Filtration_Hydraulics_Team_Lab_Report.pdf

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Florida Institute of Technology College of Engineering Department of Chemical Engineering CHE-3115 ChE Process Laboratory 1 Team Report #1 Experiment #3 Filtration Hydraulics Lab Performed By: Damian Ezell, Brock Wohlnick, Emma Mathioudakis, Zachary Teachey For: Dr. Bo Wang Experiment Performed on: October 12th, 2023 Date: October 26th, 2023 Team: F Partners: Emma Mathioudakis, Zachary Teachey Grade: Brock Wohlnick

1 Introduction: The purpose of this experiment was to get a better understanding of flow hydraulics in both fixed and fluidized beds containing a granular media, in this case, sand. During the experiment, the fixed bed received various flow rates of water between 0-10 L/s per m 2 of the column’s cross sectional area and was recorded and compared to the other flow rates. Additionally, the height of all six piezometers as well as the height of the water in the overflow tank at each varying flow rate. Using these values, the pressure drop between each layer of sand can be calculated. The procedure for the fluidized bed was very similar to the procedure for the fixed bed. Instead, the flow of the water was supplied to the bottom of the column, allowing an upward flow to occur. Similar to the fluidized bed, the varying flow rate and height was recorded and used to produce a value for pressure drop. With these values the net weight of the media within the column could be calculated. In theory, the value of the net weight would be equal to the pressure drop observed across the fluidized bed. One of the most common applications of filtration hydraulics in Florida is in a pool through a sand filter. Similar to the packed bed in this experiment, sand filters force water from the top of the tank through the sand, helping with pressure as well as removing any debris and contaminants in the water. Once the water goes through the filter, it is returned back to the pool much cleaner than it previously was.

2 Equipment and Procedures: 2.1 Description of the apparatus: The experimental apparatus used during the course of this lab was the column. The column was comprised of three main sections. On top of the column there was a reservoir tank with an inflow and overflow lines. The column itself was roughly a 1.25 meter cylindrical tube filled approximately 0.75 meters of sand. The column was marked with height indicators from the bottom to the top of the column. On the bottom of the column there was a pressure valve with an associated pressure gauge that could be manipulated to change the flow of water flowing up into the column (for the fluidized bed portion of the experiment). Next to this valve and gauge was an outflow or effluent flow pipe that drained the column during the fixed bed portion of the lab. Attached to the right of the column via various individual lines were piezometers. 2.2 General Diagram of Experimental apparatus Figure 1: Media Characterization of Experiment

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3 Figure 2: Flow Diagram for Fixed Bed portion of the Experiment Figure 3: Flow Diagram for Fluidized bed portion of the Experiment

4 2.3 Procedure: (Same as procedure) 2.31 Start up Procedure: Prior to working with the column, the media characterization was accomplished. In this portion of the lab 75 grams of sand was poured into a graduated cylinder of 50 milliliters of water. The sand was then compacted using a stirring rod. Data was gathered from this portion of the lab to characterize the density and porosity of the sand. In the fixed bed portion of the lab, water from the reservoir was used to fill the column. Any excess water flowed out of the column via the effluent line located on the bottom of the column. The flow rate of the water in the column was manipulated by adjusting how much effluent flowed out of the column. The piezometers attached to the column were then used to measure the pressure of the water at different heights along the column. In the fluidized bed portion of the lab water flowed from the bottom of the column to the top and ultimately into the reservoir where it was drained via the overflow line attached to the reservoir. The amount of water that flowed into the column from the bottom was varied by adjusting the pressure valve attached to the bottom of the column. 2.3.2 Sampling In the media characterization portion of this lab, the density of sand was calculated via: where Ws was the mass of the sand, V was the volume of the sand-water 𝐷𝑒??𝑖?𝑦 = ?? / (? − ??) mixture and Vo was the volume of just the water in the graduated cylinder. The porosity of the sand was calculated via: where Vs was the 𝑃????𝑖?𝑦 = [?? − (? − ??)]/?? volume of the sand, V was the volume of the sand-water mixture and Vo was the volume of the water. During the fixed bed portion of the experiment the flowrate was manipulated and the corresponding amount of water in each of the piezometers was collected. Then using the flowrate and the data from the piezometers the pressure could then be calculated.

