Pressure Drop for Flow Through Packed Beds Team 4 Pierre Guimard Daniel McNerny Edmund Saw Allen Yang
06-363 Transport Process Laboratory Carnegie Mellon University March 18, 2004
Abstract The experiment studies the pressure drop through a packed bed and compares the data to evaluate the validity of the Ergun equation. Two different columns of different diameters, two different types of packing and varying water flow rates were used to collect a large range of Reynolds numbers. The experimental data was plotted and compared to the theoretical curve given by the Ergun equation, a relationship between fluid velocity, the type of packing, and the pressure drop over the distance of a packed column. The void fractions were 0.37 ± 0.02 for pea gravel and 0.465 ± 0.02 for black marbles. For pea gravel, an Ergun constant deviation of 512 ± 200 and 1.65 ± 0.8 is obtained, compared to the empirical value of 150 and 1.75 obtained by Ergun. The experimental data followed the trend of the Ergun equation.
Table of Contents Introduction
1
Theory
2
Experimental
4
Results
6
Discussion
8
Conclusions
12
Nomenclature
13
References
14
Appendices
A
Introduction The packed bed is a component of important operations in chemical and other process engineering fields. The packed bed is commonly used for processes involving absorption, adsorption of a solute, distillation, filtration and separation (Geankoplis, p.125). The packed bed involves flowing one or two fluids through a tower with a fixed bed of particles. For flow of one fluid, the packing removes a specific material from the fluid through adsorption. With two fluids, liquid enters from the top of the column and flows downward, wetting the packing material. A gas enters at the bottom, and flows upward, contacting the liquid in a countercurrent fashion, initiating mass and energy transfer between the fluids (Subramanian). The pressure drop of a one fluid flow is studied in this experiment. The pressure drop can be the driving force in many of these reactions, such as filtration. Varying the pressure drop can reduce the residence time of reagents in the bed. The pressure drop is also important in determining the energy requirements to pump a fluid any given bed. This experiment studies the pressure drop through a packed bed while varying water flow rate, column diameter, and packing material. Pressure drops across the column are measured via simple differential transducers and flows are measured via calibrated rotometers (“Flow”). Two columns of different diameter and two different types of packing were used to collect data for a large range of Reynolds numbers. The experimental data was then plotted and compared to the theoretical curve given by the Ergun equation, a relationship between fluid velocity, the type of packing, and the pressure drop over the distance of a packed column. The objective of the experiment is to test the accuracy of the Ergun model and determine the range of conditions that it is valid.
1
Theory There are several approaches to treating fluid flow through packed beds. The most successful of these is the Ergun Equation, which describes flow in both the laminar and turbulent regimes. This method treats the packed column as a compact irregular bundle of tubes. Modifying the theory for straight tubes not only takes into account the irregularity of the tubes, but yields relationships similar to those derived for straight tubes as well. Geankoplis provides detailed derivations for the flow through straight pipe relations used as a basis for the following derivations (Geankoplis, p.85). This analysis assumes several conditions. First, we assume that there is no channeling in the packed bed. Channeling occurs when the fluid flowing through the packed bed finds a “preferred path” through the bed. We also assume that the diameter of the packing is much smaller than the diameter of the column as well. The maximum recommended particle diameter is one-fifth of the column diameter. We assume that velocity, particle diameter and void fraction behaves as a bulk behavior and hence we can use an average values. Just as with straight pipes, Ergun relates the flows and pressure drops to a Reynolds number and friction factor respectively. The Reynolds number for packed beds, Rep, depends upon the controlled variable vs and the system parameters ρ, ε, µ, and Dp and is defined as (Bird et al., 1996): Re p =
ρv s D p µ (1 − ε )
(1)
where Dp is the equivalent spherical diameter of the particle, vs is the superficial velocity defined as the volumetric flow rate divided by the cross-sectional area of the column, ρ is the fluid density, ε is the dimensionless void fraction defined as the volume of void space over the total volume of packing, and µ is the fluid viscosity. The friction factor, fp, depends upon vs and the pressure drop, ∆P, and system parameters, and is defined as (Bird et al., 1996): fp =
∆P D p ε 3 L ρv s2 1 − ε
(2)
where L is the length of the packed bed and ∆P is the pressure difference in the column. Bird et al use a derivation similar to one for flow through a straight pipe, using a hydraulic radius, Rh, which is the radius of a “crooked pipe” as represented by the model for packed beds. Ergun 2
relates this hydraulic radius to the void fraction and superficial velocity to arrive at the above equation. In laminar flows, viscous forces dominate the friction factor. The Blake-Kozeny equation shows a strong dependence of the friction factor to the Reynolds’s number:
f f = 75
(1 − ε ) 2
µ
ε
ρvs D p
3
(3)
which is valid for Rep < 10. In turbulent flows, kinetic forces dominate the friction factor. The Burke-Plummer equation shows that the friction factor is independent of the Reynolds number:
ff =
7 (1 − ε ) 8 ε3
(4)
The equation is valid for Rep > 1000. Ergun superimposes equation (3) and (4) into an equation that describes the friction factor for all flows:
f f = 75
(1 − ε ) 2
µ
ε
ρv s D p
3
+
7 (1 − ε ) 8 ε3
(5)
Ergun checked the equation for a variety of material and flow rates and determine that equation (5) is valid even for Reynolds’s number between 10 and 1000. For the runs that Ergun did, he simplified the parameters into (Ergun, 1952): ff =
150 + 1.75 Re p
(6)
where equation (6) shows the constants that Ergun obtained with his material and flow rates. Figure 1 shows the correlation between Ergun, Blake-Kozeny, and Burke Plummer equation. We call the values 150 and 1.75 to be the Ergun constants.
