versus Reynolds Number Graphs.............................................................. 23

1

Introduction Packed beds have many uses in the chemical industry. Examples of the various applications of

packed beds include: adsorption, gas-liquid adsorption, and catalytic reactors. Packed beds are

usually comprised of a column with various types of packing materials. Fluid flows into the

column from the bottom, passes through the packed material and exits at the top of the column.

There are two pressure nodes above and below the packing that measure the pressure drop across

the column. The dominating factors that change the pressure drop are: column diameters, liquid

properties of fluid and rate of flow, and material properties of the packing.

The purpose of this experiment was to test the validity of the Ergun Equation for pressure drop

across a packed bed. To do so, the pressure drops were obtained at various fluid flows for two

different column diameters (3.5” and 6”) and two different packing materials (black marbles and

pea gravel). To obtain the pressure drop data, the rotameters were calibrated and a relationship

between the rotameter readings and flow rate was determined. The manometer was also

calibrated to obtain a relationship between the manometer reading and pressure in units of

inH2O, and finally the void fraction for both the pea gravel and black marbles were calculated.

The raw data was compared to the pressure drop predictions from the Ergun Equation to test its

validity.

Theory

The Ergun equation is an estimation of the pressure drop through a packed bed due to the

following factors: rate of fluid flow, fluid properties (viscosity and density), density of packing

(void fraction), and physical properties of the packing material. This equation originated from

the following relationship:

2VbaVLP ρ+=

∆ (1)

In this equation, a and b pertain to column packing and fluid properties.

2

From this relationship, Blake and Kozeny found that for viscous laminar flow, the pressure drop

was proportional to the porosity of the packing material in the following manner:

3

2)1(εε− (2)

They also determined how other factors effected equation (1), which resulted in the Blake-

Kozeny equation:

223

21 )1(

ps DVk

LP

⋅⋅⋅⋅−⋅

=∆

φεµε (3)

where k1 is a dimensionless empirical constant that through many experiments was determined to

equal to 150.

Since flow in the column can become turbulent, Burke and Plummer found that for turbulent

flow the pressure change due to kinematic energy loss was proportional to the porosity of the

packing material in the following manner:

3

1εε− (4)

They also determined how other factors played an effect on equation (1), which resulted in the

Burke-Plummer equation:

spDV

LP

φερε

⋅⋅⋅⋅−⋅

=∆

3

2)1(75.1 (5)

where k2 is a dimensionless empirical constant that through many experiments was determined to

equal to 1.75.

Ergun made the assumption that the total pressure drop across a fluidized bed is due to the sum

of the viscous and kinematic forces. Through many experiments Ergun concluded that his

equation is valid for a wide range of Reynold numbers:

spps DLV

DLVP

φερε

φεµε

⋅⋅⋅⋅⋅−⋅

+⋅⋅

⋅⋅⋅−⋅=∆ 3

2

223

2 )1(75.1)1(150 (6)

In this equation the fluid properties are density (ρ) and viscosity (µ) and the packing properties

are the sphericity (φ) and the equivalent diameter (Dp). The flow rate of the fluid is in terms of

3

velocity (V), and to account for the density of the packing material, the Ergun Equation has the

void fraction (ε).

The pressure drop used in both the Blake-Kozeny equation and the Burke-Plummer equation is

directly proportional to the friction factor (fp). Equations 7, 8, and 9 express the Blake-Kozeny,

Burke-Plummer, and Ergun Equation respectively in terms of the friction factor:

sp N

fφε

ρ ⋅−⋅

=Re,

)1(150 (7)

75.1=pf (8)

75.1)1(150

Re,

+⋅−⋅

=s

p Nf

φε

ρ

(9)

The experimental friction factors can be calculated using equation 10.

)1(2

3

ερ

εφ

−⋅⋅⋅

⋅⋅⋅⋅∆=

LVDgP

f pscp (10)

In this equation gc is the gravitational conversion factor. Using this equation and the Reynolds

number equation a comparison can be established with the Ergun Equation’s friction factors.

Although the Ergun Equation is valid for a wide range of Reynold numbers, it fails once the fluid

flow surpasses the fluidization point. The fluidization point occurs when the pressure drop times

the cross-sectional area of the column equals the gravitational force of the packing material (F =

∆P*A). Past this point, the packing begins to move, and some of the parameters of the Ergun

Equation change unpredictably. This experiment aims to test the accuracy of the Ergun Equation

before reaching the fluidization point by comparing its predictions with the experimental data.

Experimental Equipment

For this experiment, the following equipment was used: two columns (3.5” and 6” diameters),

three rotameters (W1, W2, and W3), one manometer (in units of inH2O), and two packing

materials (pea gravel and 5/8” diameter black marbles).