5 In the fluidized bed portion of the experiment all of the piezometer lines were closed and the amount of water flowing up the column was adjusted using the pressure valve on the bottom of the column. The height of the fluidized bed was then noted as well as the flowrate of the water. This was repeated 10 more times with different flow rates. Two runs of this were used to solve for the minimum fluidization of the bed. The total pressure change was calculated using the flow rates. Additionally the total net weight of the media in the column was calculated and corrected for buoyancy effect. 2.3.3 Clean up The graduated cylinder with sand and water was disposed. The column was drained via the effluent line into a plastic tub. The contents of the tub were then pumped into a drain.

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6 Experimental Results: Table 1: Initial conditions: Density, ρ 996.786 kg/m 3 Dynamic viscosity, η 0.0008701 Pa*s Reference Height, h r 1.29 m Column Diameter, d c 0.083 m Initial Height Layer, h i 0.73 m Area, A 0.005410603378 m 2 Temperature, T 26 o C Initial Volume, V 0 50 mL Weight Sand, Ws 75 g Volume, V 78 mL Volume Sand, V s 40.mL density sand, ρ s 3.12g/mL porosity, e 0.3086 Table 2: Fixed Bed Experimental Data: trial Volume, V (mL) h1 (cm) h2 (cm) h3 (cm) h4 (cm) h5 (cm) h6 (cm) Time, t (sec) 1 395 90 68.5 54 40.5 30 19 10.96 2 405 91.5 71 57 44.5 34.5 24 12.12 3 395 94 75 62 50.5 42 32 12.78 4 400 96 79 68 57 49 40.5 14.22 5 405 100.5 87.5 78.5 70 63.5 57 17.56 6 395 106 98 92 86 82 77.5 24.22 7 395 111 105 101 97.5 94.5 91.5 37.25 *variable h represents height

7 Table 3: Fixed-Bed Experimental Results: Trial Volumetric Flow Rate, Q (mL/sec) Volumetric Flow Rate,Q (m 3 /sec) Velocity, U (m/s) d nominal (m) 1 36.04014599 0.00003604014599 0.006661021603 0 2 33.41584158 0.00003341584158 0.006175991706 0.000539 3 30.90766823 0.00003090766823 0.005712425413 0.000688 4 28.12939522 0.00002812939522 0.005198938687 0.000772 5 23.06378132 0.00002306378132 0.004262700426 0.000852 6 16.30883567 0.00001630883567 0.003014236035 0.000951 7 10.60402685 0.00001060402685 0.001959860316 0.001003 Table 4: Pressure Drop Experimental Data: trial h1 (cm) h2 (cm) h3 (cm) h4 (cm) h5 (cm) h6 (cm) 1 39 60.5 75 88.5 99 110 2 37.5 58 72 84.5 94.5 105 3 35 54 67 78.5 87 97 4 33 50 61 72 80 88.5 5 28.5 41.5 50.5 59 65.5 72 6 23 31 37 43 47 51.5 7 18 24 28 31.5 34.5 37.5

8 Table 5: Calculated Values from Experimental Data Trial Pressure drop expected (Pa) A B C Φ Total Pressure (Pa) U I (m/s) Re 1 14284.4741 0.000071 14258 0.00000027 15669003 0.00003780 328181 0.991911 16527.888 0.02158 464551 24.80153807 2 13352.92846 0.000076 7297439 6 0.00000020 53435919 0.00003669 607018 0.987461 14997.528 0.02001 293489 22.99558574 3 12586.27013 0.000082 9564026 0.00000013 8554541 0.00003257 517881 0.978437 12548.952 0.01851 07758 21.26955064 4 11593.76196 0.000091 1498079 0.00000010 1795173 0.00002803 23161 0.972861 9947.34 0.01684 685252 19.35764263 5 9787.414285 0.000111 169497 0.00000006 061385338 0.00001634 752503 0.958358 4897.152 0.01381 302795 15.87166852 6 7136.412027 0.000157 214716 0.00000003 634893074 -0.0000063 05689727 0.943398 -1377.324 0.00976 745313 11.22315678 7 4765.129729 0.000241 7938964 0.00000002 736879822 -0.0000388 2370691 0.931619 -5662.332 0.00635 0811134 7.297311604 *Average value of φ = 0.9662921429 Table 6: Calculated Average Particle Diameters at the top and bottom of the column particle diameter top (m) particle diameter bottom (m) 0.000626 0.000667 Table 7: Fluidized-Bed Dimensions and Initial Conditions: Column Diameter, d 8.3 cm Column Diameter, d 0.083 m Density, ρ (sand) 3.12 g/mL Density, ρ (water) 0.996786 g/m: Area, A 0.1903490989, m 2 Porosity, e 0.3086 emf Gravity , g 9.8 m/s1 l MF 0.02170418006 A 2.180847 B 5.015027

