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2 Understand the Problems of Bulk Solid Flow Perform Calculations Related to Bin Design Create Matlab Programs That Aid In Calculations Understand the Components of Effective Bin Design http://www.fil.ion.ucl.ac.uk/spm/software/spm8/ http://eng.tel-tek.no/Powder-Technology/Silo-design-and-powder-mechanics/Silo-design-based-on-powder-mechanics-overview http://bulksolidsflow.com.au/

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Storage capacity: Always keep in mind the amount of material that you are going to store because that will effect how many bins you will need to design. The location of the bin will also effect the design. Discharge Frequency & rate: How much time will the solid remain without contact? Around what range will the instantaneous discharge rate be? Does the rate depend on weight or on the volume? What is the required feed accuracy? KEY POINTS TO CONSIDER WHEN DESIGNING BINS http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf http://jenike.com/files/2012/10/BlueSiloCollapsing-41.jpg

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Temperature and Pressure: Will the material be at a low or high temperature than its surroundings? Is the material being fed into a positive or negative pressure environment? Fabrication Materials: Is the solid abrasive or corrosive? Will there be need for corrosion-resistant alloys? Are ultrahigh-molecular-weight plastic liners tolerable? Is the application subject to any regulatory compliance requirements? Safety and environmental considerations: Are there any safety environmental issues like material explosive ability or maximum dust composer limits? Bulk solid uniformity: What is the required material uniformity ( eg: size, shape, moisture content) How will particle segregation affect production and the final product? http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf KEY POINTS TO CONSIDER WHEN DESIGNING BINS http://www.proagro.com.ua/eng/research/grain/4064511.html

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UNDERSTAND BULK-SOLIDS FLOW PROBLEMS Arching or Bridging: This is when a no-flow condition occurs in which a material forms a stable bridge/dome across the outlet of a bin. Ratholing: Another no-flow condition in which material forms a stable open channel within the bin resulting in erratic flow to the downstream process. Flooding or flushing: a condition in which an aerated bulk solid behaves like a fluid and flows uncontrollably through an outlet or feeder. Flowrate limitation: Insufficient flowrate, typically caused by counter-flowing air slowing the gravity discharge of fine powder. Particle segregation: segregation may prevent a chemical reaction, cause out of spec product, or require costly rework. Capacity: As low as only 10-20% of the bins rated storage capacity. http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf Void Arching Ratholing

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MEASURE THE FLOW PROPERTIES OF THE BULK SOLID The purpose of measuring the flow properties is mainly in order to control how the fluid would behave in a bin. The table shows the most important bulk-solid handling properties. Variables that affect solid parameters: Moisture content Particle size, shape, and hardness Pressure Temperature Storage time at rest Wall surface Chemical additives http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf

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CALCULATE THE APPROXIMATE SIZE OF THE BIN The above equation is used to find the height of the cylinder section needed to store the desired capacity. This design process is iterative. H: Height m: the mass in Kg. A: the cross-sectional area of the cylinder. avg : Average bulk density in (kg/m^3) Due to the volume lost at the top of the cylinder which is due to the bulk solids angle of repose and along with the volume of material in the hopper section, a reasonable sufficient estimate for the height can be found by keeping the height of the bin between one and four times the diameter or width since values out of that range are most often uneconomical. http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf

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TYPE OF FLOW PATTERNS- FUNNEL FLOW Funnel Flow Discharge Bulk solids flow much differently than liquids in tanks. A liquid would flow in a first-in/first-out sequence, but many bins have flows in a funnel-flow pattern. Funnel-flow is defined as when some of the material flows in the center of the hopper while the rest remains stationary along the walls. Funnel- flow is the most economical choice if the bulk solid is nondegradable, coarse, free-flowing, and if the segregation during discharge is not an issue. http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf 8

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Many problems can occur when there is funnel-flow. Some problems include ratholes, arches, caking, equipment failure, etc Mass-flow occurs when all the material moves when any is discharged. Mass-flow bins work well with powders, cohesive materials, materials that degrade with time, and whenever sifting segregation must be minimized. TYPE OF FLOW PATTERNS- MASS FLOW Mass Flow Discharge http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf 9

