2 Hopper/bin design
2.2 Flow patterns
Two primary flow patterns can develop when a bulk solid discharges from a bin: funnel flow and mass flow. Funnel flow, also called core flow, is defined (Jenike 1964) as a flow pattern in which some material is stationary while the rest is moving (see Figure 2.2). According to
Figure 2.1 Subdivisions of a bin: cylinder (vertical section at top), hopper (converging section below).
Flowing material
Stagnant material
Figure 2.2 Examples of funnel flow.
this definition, a bin that contains any stagnant material during discharge qualifies the flow pattern as funnel flow.
Rotter (2005) further refined the definition of funnel flow into pipe flow and mixed flow.
The distinction between the two is whether or not the flow channel intersects any portion of the walls of the bin (usually in the cylinder section). If there is no intersection (pipe flow), the flow channel walls may be vertical (parallel pipe flow), converging from top to bottom (taper pipe flow), vertical along the sidewall of the bin (eccentric parallel pipe flow) or converging along the sidewalls of the bin (eccentric taper pipe flow) (see Figure 2.3).
If the flow channel intersects the cylinder wall (mixed flow), the converging flow channel may be symmetric about the centreline of the bin (concentric mixed flow), fully eccentric if the hopper opening is to one side of the vessel (fully eccentric mixed flow), or intersecting the cylinder walls at varying elevations because of a partially eccentric outlet (partially eccentric mixed flow) (see Figure 2.4).
In contrast to funnel flow, Jenike defined mass flow as a flow pattern in which all of the material is in motion whenever anything is withdrawn (see Figure 2.5). Note that this definition does not require that all material at a given evaluation across the cross section be moving at the same velocity. Indeed this is nearly impossible elevation to achieve, since particles near the centreline of a bin have virtually no shearing resistance, whereas particles sliding along converging hopper walls flow move slowly. This velocity distribution can be minimised by appropriate choice of hopper angle and hopper wall material of construction, or through the use of an insert.
One should beware of vendors claiming they can supply ‘mass flow’ bins with no con-sideration of the hopper wall material or stored bulk solid. The term ‘mass flow’ is often misused.
A third flow pattern, expanded flow, is a combination of funnel flow and mass flow (see Figure 2.6). Usually this is achieved by placing a small mass flow hopper below a funnel flow hopper. The mass flow hopper section expands the flow channel from the outlet up to the top cross section of the mass flow hopper. It is important to ensure that this cross-sectional area is sufficiently large so as to avoid ratholing in the funnel flow hopper section.
Expanded flow designs are generally considered only when the cylinder diameter exceeds 6 m or so.
Flowing Flowing Flowing
Internal pipe flow Eccentric pipe flow
Flowing
(a) Parallel pipe flow (b) Taper pipe flow (c) Eccentric parallel pipe flow
(d) Eccentric taper pipe flow
Figure 2.3 Forms of pipe flow. (Courtesy Rotter 2005.)
Effective transition
(a) Concentric mixed flow (b) Fully eccentric mixed flow (c) Partially eccentric mixed flow Effective transition:
Figure 2.4 Forms of mixed flow. (Courtesy Rotter 2005.)
Figure 2.5 Mass flow.
Mass flow Funnel flow
Figure 2.6 Example of expanded flow.
Mass flow
Funnel flow
Uncertain region
φw : Wall friction angle
0°
θc : Conical hopper angle, from vertical
Figure 2.7 Example of conical hopper design chart (based on data from Jenike 1964).
Jenike (1964) presented a series of graphs defining the limits of hopper angle within which mass flow can develop. The major variables that determine these limits are:
r
Wall friction angle,φw. This is the arc tangent of the coefficient of sliding friction between the bulk solid and hopper wall material. This value often changes with pressure, usually decreasing as pressure increases. In order to achieve mass flow throughout a hopper, it is important (and often sufficient) to determine the wall friction angle at the pressure level expected near the hopper outlet, since this is the region of minimum pressure during flow.r
Effective angle of internal friction,δ.r
Hopper geometry, e.g. cone, wedge.Jenike’s graphs were developed for axi-symmetric cones (hopper angle measured from vertical denoted as θc) and for infinitely long wedge-shaped hoppers (θp). Examples of design charts are shown in Figures 2.7 and 2.8. The two primary bulk solid parameters that affect limiting hopper angles for mass flowφwandδ are in turn often affected by temperature (of the bulk solid and the hopper wall material), moisture, content of the bulk solid, hopper
40° 30° 20°
10° 0°
φw : Wall friction angle
Mass flow
θp : Planar hopper angle, from vertical θp
Figure 2.8 Example of planar hopper design chart (based on data from Jenike 1964).