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9 Table 8: Fluidized-Bed Experimental Data: Trial Height, h (cm) Height, h (m) Volume, V (mL) Volume, V (L) Time, t (s) Pressure, P (psia) 1 76 0.76 405 0.405 18.66 2.65 2 76.3 0.763 405 0.405 16.82 2.75 3 77.4 0.774 415 0.415 14.82 2.8 4 79.5 0.795 400 0.4 12.66 2.8 5 89 0.89 400 0.4 8.28 2.8 6 90.9 0.909 390 0.39 7.54 2.8 7 110 1.1 395 0.395 5.06 2.81 8 114 1.14 415 0.415 5.68 2.9 Table 9: Fluidized-Bed Experimental Results: Q (L/sec) W MB W M W N ΔP theo d nom U (m/sec) I mf Length, l (m) 0.021704 18006 2.833942 036 12.829 63357 9.99569 1529 18123.23 769 0 0.00011402 3025 0.03 0.02273647671 0.024078 478 2.845128 65 12.880 27686 10.0351 4821 18194.77 679 0.000539 0.00012649 64118 0.033 0.02501012439 0.028002 69906 2.886146 232 13.065 96892 10.1798 2269 18457.08 681 0.000688 0.00014711 23279 0.044 0.03334683251 0.031595 57662 2.964452 525 13.420 47195 10.4560 1943 18957.86 048 0.000772 0.00016598 75293 0.065 0.04926236621 0.048309 17874 3.318695 28 15.024 17615 11.7054 8087 21223.26 519 0.000852 0.00025379 25266 0.16 0.1212612091 0.051724 13793 3.389543 83 15.344 91699 11.9553 7316 21676.34 614 0.000951 0.00027173 30328 0.179 0.1356609777 0.078063 24111 4.101758 211 18.569 20648 14.4674 4827 26231.00 192 0.001003 0.00041010 5651 0.37 0.2804165461 0.073063 38028 4.250913 055 19.244 45035 14.9935 3729 27184.85 654 0.001003 0.00038383 88556 0.41 0.3107318484 * Total length, l, = 0.9784263813 ** The variable W represents the Weight Fraction

10 Table 10: Sieve Analysis Height Data: H0 73 cm H1 70 cm H2 58 cm H3 45 cm H4 35 cm H5 20 cm H6 8 cm Table 10: Sieve Analysis Weight Fractions and Particle Sizing: wt frac size range 0 0 0.149 0.0014 0.0014 0.177 0.0136 0.0122 0.25 0.0157 0.0021 0.297 0.054 0.0383 0.42 0.0946 0.0406 0.5 0.1134 0.0188 0.595 0.6068 0.4934 0.841 0.9797 0.3729 1.19 1 0.0203 2

11 Table 11: Sieve Data Experimental Results: Wt Fraction d p (mm) d nom (mm) d nom (m) y0 0 0 0 0 y1 0.04109577104 0.454861 0.538946401 0.000539 y2 0.2054789547 0.638575 0.687710783 0.000688 y3 0.3835618124 0.740627 0.772172023 0.000772 y4 0.5205486623 0.805061 0.851799679 0.000852 y5 0.7260270861 0.901252 0.95064437 0.000951 y6 0.8904104071 1.002743 1.002743459 0.001003 *The variable d represents diameter Graph 1: Volumetric Flow Rate vs Total PressureThi This graph represents the relationship between the volumetric flow rate and the pressure of the fixed bed column. This correlation is linear since the graph is a straight line