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The converging hopper section must be steep enough, the wall surface friction low enough, and the outlet large enough to allow a flow without stagnant regions. This will also help prevent arching. In order to determine the wall friction angle, various wall surfaces are powder tested. These tests are conducted using a direct shear tester along the lines of ASTM D-6128. DESIGNING FOR MASS FLOW http://research.che.tamu.edu/groups/Seminario/numerical- topics/Bin%20Design.pdf http://www.dietmar-schulze.com/storage.html Sand will require a steep hopper angle in order to achieve mass flow because it is a highly frictional bulk solid. Smooth catalyst beds will achieve mass flow at a relatively shallow hopper angle because it is a low-friction bulk solid. 10

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To prevent arching you must measure the cohesive strength of the material you want to transport. DESIGNING FOR MASS FLOW http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf http://www.dietmar-schulze.com/storage.html First the flow function of the material, the is measured in a laboratory test according to ASTM D-6128 with a direct shear tester. Just like in the wall friction test, consolidating forces are applied to a material. In the test cell, the force required to shear the material is measured. Minimum outlet sizes needed to avoid arching can be calculated once the flow function is determined. 11

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This equation can be used to approximate the maximum discharge rate from a converging hopper. This can only be used if the bulk material is coarse and free- flowing. In order for a material to be considered coarse, the particles must have a diameter of at least 3 mm (1/8 in). An example of this scenario is on the next slide. In the above equation, the variables are defined as: M: mass flow rate (kg/s) : bulk density (kg/m 3 ) A: outlet area (m 2 ) g: acceleration (m/s 2 ) B: outlet size (m) : mass-flow hopper angle measured from vertical (deg.) m: outlet parameter dependent on type of hopper For conical: m = 1 for a circular outlet For wedge-shaped: m = 0 for a slot-shaped outlet http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf 12

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TYPES OF BULK SOLIDS This equation only works for coarse and free-flowing material because it neglects the materials resistance to airflow. For example, the equation would not correctly estimate the flow rate for a fine powder. The fine powder would have particles with diameters much less than 3 mm and would be greatly affected by airflow. Thus, the equation would give an answer that is much greater than the true value for the mass flow rate. Mass flowing bulk solids http://upload.wikimedia.org/wikipedia/commons/9/98/Rhodium_powder_pressed_melted.jpg Does not follow mass flow equations http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf 13

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function [ M ] = DischargeRate( rho,A,g,B,theta,m ) % DischargeRate: Approximates the maximum discharge rate from a converging % hopper. % For this function to be accurate, one must assume that the bulk material % is both coarse and free-flowing, such as plastic pellets. % Input: % rho = bulk density (kg/m^3) % A = outlet area (m^3) % g = acceleration (m/s^2) % B = outlet size (m) % theta = mass flow hopper angle measured from vertical (deg.) % m = 1 for a circular outlet and m = 0 for a slot shaped outlet % Output: % M = mass flowrate (kg/s) M=rho*A*sqrt((B*g)/(2*(1+m)*tan(theta*pi/180))); end FUNCTION THAT CALCULATES MASS FLOW RATE 14

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>> DischargeRate(10,1,9.81,1,60,1) ans = 11.8994 >> DischargeRate(10,1,9.81,1,60,0) ans = 16.8283 RESULTS The first answer is for a circular outlet. The second answer is for a slot-shaped outlet with the same parameters. http://www.inti.gob.ar/cirsoc/pdf/silos/SolidsNotes10HopperDesign.pdf 15

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FLOW RATE VS HOPPER ANGLE 16