3
Figure 1: Friction factor vs. Reynold’s number. This graph illustrates the correlation between Ergun and BlakeKozeny at Re < 10; Ergun and Burke-Plummer equation at Re > 1000.
Experimental
In order to complete the lab we used a unique apparatus that allowed us to gather the necessary data.
Water inlet
Pressure Gauge
Pressure Leads
6 inch Column Rotometers
Figure 2. Packed Bed Apparatus. This picture shows the experimental apparatus from the Rothfus Lab in Doherty Hall. In this setup, the 6 inch diameter column is shown.
As shown in the picture above the apparatus has a digital pressure gauge and three rotometers that changed the flow rate through a gate valve. When we set up the column, we attached pressure leads to the top and bottom of the column so that the apparatus could determine
4
the pressure drop. In order to determine the flow rate, we calibrated all three of the rotometers using a stop watch to measure the time it took to fill the 3.5 inch-diameter column to a predetermined height. We ran three different scenarios to ascertain a large data range of Reynolds numbers to relate to the Ergun equation. For the first scenario, we filled the 3.5 inchdiameter column with pea gravel to 3 different heights (13.35 inches, 20 inches, and 22.5 inches). Next, we filled the 3.5 inch-diameter column with black marbles to a height of 9.25 and 21 inches so that we could obtain the largest Reynolds numbers. We placed wire mesh over the packed bed with a rod that extends down from the top of the column to the wire mesh in order to prevent the fluidization of the packed bed while running tests on the 3.5inch-diameter column. For the last scenario, we added pea gravel to the 6 inch-diameter column at heights of 8.35 inches, 30.5 inches for data at low Reynolds numbers. While running our test, we found the porosity of the two packed bed materials as represented by the void fraction. The porosity of the pea gravel and black marble packed beds was determined by filling a 2000 ml graduated cylinder with one of the materials and adding a measured amount of water from a second graduated cylinder. The space filled by the water represents the void space between the packing.
5
Results
Using the calibration curves provided in the Appendix A1, we converted our arbitrary rotometer readings to superficial velocity. Figure 3 summarizes the raw data collected over the course of two lab sessions. 25000
Pea Gravel(3.5)Height22. 5
Pressure Drop(Pa)
20000
Pea Gravel(6)Height30.5
15000 `
Black Pearls
10000
5000
0 0
0.05
0.1
0.15
Velocity(m /s)
Figure 3: Sample of Raw Data. Relative Reynolds numbers are represented through the velocity. Represented is data for low, middle, and high Reynolds number ranges. Also we included data for both of our packings.
In order to compare our data to Ergun’s data, we converted our data into Reynolds numbers and friction factors. During the runs, we noticed that the temperature of the inlet water varied significantly. We accounted for the dependence of viscosity and the density of water when we calculated our Reynolds numbers.
6
Pea Gravel(3.5)Height20 Pea Gravel(6in)Height8.35
10000
Pea Gravel(3.5)Height13.35 Pea Gravel(3.5)Height22.5 Pea Gravel(6)Height30.5(W2)
1000
Black Pearls Pea Gravel(6)Height30.5(W1) Ergun Equation Ergun (high)
100 Log(Ff)
Ergun (low)
10
1
0.1 0.1
1
10
Log(Re)
100
1000
10000
Figure 3. Log-Log Plot of Reynolds number vs friction factor. The dark line represents the ideal Ergun Equation, while the dashed lines represent the range of Ergun values taking into account the measurement error of lab equipment and differences in void fraction. The error bars represent the uncertainty of the measurements.