4

An apparatus held the column in an upright position. Two pressure nodes were then fastened

above and below the packing to measure the pressure drop across the column, which was read

off the manometer display. The inlet and outlet flow rates were controlled by the three

rotameters. The setup of this experiment is shown in Figure 1:

Figure 1. Diagram of Packed Bed Setup: Water flows from the bottom to the top of the column. The fluid flow

rate was controlled using a rotameter, while the manometer gave a pressure drop reading.

The top and bottom of the column had rubber o-rings in order to ensure a secure fit, which

prevented any air from entering and water from leaking out of the system.

Experimental Procedure

The first step was to calibrate the manometer in order to have a relationship between the

manometer reading and pressure units. This was done by employing the following steps: fill

column to certain height with water, tare the manometer display, drain 10” of water and record

the pressure drop. Then the pressure drop was divided by the change in height.

Then the column was filled with packing material and settled. Next, the column was flooded

with water and the inlet lines were bled to remove air bubbles. Then the rotameters were

calibrated in order to have a relationship between the rotameter readings and flow rates.

The final step of our experimental procedure was to record the pressure drop readings against the

volumetric flow rate. While increasing and decreasing the volumetric flow rate using the

rotameters, the respective pressure drops for each flow rate were recorded.

Water In Rotameter

Manometer

Water out

Column

Packing Materials

5

To use the Ergun equation, the void fractions for both packing materials were calculated. The

void fraction was calculated by adding one of the packing materials to a graduated cylinder and

recording its volume. Water was then added to the same graduated cylinder until the entire

volume of packing material was saturated. This volume of water was also recorded. The void

fraction (ε) equals the volume of water divided by the volume of packing material.

Results

In order to use the Ergun Equation, void fractions were calculated using each of two graduated

cylinder diameter sizes, 1.5” and 4”. Using the linear regression equation, the void fraction for a

3.5” diameter column was found. The void fraction for the 6” diameter column was assumed to

be the 4” diameter case. The linear regression for the calculation of the void fraction, was found

using Excel to be y = -0.0636x + 0.5935, as shown in Figure 2.

y = -0.0636x + 0.5935

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

Diameter of Graduated Cylinder (in)

Void

Fra

ctio

n

Void fraction

Linear (Void fraction)

Figure 2. Void Fraction Pea Gravel: This graph plots the two graduated cylinder sizes and the linear regression for the two known void fractions.

Data calibrations for the W-2 and W-3 rotameter were conducted. Using this information,

graphical representations of the volumetric flow rate as a function of the rotameter reading were

6

created. Figure 3 shows an example of our W-2 rotameter calibration curve. The linear

regression of the rotameter data was found to be y=0.004x-0.0174.

y = 0.004x - 0.0174

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 10 20 30 40 50 60 70 80 90 100

Rotometer Reading (R)

Flow

rate

(L/s

)

Figure 3. W-2 Rotameter Calibration Curve: This graph shows W-2 rotameter readings and corresponding calculated flow rates. The relationship was obtained by calibrating the rotameters.

Then, the pressure drop at various flow rates were tabulated and graphed for each column

diameter and each type of packing material. For the Ergun Equation, the velocity was calculated

by dividing the volumetric flow rate (Q) by the area of the column. The experimental data and

the predictions using Ergun’s Equation were plotted on the same graph. The graph of the

pressure drop versus the volumetric flow rate using pea gravel as the packing material in the

large column is shown in Figure 4:

7

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004

Q W-2 (m3/s)

dP (P

a)

Figure 4. Pressure Drop vs. Volumetric Flow Rate for Pea Gravel in 6” Diameter Column: the graph displays the

pressure drop in the large column with a .338 void fraction. Fluidization can be seen where the experimental data

levels off.

The friction factors and Reynold numbers for the packing material were calculated. The Ergun

Equation friction factors and Reynold numbers were also calculated. Figure 5 shows the friction

factor versus Reynolds plot for pea gravel with the Ergun equation plotted as well.

8

0

20

40

60

80

100

120

0 20 40 60 80 100 120

Re

Fric

tion

Fact

or

Small ColumnErgun's EquationLarge Column

Figure 5. Friction Factor vs. Re for Pea Gravel: This graph compares friction factors for pea gravel the small and

large columns against the predicted values. For the error bars the small column had a standard deviation of ±21.82,

and for the large column the error bars had a standard deviation of ±3.94.

Discussion

When comparing Ergun’s equation to the experimental data, it is seen that the Ergun equation

deviates from the experimental data. The factors that can be attributed to this deviation could be

due to the assumptions made about the apparatus. When deriving his equation, Ergun used

packing material with a rough surface whereas in this experiment the packing materials were

rather smooth.

