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Day students our lecture. Today is on hopper design. Hopper is usually used as a a short term storage for raw materials intermediates and products in the form of particulate in a process plant interruption of flow from discharge hopper may frequently be responsible for production hot therefore hoppus should be designed to ensure flow of the intended particles or powder.

Hopper design is a white subject in this lecture. We only cover design philosophy to ensure flow from conical hoppers. When required in hopper design.

The four big questions are what is the appropriate flow mode hopper angle. How large is the outlet for a reliable flow. And what type of discharge is required and what is the discharge rate.

So lets look at this elements. First lets look into hopper flow modes that are available. So the main hopper flow mode.

That has been identified a mass flow and call or funnel flow somaek in mass flow. All the materials in the hopper is in motion. So when we open the discharge orifice.

All the materials will start to move will start to move. But not necessarily at the same velocity. Not all layers or all particles will be moving at the simulus velocity in core or funnel flow.

The central part will be moving and the sights will be dead. Then we have another flow mode. Which is expanded flow and this is a combination of mass flow call with funnel flow of of it.

So. This is a diagram of cough low or funnel flow and mass flow income flow. The top solids.

The solids at the top will start moving first once the discharge orifice is opened. So the sights will remain stationary. Theres only contacts between these particles and these particles okay.

So once it will empty from the top. So once all these powders are out so finally these powders here will be discharged. So this is call of funnel flow in mass flow.

All particles will start moving at the same time at different velocity. So this particles will have particles particles interaction as well as particles moving along the wall of the hopper or silo so. This is the graphical image of the core flow and mass flow so in mass flow.

These lines. If you notice as they go down it moves as a straight line until it reaches the core. So this is the conical part okay so this is called the silo a straight.

One okay so it it moves in a straight line until it reaches the conical part and then still the lower part of the materials dot will be discharged first so it follows its turn until finally all the particles are discharged as in call flow pattern. If you see if you have particles with different layers. The middle part will start to flow first so it will start flowing up.

And then finally is the first layer that went in will be the last layer to leave the copper as for expended flow. It is a combination of mass flow at the bottom. Here.

The corn is not a direct comb. So it has a cone shape here and then a narrow discharge point. Here.

So. Therell be a mass flow here and a funnel flow taking place at this area. So that is expanded flow.

This are a list of characteristics of mass flow so in a mass flow. The motion powder is the motion of powder is uniform so it moves uniformly and at steady state and the bulk density of the discharged powder is constant and independent of the silo height stresses among the solids are low. So there are low compassion compaction of powder.

So it actually allows movements smooth movement of powders. There is no second regions in a mass flow and the risk of product. Degradation is also small compared with cough low the first in first out flow pattern of mass flow.

Hopper ensures. Narrow range of residence time for solids in the silo. So so the first product that came in would be the first product that goes out so and we was that that will be used first so that means we dont store a product for extended period.

Okay and in segregation of particles. According to size is also less of a problem in mass flow. However the friction between moving solids and hopper walls can result in erosion of walls and then when this erosion of wall this can give rise to contamination of solids by the material of the hopper wall as well as it can also cause degradation of the hopper wall.

So that will require some mechanical repairing as well so when erosion becomes a problem when that powder usually causes erosion. It doesnt get along very well with a material of the hopper. Then we can consider half flow for that particular powder production powder used in production next look into the next.

Lets look into the second question. Which is hopper angle for conical hopper the slope angle required to ensure mass flow depends on powder powder friction and powder wall friction so app hopper which gives mass flow with one powder may give a cough low with another so if we look at this figure as the compaction stress increases the strength of the powder increases so this is for a weak powder as the compaction increases the strength of the powder increases faster and this is for strong powder. Which is cohesive.

Which is easily flowable powder. Okay so this one the the strength when we change the compare compacting strength. When there is a small change.

Theres also a small change in the strength of the powder. So what does this strength of the powder do to affect a flow from hopper. So it actually produces arching in the flow.

So if this arch as it builds up so when you have powders here or in small powders settles and then as it builds up so it will start to compact one another so when the compaction increases. So once the strength is higher than the strength among the flowing powders. So the arching will happen and flow will be stopped if the strength of the powders.

Which are flowing are stronger the actual form. But it will collapse because this side is stronger than the arch itself. So the flow will still continue to happen so the hope of flow vector ff relates.