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12 Discussion: This experiment was considered to be both accurate and consistent with respect to data collection. The data collection methods followed in this experiment were quite simple. The simplicity of the data collection allowed for multiple trials with generally the same conditions and environment to be tested. The results, however, did not remain accurate throughout the experiment. The values collected during the beginning of the experiment were much more accurate than that of the later trials and experiments. This drop in accuracy could stem from the tediousness of the experimental procedure. It is highly likely that this drop in accuracy could be attributed to small mistakes made while performing the experiment due to group members getting distracted, bored, or even slightly careless about reading and taking measurements. These small issues can be seen in the form of; letting small amounts of water escape the graduated cylinder while measuring the flow rate, misreading or quickly estimating the volume of the graduated cylinder, adjusting the flowrate of the tower too aggressively, etc. During the data processing stage of the experiment, there was a graph created to show the relationship between the pressure of the water and the flowrate of it. This relationship is important to validating the accuracy of the data because it is known that this relationship should be linear. When looking at the data points on the graph, it is easy to see that the experimental data follows this by trending to be linear. The differences between the experimental and theoretical values for the rest of the experiment, however, were far from each other. If given the choice to change the medium for filtration to wood chips instead of the granular sand used in the experiment, it would be a mistake for the following reasons: the diameter and particle size of a column of wood chips would not allow for the greatest filtration of water. Wood chips would also soak water up, this would cause mold and other bacteria to begin to grow in the column while water isn't flowing. These issues are not as apparent in a sand column since sand will not absorb as much water as wood chips and will dry quicker. The particle size and diameter of sand is also more useful for the filtration of water.

13 Conclusions Given these results, only some values were consistent with the theoretical values. The plot of total pressure versus flowrate was accurate, as it’s a linear plot and agrees with laminar approximation of the modified Ergun equation. Using the measured pressure drop and the calculated sphericity, the ratio between the average particle diameter for the top and bottom layers was calculated but were aligned with the middle values of the sieve analysis, not the top or the bottom indicating a margin of error due to human error and not waiting long enough for the system to stabilize in combination with some likely lifting of sand particles by the downfalling water. This means there were experimental errors. However, the percent difference between the particle diameter and nominal diameter was much smaller, The percent differences between theoretical pressure drop and calculated pressure drop were relatively small for higher flowrates, however when the flowrate was low the percent difference spiked to roughly 100% or greater. This is likely caused by smaller mistakes leading to far higher relative errors. In future cases, allowing the column to settle for longer and using a more precise method of getting flowrate would potentially greatly improve the accuracy of the results. Table 12: percent differences between the measured and calculated theoretical pressure drop percent difference in pressure drop for fixed bed 13.57350617 10.96580408 -0.2973804485 -16.55137914 -99.8593118 618.1360396 184.1548982

14 Table 13: Percent Difference Between the Particle Diameter and Diameter nominal for y1-6 y1 y2 y3 y4 y5 y6 %difference between particle diameter and nominal diameter 18.485955 27 7.69459859 8 4.25923211 5.8056071 53 5.480417242 0.00004577444 071 Contribution of Each Team Member: Throughout the experimental procedure, each team member played a vital role gathering efficient and accurate data. The breakdown of the team was as follows: Team Leader - Damian Ezell Scribe - Emma Mathioudakis Engineer I - Zachary Teachey Engineer II - Brock Wohlnick Taking these roles into account, the breakdown of the work for each team member was simple. Emma recorded all data values found during the experiment into a spreadsheet so that the data could be easily processed post-lab. Damian, Zachary, and Brock all took a role in the data collection procedure. Damian was responsible for adjusting the flow rate of the feed by adjusting the position of the valve to the right of the columns. Brock was responsible for collecting water from the exit stream into a graduated cylinder while Zach timed him. The volume of water that was collected by Brock would then be divided by the time that he collected it, recorded by Zach. For the report, each section was written by a separate member of the team. The introduction and conclusions were written by Emma. The equipment, procedure, and discussion sections were written by Zachary. Damian compiled the experimental results and documented the contributions each team member made to the experiment/ report. The experimental and sample calculations and references were done by Brock. Each member of the team adequately fulfilled their roles and responsibilities that were given to them in order to evenly divide the experimental procedure and presentation of the experimental results.

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15 References: Lab Manual, Chemical Engineering Processes Laboratory-Manual, Dr. Bo Wang, Florida Institute of Technology College of Engineering and Sciences. (2023)