Another source of error is the accuracy of the calculated void fraction. When the void fraction

was changed and all other variables were left unchanged, the predicted values showed a better fit

with the data with a lower void fraction, see Appendix A6 and A7. It is most likely that this

overestimation of the void fraction occurred due to the fact that the small column void fraction

for pea gravel was tabulated and the large column void fraction was assumed for both the pea

gravel and black marbles. Table 1 compares the drop in pressure using the calculated and

adjusted void fractions versus the experimental pressure drops:

9

Table 1.: This table shows the pea gravel pressure drops using the calculated and the adjusted void fractions. These pressure drops are compared to the experimental data. The void fraction used in the original analysis was .3709, and the adjusted void fraction was .265. The experimental pressure drops show a better fit to the modified Ergun pressure drop.

Experimental Experimental (ε) Adjusted (ε) ∆P (Pa) ∆P (Pa) Error ∆P (Pa) Error 7472.40 1322.60 82.30 4839.66 35.23 8219.64 1735.80 78.88 6320.91 23.10 10461.36 2176.95 79.19 7891.69 24.56 11955.84 2646.05 77.87 9552.00 20.11 14197.56 3143.10 77.86 11301.83 20.40 15692.04 3668.09 76.62 13141.19 16.26 17186.52 4221.04 75.44 15070.08 12.31 18681.00 4801.93 74.30 17088.49 8.52

As the system approaches fluidization, the Ergun Equation starts to fail. This is because the void

fraction changes in fluidization since the pea gravel starts to unsettle so its density decreases,

thus yielding a higher void fraction. The length of the packing increases as well, giving a higher

pressure drop than the actual.

Another source of error arose from the calibration of the rotameters. The rotameter levels were

unsteady, so the flow rates were not constant. The error in the calibration can be seen in

Appendix A.1. The flow rate approximation used for calculations contributed to the experimental

and theoretical data deviation. This error would propagate to the calculation of the predicted

Ergun pressure drops and cause them to change.

Lastly, the assumption that sphericity (phi) is equal to 1 for both the black marbles and the pea

gravel could provoke problems. Pea gravel is highly non-spherical, so the sphericity of the pea

gravel should be less than one. Since sphericity is in the denominator, a lower phi value will

yield a greater pressure drop than the experimental values presented. This explains the higher

calculated Ergun values.

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Conclusion

The experimental data for pea gravel and the black marbles does follow the trend of the Ergun

Equation. The black marbles showed a closer fit with the Ergun equation. This can be explained

by the uniform shape and size of the marbles.

At higher flow rates, there were small pressure peaks once the bed had fluidized. This is due to

the variation in the void fraction which was caused by the fluidization. During this time, both

materials deviated from the Ergun equation. This showed that the Ergun equation no longer

applies to the system after this pressure peak has been reached.

The Ergun equation in this experiment is assumed to be valid while using approximated values in

order to validate it to experimental data. Without better ways to measure error other than

qualitatively there will be some uncertainty in the calculations and calibrations done during this

experiments.

In the future, groups may want to run their experiments at a higher starting level in the column,

which would reduce the error. Also, it would be best to keep a constant height for each run, as it

would standardize the results. Another idea for the future would be to use finer particles,

opposed to pea gravel or marbles, as this would reduce the structured void space between each

particle. This would thus decrease the void fraction and may be a better way to check the

validity of Ergun’s equation. Lastly, groups should not only increase the flow rates, but

decrease it as well, as this will show a higher pressure drop as the void fraction will decrease.

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Nomenclature and Abbreviations

Lower-case letters

a representation of packing and fluid characteristics at laminar flow

b representation of packing and fluid characteristics at turbulent flow

fp friction factor

gc gravitational conversion factor

k1, k2 dimensionless empirical constant

Upper-case letters

As surface area of packed bed (m2)

Dp equivalent diameter (m)

L depth of column (m)

NRe, ρ Reynolds number

∆P pressure (Pa)

V velocity of fluid through entire column (m·s-1)

Greek Letters

ε porosity (dimensionless)

µ dynamic viscosity (kg·m·s-1)

ρ density (kg·m-3)

τw shear force per unit area (Pa)

φS sphericity (dimensionless)

12

References 1. McCabe, smith, and Harriott, Unit Operations of Chemical Enginerring, 6th edition,

McGraw-Hill, 2001.

2. Perry, R.H. and D.W. Green (eds.), Chemical Enigineer’s Handbook, 7th ed., McGraw-

Hill, 1997.

3. Packed and Fluidized Beds, CE427-Chemical Engineering Laboratory III, Fall 2004

Available: http://www.eng.buffalo.edu/courses/ce427/fluidized%20bed.pdf

4. Ludwick, R., S. Marshall, T. O’Dowd, G. Pan, and H. Yun, Flow in Packed Beds,

Team 5, Spring 2004

5. Back, S. A. Beaber, E. Boudreaux, K. Paavola, Pressure Drop for Flow in Packed

Beds: An analysis using Ergun’s Equation, Team 3, Feb. 18, 2004.