The stress developed in a particular solely with the compacting strikes acting in a particular hopper. So its as so the flow factor ff is just a ratio. Okay so a high value of effect.

Means low probability. Why because high compacting stress factor. Means greater compaction.

A low value of stress in powder means more chance of an arch forming so high value of stress compacting stress and low value of stress in powder will give you high value of hopper flow factor. Okay. So thats why high value means the flow will not happen.

Because there is an arch so hopper flow factor. Depends on nature of the solid nature of the wall material and slope of the hopper wall. The hopper flow factor can also be associated with yield stress.

Okay yield stress is a stress. Which causes flow them is it allows flow. So the powder in the exposed surface of the arch is sigma y.

Which is also known as and confined yield stress of the powder. So if the stresses developed in the powder forming the arch a greater than the uncon. Find you stress flow will occur okay.

So. If the powder stress is bigger than de and confined to yield. Stress.

So the fold flow will occur. If we look back to the slides. Yeah.

So this is where the uncon find yield. Stresses and this is where the powder stresses are so obviously. If this stresses are stronger.

The powder flow. This criterion can be rewritten. As this okay.

So this sigma. D. Is replaced by sigma c.

Ova. Flow factor. So the unconfined yield stress of the solids varies with the compacting stresses of different powders.

So this yield stress is a function of the compacting stress of a specific powder. So lets look into the critical flow flow. Condition the limiting condition for flow.

Is when sigma c. Over flow factor. Is equals to sigma y.

The unconfined yield stress. So lets look at this figure. Ten point seven.

This is a plot of stress developed. Which is six mardy or on confined yield stress. Which is sigma y against compacting.

Stress. Sigma c. So.

This is the same graph f. As we have same figure that we have seen previously its this graph this figure will actually tell us at which condition flow will occur for a specific specific powder in the hopper. If lets say.

The powder has a yield stress oki. The sigma y or sigma d. Greater than sigma.

C. Over f f. Okay.

That means. If it is greater than this line. So this line indicates.

When its equals. So if the yield stress is greater than this value. Which is actually a no flow will occur that is the flow function for powder.

A if the powder has a yield stress lesser than sigma c ff. Which is function c the flow occurs now lets look as powder be okay powder b. Has a critical condition this point because it coincides with the limiting condition of flow.

When is equals to so at this point. The unconfident yield stress is equal to stress developed in the powder. So therefore.

If actual stress develop is smaller than the critical flow. Okay when its smaller then there is no flow. If it is bigger then place flow okay.

So that is the condition with outer v. The third big question in the hopper design is outlet size. So the stress developed in the arch is related to the size of the hopper outlet.

Gsturm tasbeeh and the bulk density of the powder. So a minimum outlet dimension of the hopper is defined by this equation. And the parameters are if you can see there is a 6th ma.

Critical and then there is a gravitational flow. And then theres density of particles. As well as h.

Theta. So what is h theta h. Theta is the factor determined by the slope of the hopple wall.

So the slope that we did mean because its conical in shape right so. There will be a slope angle. So that is h.

Theta and h. Theta can be estimated using this equation. So this is particularly for conical hoppers.

This slide. Gives a summary of a conical hopper design. If you notice.

We did not discuss the big question for which is this touch. How the particles will be discharged. And what is the discharge rate.

So that is not discussed in this lecture. So we will we have only discussed up until. The first three big questions okay so in summary.

The relationship between strength of powder in the arch. Yeah. The unconcealed strength with compacting stress acting on the powder plays an important role on the flow characteristics of the powder.

So variation of hopper flow vector is affected by nature of the powder nature of the hopper wall slope of the hopper wall. Yeah hopper flow vector is a function of powder properties. As well as the hopper properties and knowing the hopper flow vector and powder flow function.

Critical stresses in the arch can be determined and minimum size of the outlet. Can be formed so so to design a hopper. We need to measure the powder cohesiveness.

Which gives us the interparticle friction wall. Friction. The powder and wall friction.

Compressibility and permeability of the powder and then we calculate the hop outlet size upper angle for mass flow and the discharge rate. So this one was is not covered in this lecture. So you are expected to know up until this.

But lets look into the worked example to understand how we use the equations that we have discussed in designing the so. The question states share cells test test on a powder gives the following information. So we have kinda matic anger.

