Hopper half-angle

The mass-flow and funnel-flow limits in silos are well known and have been used extensively in proper design. The limits for conical hoppers and plane hoppers depend on the hopper half-angle 9, the effective angle of internal friction 5 and the wall friction angle ( ). Once the wall friction angle and effective angle of internal friction have been determined by experimental means, the hopper half angle 0 may be determined. In function form it can be expressed as... 

The hopper half-angle is chosen from the mass-flow limits as shown in Figure 3.7, while the condition for Cj = is obtained from the intersection... 

Hopper half-angle 0 and mass flow rate versus hopper opening B. 

To evaluate the discharge mass flow rate Q, the discharge velocity needs to be determined. The hopper half-angle is read of from Figure 3.13 as 28°, so that tan (28°) = 0.5317. For conical hoppers m = 1, and ff/ff approaches zero. With all these figures and considerations, substituting into Equation 3.10 ... 

Flow factor graph (similar to Figure 3.11) showing the value of ff and the hopper half angle. 

Hopper half-angle, hydrocyclone cone angle... 

If steady flow is desired, the hopper geometry should be designed such that mass flow will take place no stagnating regions should occur. In this case, the solids flow along the walls of the hopper. The wall, therefore, must be sufficiently steep and the flow channel must not have any sharp corners, abrupt transitions, or discontinuities in frictional properties at the wall. As a rule, the hopper half-angle a should not exceed with a ax determined from ... 

Comparison with Mechanical Blenders or Homogenizers. Mechanical blenders normally have recirculation systems either within the blender or placed outside it. There are several types, but they have in common steep hopper half-angles with respect to vertical as compared to fluid bed blenders. This is required to allow all the mixed material to flow out of the blender. The silo geometry has to ensure mass flow conditions, which accounts for even withdraw of products, but at the same time, the angles have to be somewhat shallower to allow shear between the flowing layers of products to cause blending. If the outer hoppers are shallower, then normally, blenders are equipped with inserts, such as hopper-in-hopper types. These inserts can work in both axisymmetric and plane flow blenders, although the former types are more common. 

A comparison of the experimentally determined critical arch spans /outlet widths and the critical outlet widths calculated from the Jenike method [2] are given in table 2, for fly ash, hydrated lime and olivine sand. In table 2 at filling represents the measured maximum arch span which occurred prior to sustained flow of the stored bulk solid. The intervals on the predicted outlet widths are at the 95% confidence limit determined from statistical analysis of the failure function data. The recommended mass flow design lines [2] correspond with hopper half angles of 37, 28, and 30 degrees respectively for fly ash, hydrated lime and olivine sand. 

Therefore, to ensure mass flow of the given powder in a conical hopper, the half angle of the hopper should be no more than 35° and the circular opening should be larger than 0.21 m to prevent the formation of a stable arch that can span the opening. 

Example 8.2 Consider a conical feed hopper with half angle 10°. The inlet and outlet diameters of the hopper are 1.0 and 0.5 m, respectively. The particles, 200 pm beads with a density of 2,300 kg/m3, are fed into the hopper at the mass flow rate of 100 kg/m2 s under an inlet pressure of 105 Pa. The particles are in a moving bed motion with a particle volume fraction of 0.5. The pressure at the hopper outlet is 1.2 x 105 Pa. Determine the air flow rate in this hopper flow. The coefficient is given as 102,700 kg/m3 s. 

As mentioned, the flow rate in a standpipe depends on the solid feed device as well as the flow control valve. In this section, we discuss the gas-solid flows in a simple standpipe system where the feed device is a mass flow hopper and the solid flow regulator is a discharge orifice [Chen et al., 1984]. As shown in Fig. 8.15, the entrance of the vertical standpipe is connected to a conical hopper feeder of half angle solids flow patterns are considered. One is a dilute suspension flow, and the other is a solid moving bed. In this case, the following additional assumptions are needed ... 

A conical hopper of a half angle 20° and an angle of wall friction 25° is used to store a cohesionless material of bulk density 1,900 kg/m3 and an angle of internal friction 45°. The top surface of the material lies at a level 3.0 m above the apex and is free of loads. Apply Walker s method to determine the normal and shear stresses on the wall at a height of 1.2 m above the apex if the angle between the major principal plane at that height and the hopper wall is 30°. Assume a distribution factor of 1.1. 

Determine the half angle of the hopper and the minimum outlet diameter for the two cases where the hopper wall material has an angle of wall friction of (i) 10° and (ii) 30°. 

Hopper Index [degrees], HI = the recommended conical half-angle (measured from the vertical) to ensure flow at the walls. Usually add 3° to account for variability. Values range 14-33° with 304 s/s. 

In order to apply Fick s Lrst law to the low of particulate material, we refer again to the above case study, that is, the discharge of the particulate material out of a conical-shaped hopper with an oriLce of 5 mm and a half-center angle of fee Figure 20.17). 

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