# Experiment 8:
Sumedh Deshkar, 21MI3EP18
### Studying the effect of design considerations of a Hopper bin on material outflow characteristics
Sumedh Deshkar | 21MI3EP18

### Objective:
To study the flow pattern of material in a hopper bin with different hopper design considerations.

### Observation:
Hopper cylinder diameter: 38 cm
Hopper cylinder total length: 99 cm

#### Table - 1
#### Table - 2
Diameter (d) =38cm
At t=0, empty height= 38.5 cm
Total volume= 81022.92 cm3


### Calculations:

#### For table -1

V1= $𝝅 d^2 * h/4$ , d= hopper cylinder diameter
h= height of sand

V2= $(⅓)*𝝅*(r1^2+r2^2+r1r2)*(r1-r2)*cot(θ/2)$, r1= larger radius of fructum
r2= smallar radius of fructum
θ= angle of cone
1.
V1= $𝝅*38^2*(99-32.5)/4$ = 75418.64 cm3
V2= $(⅓)*𝝅*[(19)^2+(4.85)^2+(19)*(4.85)]*(19-4.85)*cot(25/2)$=31860.30 cm3

V=V1+V2= 107278.94 cm3
Flow rate= V/time = 4126.11 cm3/s
2.
V1= $𝝅*38^2*(99-34.5)/4$ = 73150.41 cm3
V2= $(⅓)*𝝅*[(19)^2+(7.5)^2+(19)*(7.5)]*(19-7.5)*cot(30/2)$=25157.54 cm3

V=V1+V2= 98307.95 cm3
Flow rate= V/time = 5498.21 cm3/s
3.
$V1= 𝝅*38^2*(99-42)/4$ = 64644.55 cm3
V2= $(⅓)*𝝅*[(19)^2+(2.6)^2+(19)*(2.6)]*(19-2.6)*cot(60/2)$=12408.97 cm3

V=V1+V2= 77053.52 cm3
Flow rate= V/time = 733.84 cm3/s

#### For table -2

Initial filled volume =[(𝝅*38^2*(99-38.5)/4 + 12408.97] cm3
= 81022.92 cm3

$𝚫V= 𝝅*d^2*𝚫H/4$ , Spontaneous flow rate= 𝚫V/t

1. $𝚫V= 𝝅*38^2*3.5/4= 3969.40 cm3$
Total volume = (81022.92-3969.40) = 77053.52 cm3
Flow rate = 516.17 cm3/s

2. $𝚫V= 𝝅*38^2*4.5/4$= 5103.51 cm3
Total volume = (77053.52-5103.51)cm3 = 71950.01 cm3
Flow rate = 985.23 cm3/s

3. $𝚫V= 𝝅*38^2*1.5/4$= 1701.17 cm3
Total volume = (71950.01-1701.17)cm3 = 70248.84 cm3
Flow rate = 309.86 cm3/s

4. $𝚫V= 𝝅*38^2*3.5/4$= 3969.40 cm3
Total volume = (70248.84-3969.40)cm3 = 66279.44 cm3
Flow rate = 638.17 cm3/s

5. $𝚫V= 𝝅*38^2*4.8/4$= 5443.75 cm3
Total volume = (66279.44-5443.75)cm3 = 60835.69 cm3
Flow rate = 733.66 cm3/s

6. $𝚫V= 𝝅*38^2*5.2/4$= 5897.39 cm3
Total volume = (60835.69-5897.39)cm3 = 54938.3 cm3
Flow rate = 842.48 cm3/s

7. $𝚫V= 𝝅*38^2*5.8/4$= 6577.86 cm3
Total volume = (54938.3-6577.86)cm3 = 48360.44 cm3
Flow rate = 822.23 cm3/s

8. $𝚫V= 𝝅*38^2*7.2/4$= 8165.63 cm3
Total volume = (48360.44-8165.63)cm3 = 40194.81 cm3
Flow rate = 1038.88 cm3/s

9. 𝚫V= $𝝅*38^2*3/4$= 3402.34 cm3
Total volume = (40194.81-3402.34)cm3 = 36792.47 cm3
Flow rate = 405.04 cm3/s