No effective angle of internet internal friction is what it decrees kinematic angle is this 116 degree bulk density mm and the powder flow function is represented by this so determine the maximum semmy included angle of a conical mild steel hopper that will confidently ensure flow and minimum diameter of the outlet right we have not discussed this part the semi included angle in the lecture part so ill show you in this tutorial in this question. How we find the semi included angle. This is actually the h.

Theta. So we are going to find this. Data menu and then we are going to solve for the minimum diameter of the circular outlet to do that we need a set of figures which im going to show you now and it is actually present in the appendix part of the lecture.

Okay. Its here. So there are its actually known as hopper flow factor values or conical channels at different angles.

So we have one figure for 30 degrees. One for 40 degrees. And then one for 50 in another for 60.

So if you notice. These are the flow factors and then this is a so this is for when the conical channel is has a effective angle of internal friction. Which is 30 degrees.

So this is the data. Okay so this is the semi included angle of a conical part. Okay.

The theta in the h h. Theta. And this is the syw is the kinematic angle of wall friction.

Okay so if we know this value and we know the flow factor. We can find the theta value okay. Thats what we are going to do for the example question right since the question mentioned that the internal friction angle is 40 degrees.

So we will be using the figure of 40 decrease okay. Which is thicker be 40 degrees is weaker ten point one eight b. So its given the kinematic angle is 16 degrees.

So this is about fifty sixteen degrees. So we draw a line until we touch flow factor value okay and then we give a safety margin okay so there should be a 350 degree safety margin so we obtain the hop of flow factor as fifteen because it only crosses the ff 15. At all points right and the semi included angle for mass flow is thirty degrees.

So now we have no we have identified the data. So that is the answer for the question of the maximum semi included angle of a conical mice still hopper that will confidently ensure mass flow. So that is the answer so theta is 30 degrees.

So now with that answer we can calculate the part b minimum diameter of circular outlet to ensure flow. When the outlet is opened so we use this equation. And so we have a leash function here given so if we substitute the sigma c.

So we get this so the critical value of uncombined yield. Stress. Is found when it is equals.

Okay. This symbol here should be equal. So why are we looking for a critical value because the equation for minimum outlet.

Diameter. Requires the critical value so then we saw for critical value. And then we also calculate the.

H. Theta so at theta is equals to 30 the h solves to 25. And then we can find b okay.

Therefore. The mass flow without blockages can happen when the semi included cone angle is 30 degree and circular outlet. Diameter is at least twenty three point four in the final parts of the lecture.

Ive included the problems that can occur in hoppus. So we can have these problems happening in a hopper. So lets look into each one of it so we can have rattling or piping.

So this is the half flow. So this is where the white is where the particles are keep moving and this powders here continues to be stagnant thats little red cooling or piping. Theres a youtube video link that ive attached at the end of the slides for you to watch.

So one of the youtube video shows red cooling animation then funnel flow okay so in funnel flow segregation can happen so if we put in a mixed powder. So the cost of wants will build up and the finer ones may be flowing down. Okay.

So this can happen. Arching and doming. So this is what we studied just now so we can have arching where especially when the powders are very fine.

Then we can have insufficient flow. So this could be when the outlet size is too small we can have flushing when the outlet size is too big we can have inadequate emptying. Which means every time we empty our hopper.

This parts just remains okay. So this one can cause quality issue. Then mechanical arching.

So this is similar to traffic jam at the outlet of bin. So too many large particles are competing for the small outlet. So this can cause mechanical arching and time consolidation.

Consolidation or kicking. So many powders will tend to kick as a function of time humidity and pressure and temperature. So this one is like example is milo.

So. If lets say mullet milo. Is passing through a funnel and then if there are some milo particles that are left in a funnel after some time it will attract humidity and it can actually start kicking building up and sticking to the wall.

Okay. So this can also happen so these are the problems that can happen in a storage bin or hopper. We call it right so these are some youtube videos.

Interesting youtube videos that you can watch that explains about about the hopper flow. So this one is on corinne funnel or cough flow. So im sorry this one is on ko flow.

This one is on mass flow. And this one is on red holi. So enjoy watching the videos and these are the appendix.

Which also present in your textbook for different angle values right. Thats all for copper design. See you in the next lecture.

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