Flows from Bins: New Results - Selected Topics of Computational and Experimental Fluid Mechanics - page 165

ʲ

quantify the dependence of themass flow rate on , the angle of tilt. To our knowledge,

investigations of this character are very scarce. Franklin and Johanson (1950) studied
systematically the effect of the inclination on the flow rate by taking into account the
horizontal and vertical cases and two other intermediate angles. Later, Sheldon and
Durian Sheldon and Durian ( 2010 ) reported similar studies where the Franklin and
Johanson's correlation was maintained. Here, we consider the fact that the bin walls
have a finite thickness, and as a consequence this will bias the determination of the
mass flow rate at a given angle

.
In summary, we will contrast through experiments the new findings for the flow
rate in tilted bins with those recently published by other authors (Sheldon and
Durian 2010 ;Liu 2014 ). A general correlation valid for the angles of inclination
will not be reached, but, instead solid arguments related to the flow occurrence will
be featured.

ʲ

4.2 The Franklin and Johanson Formula

Near 60years ago, Franklin and Johanson (1950) established, by using a logical
reasoning, that the mass flow rate from an inclined, circular orifice at an angle ʸ with

the horizontal, is given by the relation

m 0 cos

ʸ r + cos ʸ

m ʸ =

, (5)

cos ʸ r + 1

ʸ r is the angle of repose and m is the flow
rate through a horizontal orifice, which essentially is given by the Hagen's law

where ʸ is measured counterclockwise,

m 0 = g 1 / 2 D 5 / 2 a ˁ , (6)

where a is the discharge coefficient.
From Eq. ( 5 ) it is clear that if m ʸ = m 0 , meanwhile if ʸ = 0 , ʸ = ˀ − ʸ r

then m ʸ = /2Eq.( 6 ) suggests that the flow rate for the vertical

case is directly obtained from the flow rate in the horizontal case by simply using
the multiplying factor cos

0. When ʸ = ˀ

. As mentioned above, this result is not
correct because in the vertical case the wall thickness determines the existence of the
flow, even for the ideal case where the wall thickness is zero (sharp-edge hole). As
was mentioned previously, in our formula we have that

ʸ r /( cos ʸ r + 1 )

ʱ = ˀ/ 2 and consequently,

m m 0 (ˀ/ which fit the experimental data very well. Thus, a correct

correlation for the flow rate among the horizontal and the vertical cases must include
information about the wall thickness.
In order to derive a correct formula valid for a wide range of tilt angles we have
performed a set of experiments that also involve changes in D , w and

= 2 − ʸ r )

ʸ r . We describe

such experiments in the next section (Fig. 12 ).