Figure 4 is a log-log plot comparing our data to Ergun’s data. The high and low Ergun data lines represent the range of theoretical values taking into account differences in void fraction and measurement error. We found the void fraction for pea gravel to be 0.37 ± 0.02 and 0.465 ± 0.02 for the black marbles. Similarly, we found a variation of 1 unit in the W2 rotometer readings, and a deviation of 0.2 in the W3 rotometer readings.
7
Discussion
Our data for packed beds in both columns, using varying packings fits the general trend of the Ergun Equation. Data at lower Reynolds number was higher than expected, however, some data between Reynolds numbers of 100 and 1000 falls within the estimated deviation range of the Ergun Equation. Our friction factor values for the black marble fell below the theoretical values, with a similar deviation as the pea gravel. Ergun's Equation
90
Day 1 Pea Grave (both columns)
80 y = 712.7x + 1.1486 R2 = 0.996
70
Day 2 Pea Gravel (both columns)
y = 504.74x - 0.1475 R2 = 0.9756
60
Ff
50 40 30
y = 150x + 1.75
20 10 0 0
0.05
0.1
0.15
0.2
0.25
1/Re
Figure 4. Day to Day Deviation of Pea Gravel Data. This data comprises all of the pea gravel data, grouped by day. Ergun’s Equation is provided for comparison.
Figure 4 illustrates the deviation of replicate data for the pea gravel. There is a considerable deviation from the different lab periods. The data fits the general trend of the Ergun equation if data in this plot is linear. Our data deviates in the Ergun constants, with slopes larger than the theoretical values. The intercept for the run of Day indicates a negative intercept; however, since we are dealing with a log-log plot, this is still a positive number since the inverse-log function gives positive values. In deriving the Ergun equation, Ergun assumed that the particle diameter is at least a fifth of the size of the column diameter. We determined that the black marbles in the 3.5” column had a particle to column diameter ratio of 0.18, very near to the maximum particle size for the column. For a particle larger than this recommended size, the wall affects the void fraction of the particle. We define the void fraction as the space between particles, however for large particles in a small column; the wall presents an artificial boundary that alters the void
8
fraction. In this situation, the void fraction appears to be smaller than its true value, and the data appears lower than its true value. However, since we are near the recommended size, this shift is minimal. Black Pearls
10
Ergun Equation
Log(Ff)
Ergun (high) 1 Ergun (low)
0.1 100
1000 Log(Re)
10000
Figure 4. Black Marble Void Fraction Shift. This graph represents a shift in values due to a large particle diameter.
Figure 4, illustrates the effect of increasing the void fraction by 5%, which shows this effect is minimal and does not account for the deviation completely. We also made the assumption that the packing was mono-dispersed. In reality, the black marble packing is comprised of marbles of varying diameter. Since some particles were smaller than others, the overall void fraction is lower than if the particles were the same size. Since the packing is roughly spheres, an average of the particle diameters could correct this.
9
Black Marble Avg
10
Ergun Equation Ergun (low )
Log(Ff)
Ergun (high)
1
0.1 1000
Log(Re)
10000
Figure 6. Black Marble Average at 95% Confidence. The error bars are from the deviation of a replicate run.
For our black marble runs, we replicated our data through a second run at a different bed height. There was a significant deviation in our replicate data. Figure 5 is a plot of our average values at a confidence interval of 95%. We conducted both runs on the same day. Therefore, we cannot make any conclusions of the long term precision of our data. More replicate runs on different days in this range of Reynolds numbers would allow for a stricter statistical analysis. We are also limited by the precision of the measurements. For the packed bed apparatus, the limiting factor is the precision of the rotometer, pressure gauge, and graduated cylinders (for void fraction analysis). Since the pressure gauge is a digital gauge reading up to 0.1 inches of water, we reduced the error by reading our data once we reached a stable digital value. The rotometer and cylinder are graduated, and therefore are subject to measurement error, both human and uncertainty. These translate to an uncertainty in our calculated velocities and void fractions and propagate into our calculations for Reynolds number and friction factor. Figure 7 contains an estimation of the effect that the measurement uncertainty has on our data.
10
Pea Gravel(3.5)Height20 100
Pea Gravel(6in)Height8.35 Pea Gravel(3.5)Height13.35
Log(Ff)
Pea Gravel(3.5)Height22.5 Pea Gravel(6)Height30.5(W2) Pea Gravel(6)Height30.5(W1) Ergun Equation
10
Ergun (high)
1 10
100 Log(Re)
1000
Figure 7. Measurement error and uncertainty. This data is in the range for which we have the most values. All data is subject to a similar uncertainty, but the bars are not included for clarity purposes.