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% Creates a graph that shows the comparison of circular and slot- shaped % outlets. The mass flow rates are plotted versus the mass flow hopper % angle measured from vertical. % rho = bulk density (kg/m^3) rho=10; % A = outlet area (m^3) A=1; % B = outlet size (m) B=1; % g = acceleration (m/s^2) g=9.81; % The values for theta are from 1 degree to 90 degrees. theta=(1:1:90); % Mc = mass flow rate for a circular outlet (kg/s) Mc=rho*A*sqrt((B*g)./(2*(1+1)*tan(theta*pi/180))); % Ms = mass flow rate for a slot-shaped outlet (kg/s) Ms=rho*A*sqrt((B*g)./(2*(1+0)*tan(theta*pi/180))); plot(theta,Mc,'-b',theta,Ms,'--r') title('Comparison of Outlets') xlabel('mass flow hopper angle measured from vertical (deg.)') ylabel('mass flowrate (kg/s)') legend('Circular','Slot-shaped') THE PREVIOUS PLOT COMPARES THE TWO SHAPES OF OUTLETS AND ALSO THE MASS FLOW WITH RESPECT TO A CHANGING HOPPER ANGLE. AS THE PLOT SHOWS, THE SLOT-SHAPED OUTLET HAS A LARGER MASS FLOW FOR ALL VALUES OF THE HOPPER ANGLE THAN THE CIRCULAR OUTLET. THE PART OF THE GRAPH BETWEEN 20 AND 70 IS WHERE A REALISTIC HOPPER ANGLE WOULD EXIST. IN THIS REGION, AN INCREASING LEADS TO A DECREASE IN MASS FLOW. ESSENTIALLY AS THE SLOPE OF THE BIN DECREASES, LESS MASS EXITS THE BOTTOM OF THE BIN PER UNIT TIME. THE CODE THAT CREATED THE PLOT IS GIVEN BELOW: 17

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The main factors for funnel flow are making the hopper slope steep enough to be self-cleaning, and sizing the hopper outlet large enough to overcome arching and ratholing. For the bin to capable of self-cleaning, the hopper slope must be 15-20 degrees steeper than the wall friction angle, assuming that a rathole has not formed. Knowledge of the materials cohesive strength and internal friction is needed in order to determine the minimum dimensions to overcome ratholing and arching. For funnel flow, the design of the mass-flow bins is independent of scale, but the overall size matters. Thus, large funnel flow bins have a higher ratholing tendency, while mass flow bins have no chance of ratholing. DESIGNING FOR FUNNEL FLOW Flow Channel Non-flowing region http://research.che.tamu.edu/groups/Seminario/numerical-topics/Bin%20Design.pdf 18

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Unfortunately, some fluids have properties that can make flow calculations difficult. In these cases, collecting experimental data and interpolating can be the next best thing. For example, this data was generated to simulate storing a very viscous, shear-thickening, non Newtonian fluid. This liquid rapidly thickens and becomes more adhesive when exposed to a high pressure gradient. While the exact calculations are beyond the scope of this project, the data shows that at any angle less than 30 degrees from vertical, flow rate drops rapidly as the fluid hardens into a gooey solid. The question is, how do we model this flow and find a theoretical maximum rate? EXPERIMENTAL FLOW CALCULATIONS: Angle From Vertical Flow (in^3/s) 54.1 104.7 156.1 208.2 2527.3 Angle From Vertical Flow (in^3/s) 3086.2 3560.3 4076.4 4566.1 5054.1 19

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We use splines to interpolate the data and provide a model fit. Our matlab code was: Our graph provides estimated flow values at any angle from 5 to 50 from vertical. It shows our theoretical maximum flow is around 90 in^3/s at approx 32 degrees from vertical. EXPERIMENTAL FLOW CONT. AN=[5 10 15 20 25 30 35 40 45 50];F=[4.1 4.7 6.1 8.2 27.3 86.2 80.3 76.4 66.1 54.1]; EF=spline(AN,F,linspace(5,50,250)); ANE=linspace(5,50,250); plot(ANE,EF);hold on;plot(AN,F,'*k'); xlabel('Angle from vertical'),ylabel('flow rate (in^3/s)'),title('Experimental flow calculations') legend('Experimental fit','Table values') 20