Figure 2.9 Cohesive arch.
wall material including its surface finish, and time of storage at rest. A secondary effect is the particle size distribution of the bulk solid.
It is important to emphasise thatφwvalues can be determined only by shear testing (as described in Chapter 1). No other so-called flow tests (e.g. angle of repose) can provide this information.
2.3 Arching
Arching (sometimes called bridging or doming) can result from either mechanical interlock-ing or cohesive strength (see Figure 2.9). Mechanical interlockinterlock-ing occurs when particles are large relative to the outlet opening, whereas cohesive arching occurs because of bonding between particles.
To avoid interlocking arching, the outlet dimension must exceed some critical multiple of the characteristic particle dimension. While this characteristic dimension is not well defined, a conservative approach is to take this as the maximum length of a chord that spans the particle in any direction. The minimum values for outlet size are as follows:
r
For a circular or square outlet, the outlet size should be at least 6–8 times the charac-teristic particle dimension.r
For an elongated outlet, its width must be at least 3–4 times the characteristic particle dimension and its length must be at least 3 times the width.Mechanical interlocking usually governs minimum outlet dimensions only when the bulk solid’s mean particle size is greater than about 6 mm, there are few ‘fines’, and the material has little to no surface moisture or other condition that would cause particles to adhere to each other.
Sizing an outlet to avoid cohesive arch formation is not as simple as considering only particle size. Flow properties tests need to be run to determine the material’s flow function, which is the relationship between cohesive strength and consolidating pressure. Once this has been obtained, the hopper’s flow factor must be determined. The original source for
BC = Minimum outlet diameter Figure 2.10 Mass flow conical hopper.
flow factors, and still a convenient way of obtaining them, is through graphs presented by Jenike (1964).
To avoid arching in a mass flow hopper, the outlet diameter BC (see Figure 2.10), or the width of wedge-shaped outlet BP (see Figure 2.11), must be at least the following:
B= σ¯1H (θ) gρb
(2.1)
where B is BC or BP, m; ¯σ1is the critical stress required to cause arch to fail, N/m2; H (θ) is the dimensionless function dependent on hopper angle (Jenike 1964). Value typically ranges from 2.0 to 2.3 for conical mass flow hoppers, half this for wedge-shaped hoppers; g is the acceleration due to gravity, 9.81 m/s2;ρbis the bulk density at consolidating pressure calculated at hopper outlet, kg/m3.
For a wedge-shaped hopper, the outlet length must be at least two times its width if the end walls are vertical and at least three times its width if the end walls are converging.
To avoid arching in a wedge-shaped funnel flow hopper, the outlet width, BF, must be at least that calculated using Equation (2.1) with the appropriate values of ¯σ1andρb and
H (θ) = 1.
A material’s flow function, which can be determined only by shear tests, is often strongly influenced by its temperature, time of storage at rest, moisture content and particle size distribution. A hopper’s flow factor is a function of the effective angle of internal friction (δ), wall friction angle (φw) and hopper geometry.
BP = Minimum solt width Figure 2.11 Mass flow transition hopper.
2.4 Ratholing
Also called piping, this is a phenomenon in which a more or less vertical flow channel develops above the hopper opening and, once emptied, remains stable (see Figure 2.12).
In order to avoid the formation of a stable rathole, it is necessary that the size of the flow channel exceed the critical rathole diameter, DF, which is calculated as follows:
DF=σ¯1G(φ) gρb
(2.2) where ¯σ1is the cohesive strength of bulk solid at major consolidating pressure calculated at bin outlet, N/m2; G(φ) is the dimensionless function dependent on bulk solid’s angle of internal friction (Jenike 1964). Typical values range from 2.5 to over 7.
The two key parameters in this equation, flow function and angle of internal friction, can be strongly affected by the bulk solid’s temperature, time of storage at rest, moisture content and particle size distribution.