In addition to measurement uncertainty, our pea gravel data is about as precise over replicate runs as our black marble data. We expect a similar deviation over replicate runs conducted under the same conditions. We see in Figure 7 also that the deviation increases as the Reynolds number decreases. This is due to the increasing uncertainty of the W1 over the W2 rotometer, and the W2 over the W3 rotometer. Data measured using more imprecise equipment tends to deviate more over replicate runs. More data in this range would allow for a more rigorous statistical analysis of the deviation low Reynolds number ranges. Our Ergun constants is significantly different from what Ergun obtained (512 ± 200, 1.65 ± 0.8) vs. (150, 1.75). This could be explained by the different sets of material, different equipments that Ergun used, and the experimental errors previously discussed. Also, there is surface roughness on the particle that contributes to the systematic errors with the assumption of straight smooth tubes. If we have a smoother path, the Reynolds number will increase. (Geankoplis, p.94)
11
Conclusion
The Ergun equation is a collective model for all packings. The pressure drop data is shown to model the general trend of the Ergun equation for the range of Reynolds number obtained on the packed bed. The experimental data’s deviation from the Ergun equation is contributed to the non-uniform size and shape of the packings tested, and the surface roughness of the particles. The majority of data in this experiment relates to pea gravel and only one run was used on spherical black marbles. Future experiments could focus on multiple packings to determine how well different packings follow the Ergun equation. In addition, future work is necessary to prove that the experimental data follows the Ergun equation at higher Reynolds numbers. A new apparatus that allows higher flow rates and high pressure drops is needed since the apparatus in the Rothfus Laboratory maximizes at less than 100 inches of water.
12
Nomenclature
Lower case fp – friction factor of the packed beds [=] dimensionless vs – superficial velocity [=] m3/s Upper case Dp – Diameter of the particle [=] m L – column height [=] m ∆P – Pressure drop [=] Pa Rep – Reynold’s number of the packed beds [=] dimensionless Greek ε – void fraction [=] dimensionless µ – viscosity of the fluid [=] Pa-s ρ – density of the fluid [=] kg/m3
13
References Bird, R. Byron, Transport Phenomena. Madison, Wisconsin: John Wiley & Sons, 1996 Ergun, Sabri, “Fluid Flow through Randomly Packed Columns and Fluidized Beds.” Industrial and Engineering Chemistry, Vol. 41, No. 6 (1949): 1179-1184 Ergun, Sabri, “Fluid Flow through Packed Columns.” Chemical Engineering Progress, Vol. 48, No 2. (1952): 89-94 Geankoplis, Christie J., Transport Process and Unit Operations. 4th ed., New Jersey: Prentice Hall, 2003. R. Shankar Subramanian, “Flow through Packed Beds and Fluidized Bed.” www.clarkson.edu/subramanian/ ch301/notes/packfluidbed.pdf
14
Appendix A1: Calibration Curve for the Rotameters
In order to convert the value of FR reading from the rotameters into actual volumetric flow rate, we need calibration curves for all three rotameters. We assume that the volumetric flow rate is linearly related to the FR reading from the rotameter. y = 1.6466E-06x
8.00E-04
R 2 = 9.3398E-01
7.00E-04
Volumetric rate (m^3/s)
6.00E-04 5.00E-04 W1 Reading
4.00E-04
Linear (W1 Reading)
3.00E-04 2.00E-04 1.00E-04 0.00E+00 0
5
10
15
20
25
30
Rotam eter reading
Figure A1: Rotameter reading vs. Volumetric rate for W1.
A
y = 3.9932E-06x R 2 = 9.9616E-01
0.0008 0.0007
Volumetric rate (m^3/s)
0.0006 0.0005 W2 Reading
0.0004
Linear (W2 Reading)
0.0003 0.0002 0.0001 0 0
20
40
60
80
100
Rotam eter reading
Figure A1: Rotameter reading vs. Volumetric rate for W2. y = 5.1936E-05x R2 = 9.8935E-01
0.0008 0.0007
Volumetric rate (m^3/s)
0.0006 0.0005 W3 Reading
0.0004
Linear (W3 Reading)
0.0003 0.0002 0.0001 0 0
5
10
15
Rotam eter reading
Figure A1: Rotameter reading vs. Volumetric rate for W3.
We obtain three equations for the rotameters reading, where we set our intercept to 0 since at 0 FR reading the volumetric flow rate is 0.
B