16 Appendix A: Sample Calculations Variables: V 0 = Initial Volume of Water W s = Weight of Sand V s = Volume of Sand V= Final Volume with Sand Added ρ s = Density of Sand ρ w = Density of Water at 26 ℃ e = Porosity of the sand μ = Viscosity of Water at 26 ℃ V m = Measured Volumes used for Flowrates t = Time measured for Flowrates Q = Volumetric Flowrate h n = Height of n th piezometer h 0 = Initial Height of Water h 0B = Initial Height of Sand Bed d col = Diameter of the Column A col = Area of the Column φ = Average Sphericity ΔP NS = Non-spherical Pressure Drop P T = Total Pressure Re = Reynold’s Number d p,n,theo = Theoretical Particle Diameter of Layer n r dp = Ratio between Top and Bottom Layer Particle Diameters a, b = Constants Obtained using Given Sieve Analysis Data in DataFit; found to be 2.18 and 5.02, respectively d p,1 = Particle Diameter of layer n from Sieve Analysis d p = Average Particle Diameter from Sieve Analysis d nom = Nominal Diameter from Sieve Analysis l n = Layer Thickness of Layer n g = Gravitational Constant = 9.81 m/s 2 W t,n = Weight Fraction of Layer n U = Velocity U I = Initial Velocity ΔP m,n = Measured Pressure Drop of Layer n ΔP C,n = Calculated Pressure Drop of Layer n h B = Height of the Bed P = Pressure from the Gauge W MB = Weight of Media from Buoyancy W M = Weight of Media W N = Net Weight ΔP theo = Theoretical Pressure Drop e mf = Porosity at Minimum Fluidization = 0.476 l mf = Layer Thickness at Minimum Fluidization l eb = Expanding Bed Layer Thickness Calculation for Density of the Sand: ? ? = ? ? (? − ? 0 ) = 75.000 ? (78 − 50) ?𝐿 = 3.120 ? ?𝐿 ∗ 10 6 ?𝐿 1 ? 3 ∗ 1 ?? 1000 ? = 3120 ?? ? 3 Calculation for Porosity: ? = ? ? − (? − ? 0 ) ? ? = 40.5 ?𝐿 − (78 − 50) ?𝐿 40.5 ?𝐿 = 0.3086 Downflow Calculations: Calculation for Flowrate: ? = ? ? ? = 395 ?𝐿 37.25 ? = 10.604 ?𝐿 ? ∗ 1 ? 3 10 6 ?𝐿 = 1.06 ∗ 10 −5 ? 3 ?

17 Sample Calculation for Top (Piezometer 1) Layer Thickness: ? 1 = ℎ 0 − ℎ 1 = 129 ?? − 90 ?? = 39 ?? ∗ 1 ? 100 ?? = 0.39 ? Calculation for Weight Fraction of Top Layer: ? ?,1 = ℎ 0 − ℎ ? ℎ 0 = 39 ?? 129 ?? = 0.3023 Calculation for Particle Diameter of Top Layer: ? ?,1 = [ − ln(1 − ? ?,1 ) 𝑎 ] 1 ? = [ − ln(1 − 0.3023) 2.18 ] 1 5.02 = 0.696 ?? ∗ 1 ? 1000 ?? = 0.000696 ? Calculation for Nominal Diameter: ? ??? = √? ?,1 ∗ ? ?,6 = √0.000696 ? ∗ 0.001 ? = 0.000836 ? Calculation for Column Area: 𝐴 ??? = ?? ??? 2 4 = 3.14(0.083 ?) 2 4 = 0.00541 ? 2 Calculation for Velocity: ? = ? 𝐴 ??? = 1.06 ∗ 10 −5 ? 3 ? 0.00541 ? 2 = 0.00196 ? ? Calculation for Measured Pressure Drop of Top Layer: Δ? ?,1 = ? 1 ∗ ? ∗ ? ? = (.73 − .19) ? ∗ 9.81 ? ? 2 ∗ 3120 ?? ? 3 = 16527.88 ?𝑎 Calculation for Calculated Pressure Drop of Bottom Layer using Equation 4.20:

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18 Δ? ?,1 = [150 (1 − ?) 2 ? 3 𝜇? ? ? 2 + 1.75 (1 − ?) ? 3 ? 𝑤 ? 2 ? ? ] ∗ ? 1 = [150 (1 − 0.3086) 2 0.3086 3 0.0008 ?𝑎 ? ∗ 0.00196 ? ? (0.000836 ?) 2 + 1.75 (1 − 0.356) 0.356 3 996 ?? ? 3 ∗ (0.0067 ? ? ) 2 0.000836 ? ] ∗ .39? = 14284.47?𝑎 Calculation for Percent Difference Between Measured and Calculated Pressure Drop: % 𝐷𝑖???????? = Δ? ?,1 − Δ? ?,1 Δ? ?,1 ∗ 100 = 16527.89 ?𝑎 − 14284.47 ?𝑎 16527.89 ?𝑎 ∗ 100 = 13.57% Calculation for Average Sphericity: Done via the modified Ergun equation for non-spherical particles and plugged into Wolfram alpha Solved for ϕ at lowest flowrate −∆? ? = 150 𝜇?(1 − ?) 2 ? 2 𝜑 2 ? 3 + 1.75 ?? 2 (1 − ?) ? ∗ 𝜑 ∗ ? 3 𝜑 = .931619 Calculation for Non-spherical Pressure Drop: Δ? ?? = [150 (1 − ?) 2 𝜑 2 ? 3 𝜇? ? ? 2 + 1.75 (1 − ?) 𝜑? 3 ? 𝑤 ? 2 ? ? ] ∗ ? 1 = [150 (1 − 0.356) 2 0.098 2 ∗ 0.356 3 0.001 ?𝑎 ? ∗ 0.0067 ? ? (0.00086 ?) 2 + 1.75 (1 − 0.356) 0.098 ∗ 0.356 3 996 ?? ? 3 ∗ (0.0067 ? ? ) 2 0.00086 ? ] ∗ 0.381 ? = 5111.51 ?𝑎

19 Calculation for Initial Velocity: ? 𝐼 = ? ? = 0.00666 ? ? 0.356 = 0.0216 ? ? Calculation for Reynold’s Number: ?? = ? 𝐼 ∗ ? ??? 𝜇 ? 𝑤 = 0.0216 ? ? ∗ 0.000836 ? 0.0008 ?𝑎 ? 996 ?? ? 3 = 24.80 Sample Calculation for Theoretical Particle Diameter of Top Layer: ΔP m,1 = [150 (1 − ?) 2 𝜑 2 ? 3 𝜇? ? ?,1,?ℎ?? 2 + 1.75 (1 − ?) 𝜑? 3 ? 𝑤 ? 2 ? ?,1,?ℎ?? ] ∗ ? 1 → 16527.89 ?𝑎 = [150 (1 − 0.3086) 2 0.991911 2 ∗ 0.3086 3 0.0008 ?𝑎 ? ∗ 0.00666 ? ? ? ?,1,?ℎ?? 2 + 1.75 (1 − 0.3086) 0.991911 ∗ 0.3086 3 996 ?? ? 3 ∗ (0.00666 ? ? ) 2 ? ?,1,?ℎ?? ] ∗ 0.39 ? Solved in Wolfram Alpha: d p,1,theo = 0.000626 m Sample Calculation for the Ratio of Particle Diameter of Top Layer to Bottom Layer: ? ?? = ? ?,1 ? ?,6 = 0.000626 ? 0.000667 ? = 0.939 Calculation for Total Pressure at First Flowrate: ? ? = ? ? ∗ ? ∗ (ℎ 0 − ℎ 6 ) = 3120 ?? ? 3 ∗ 9.81 ? ? 2 ∗ (1.29 ? − .19?) = 33667.92 ?𝑎 Up-flow Calculations: Calculation for Weight of Media from Buoyancy:

20 ? ?? = 𝐴 ??? ∗ ℎ 0? ∗ (1 − ?) ∗ ? 𝑤 = 0.0054 ? 2 ∗ 0.76 ? ∗ (1 − 0.3086) ∗ 996 ?? ? 3 = 2.83 ?? Calculation for Weight of Media: ? ? = 𝐴 ??? ∗ ℎ 0? ∗ ? ? = 0.0054 ? 2 ∗ 0.76 ? ∗ 3120 ?? ? 3 = 12.83 ?? Calculation for Net Weight: ? ? = ? ? − ? ?? = 12.83 ?? − 2.83 ?? = 10.00 ?? Calculation for Theoretical Pressure Drop: Δ? ?ℎ?? = ? ? ∗ ? 𝐴 ??? = 10.00 ?? ∗ 9.81 ? ? 2 0.005 ? 2 = 18123.24 ?𝑎 Calculation for Layer Thickness at Minimum Fluidization for Largest Flowrate: ? ?? = ℎ ? − ℎ 0? = 1.14 ? − 0.73 ? = 0.41 ? Calculation for Expanding Bed Layer Thickness for Largest Flowrate: ? ??,6 = ? ??,5 + ? ?? (1 − ? ?? ) (1 − ?) = 0.668 ? + 0.41 ? ∗ (1 − 0.476) (1 − 0.3086) = 0.978 ?

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