10. 𝚫V= $𝝅*38^2*21/4$+ volume of conical part = ($𝝅*38^2*21/4$ + 2136.51) cm3
= 25952.92 cm3
Total volume = 0 cm3
Flow rate = 589.24 cm3
#### Graphs :
Through graph, using the 7th and 8th reading as it coincides with the trend line
𝚫V = (54938.3-48360.44) = 6577.86 cm3
𝚫T = (47-39) sec = 8 sec
Flow rate = 822.23 cm3/sec
### Results:

The average flow rate is lowest for the smallest diameter frustum orifice and highest for the larger diameter. According to Table 2, the instantaneous flow rate ranged from around 550 cm3/sec to 850 cm3/sec, with some values showing more variation. The volume vs. time graph was nearly a straight line with a slope of 822.23 cm3/sec. The average flow rate was 733.84 cm3/sec when the bin was initially completely filled for a 60-degree angle frustum (3rd case in Table 1). In contrast, it was 589.24 cm3/sec when it was partially filled initially (10th case in Table 2).

### Discussion:

Based on our experimental results and observations:
* Flow Type: In all three cases, the flow was funnel-like. This happened because the height of the hopper bin wasn't very high. If it were taller, we might have expected a more uniform mass flow, similar to how water flows.
* Residue: Residue was left inside the hopper bin in the first two cases. However, when the angle of the conical end was steep (60 degrees), we noticed that no residue was left. Steeper walls ensured that all the material flowed out, while less steep walls caused some material to get stuck.
* Impact of Orifice Diameter: The flow rate was higher for a larger diameter orifice because it allowed more material to flow out easily. Conversely, with a smaller opening, material couldn't flow out as smoothly.
* Impact of Frustum Angle: We expected the flow rate to be higher for a steeper angle of the frustum. However, in our case, the flow rate was lowest for the highest angle of the frustum because the frustum's diameter was very small. The flow rate was highest for the 2nd case, where the angle was steeper (30 degrees) and the orifice diameter was larger than in the 3rd case.
* Variations in Flow Rate: In some cases, the instantaneous flow rate varied significantly, reaching values as high as around 1000 cm3/sec or as low as around 400 cm3/sec. This variation could be due to inconsistent flow of sand through the hopper bin, caused by the funnel flow and hopper bin surface.
* Initial Fill Height: The average flow rate was higher when the bin was initially filled to a greater height (733.84 cm3/sec) compared to when it was partially filled (589.24 cm3/sec). This occurred because a higher initial height resulted in smoother material flow with fewer void spaces and increased pressure, leading to a higher flow rate.

### Questions
1. **Impact of Discharge Opening Size:**
The size of the hole where material flows out matters. A big hole allows more material out quickly, but it can make the flow uneven because it's harder to control. A small hole slows the flow, but it's more controlled. We found the best flow rate with a medium-sized hole.
**Impact of Discharge Chute Angle:**
If the chute is steep, material flows out faster because gravity helps. There's less leftover material with a steep chute. A shallow chute means slower flow.
Both factors work together to determine the flow rate. We found the best flow rate with a medium angle and diameter.

2. **Flow in Liquids vs. Granular Materials:** Liquids flow smoothly and predictably, while granular stuff like sand can be inconsistent and sometimes clog up. With a shallow bin, granular material can leave some behind. This doesn't affect liquids as much. The hole size matters more for granular stuff.

3. **Ball Mills and Hopper Bins:**
* Ball mills grind things into smaller pieces, and hopper bins store stuff.
* Based on our experiments, the flow rate and material characteristics help us choose the right shape, height, hole size, and chute angle for a hopper bin. How material flows out helps us design the bin.
* When it comes to ball mills, we need to think about how the material moves, if any stays still, and how the chute angle affects how much material goes in. This all affects how efficiently the grinding works. Separating different-sized material based on how it flows also affects grinding efficiency.
### References:
* Material Handling and Mineral Engineering Lab Manual
* www.wikipedia.com