13

Appendix

A1. Calibration Data Calibration Data for W-2

rotatmeter reading (W-2) time (s) flowrate (L/s)

28 102 0.098039216

60 47 0.212765957

90 29 0.344827586

Calibration Data for W-3 rotatmeter reading (W-3) time (s) flowrate (L/s) 5 35 0.28571429 12.1 20 0.5 15.2 13 0.76923077

W-3 Calibration y = 0.0444x + 0.04R2 = 0.9192

00.1

0.20.30.4

0.50.60.7

0.8

0.9

0 2 4 6 8 10 12 14 16Rotometer Reading. R

14

W-2 Calibration y = 0.004x - 0.0174R2 = 0.9965

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 20 40 60 80 100Rotometer Reading, R

A2. Rotameter Pressure Drop Graphs Pea Gravel

1st Trial small column pea gravel Rotameter velocity vs. Pressure (in H20)

0

10

20

30

40

50

60

70

80

90

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Rotameter Velocity W-2

15

2nd Trial small column pea gravel Rotameter velocity vs. Pressure (in H20)

0

10

20

30

40

50

60

70

80

90

0 0.02 0.04 0.06 0.08 0.1 0.12

Rotameter W-2 Velocity

Small column pea gravel Rotameter velocity vs. Pressure (in H20)

73.5

74

74.5

75

75.5

76

76.5

0 0.05 0.1 0.15 0.2 0.25 0.3

Rotameter W-3 Velocity

16

large column pea gravel Rotameter velocity vs. Pressure (in H20)

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

Rotameter W-2 Velocity

Large column pea gravel Rotameter velocity vs. Pressure (in H20)

0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Rotameter W-3 Velocity

17

A3. Rotameter Pressure Drop Graphs Black Marbles

1st Trial small column black marbles Rotameter velocity vs. Pressure (in H20)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Rotameter W-2 Velocity

2nd Trial small column black marbles Rotameter velocity vs. Pressure (in H20)

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Rotatmeter W-2 Velocity

18

2nd Trial small column black marbles Rotameter velocity vs. Pressure (in H20)

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Rotatmeter W-3 Velocity

1st Trial small column small black marblesRotatmeter velocity vs Pressure

0

10

20

30

40

50

60

70

80

90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Rotameter W-3 Velocity

A4. Void Fraction Calculations 1.5 in Diam. 4in diam. Cylinder 4in diam. Cylinder

500ml pea gravel unsettled 1800 ml pea gravel 1000mL black glass marbles 249 ml added 610 ml added 385 mL added void fraction 0.498 void fraction 0.338888889 void fraction 0.385

19

A5. Particle diameters Packing Type Particle diameter (m) Pea gravel .002 Black Marbles 0.0178562

A6. Pressure Drop versus Volumetric Flow rate with initial void fraction Pea Gravel Large Column void fraction, .338

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 5 10 15 20 25 30Q W-3 (m3/s)

ExperimentalErgun

Pea Gravel, Large Column void fraction .338

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004Q W-2 (m3/s)

ExperimentalErgun

20

Black Marbles Small column void fraction, .385

0

1000

2000

3000

4000

5000

6000

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004Q W-2 (m3/s)

ExperimentalErgun

Black Marbles Small column void fraction .385

0

5000

10000

15000

20000

25000

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008Q W-3 (m3/s)

ExperimentalErgun

21

A7. Pressure Drop versus Volumetric Flow rate with modified void fraction

Pea Gravel Large Column void fraction, .25

0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30

Q W-2 (m3/s)

Experimental

Ergun

Pea Gravel Large Column void fraction .25

0

5000

10000

15000

20000

25000

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004Q W-2 (m3/s)

ExperimentalErgun

22

Black Marbles Small column void fraction, .3

0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12 14 16 18 20

Q W-2 (m3/s)

Experimental

Ergun

Black Marbles Small column void fraction, .3

0

5000

10000

15000

20000

25000

30000

0 2 4 6 8 10 12 14 16 18

Q W-3 (m3/s)

Experimental

Ergun

23

A8. Friction Factor versus Reynolds Number Graphs

Black Marble with Ergun Equation

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500Re

Ergun Equationblack marbles, small column w-2black marbles, small column w-3

Pea Gravel Data with Ergun Equation

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200Re

pea gravel, small column w-2Ergunpea gravel, large column w-2pea gravel, large column w-3

24

Ergun equation with all packing and columns

0

25

50

75

100

0 500 1000 1500 2000 2500 3000Re

Ergunequation

pea gravel,small columnW-2

pea gravel,large columnw-2

pea gravel,large columnw-3

blk marbles,small columnw-2

blk marbles,small columnw-3