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Ratholes can cause serious problems with flow. To better understand the issues they cause, this function calculates the fraction of usable flow area left by a rathole, and the fraction of the total volume of the bin the rathole takes up. It makes the assumption that you are using an economically designed (H=(1:4)*max diameter) cylindrical hopper with a centered cylindrical rathole and a circular outlet. Additionally, it assumes the material is not significantly large and has negligable tendency to clump together. The function is as follows: RATHOLE CALCULATIONS function [FA,FV]= Rathole(DI,DO,Hbin,DR,BA); %Inputs: %DI is the input diameter, or the diameter of the cylindrical bin %DO is the output diameter, or the diameter of the circular outlet %Hbin is the height of the cylindrical bin area %DR is the diameter of the rathole %BA is the bin angle in degrees. %Outputs %FA is the usable fractional area of the outlet for flow %FV is the fraction of the total volume of the bin the rathole takes up %In function %AI,AO,RA are the input, output, and rathole area %HC and HT are the height of the conical bottom section and the total area %Vtotal and VR are the total volume of the bin and the rathole volume 21

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if Hbin 4*DI, error('Bin height should be 1 to 4 times bin diameter to be economical.') end if DR > DO | DO > DI, error('Diameters should be: DI>DO>DR') end %Our article stated that H should be DI*(1-4); this step checks that condition and other logical conditions AI=pi.*DI.^2./4;AO=pi.*DO.^2./4;RA=pi.*(DR.^2)./4; %This step calculates the input, output, and rathole area. UA=AO-RA; %This step computes the usable area by subtracting rathole area from output area. FA=UA./AO; %The fractional area is computed by dividing the usable area by the output area. HC=(DI-DO)./2.*tand(BA);HT=HC+Hbin; %The height of the bottom section is computed by the slope of the bin and the difference of the %input and output diameters, assuming the bottom section is approximately a frustrum of a cone. Vtotal=pi.*HC./3.*(DI.^2+DO.*DI+DO.^2)+AI.*Hbin; %The volume total is a combination of the formula for the volume of a %cylinder for the top combined with the volume of the bottom frustrum. VR=HT.*RA; %The volume of the rathole is calculated by multiplying the bins total height by the ratholes area. FV=VR./Vtotal; %The fractional volume of the rathole is calculted by dividing the rathole volume by the total volume. RATHOLE FUNCTION CONTINUED: 22

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Assuming a bin with a 10 foot diameter inlet, a cylindrical bin height of 25 ft before the conical section, a bin angle of 60 degrees from horizontal, and a varying output diameter, this graph shows the effect of ratholes on fractional output area. As you can see, even small ratholes cause immediate drops in the usable flow area, even when the output diameter is very large (1/2 inlet diameter) While it is not shown, the fraction of the total volume taken up by these ratholes is very low; the maximum was just over 20% for a rathole that was 5 ft across, or half the diameter of the input. For outlets that are small fractions of the inlet diameter, less than 5% of the total volume will cause near complete loss of usable flow area EXAMPLE RATHOLE CALCULATIONS 23

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The Janssen equation, as seen in (CITE OTHER POWERPOINT HERE), calculates the pressure on a bin as a factor of bin major diameter, bin height, gravity, material density, Janssen coefficient, and bin angle from vertical. *INSERT FIGURE WITH EQUATION HERE* But what if we know the maximum pressure our bin can support, but want to figure out the minimum deviation from vertical our bin can support? We can use Matlabs fzeroes function, the Janssen equation, and our maximum pressure to solve for the minimum angle from vertical. JANSSEN CALCULATIONS: function AD = Amin(D,H,y,g,K,pmax); %This function calculates the minimum angle from the vertical a hopper must be using the Janssen equation. %The function calculates angle using US units. % We use.8 pmax in our calculations as a safety factor, so that fluctuations during use do not go over our maximum tolerance. %D=Diameter, H=Height, y=density, g=gravitational acceleration %K=Janssen coefficient,pmax=max pressure %AD is the minimum bin angle from vertical in degrees. AD=fzero(@(x) ((y.*g.*D./(4.*tand(x).*K)).*(1-exp(-4.*H.*tand(x).*K./D))-.8.*pmax),45); %This finds the zeroes of an anonymous Janssen function of angle, minus the (practical) pmax. %It guesses an intermediate angle of 45 degrees to start. if AD>70, error('Pmax is too low to be practical') elseif AD