16 / June 2021 powderbulk.com
der Flow," Chemical Engineering, Vol.
121, No. 1 (January 2011).
2. G. Mehos and D. Morgan, "Hopper
Design Principles," Chemical Engi-
neering, Vol. 126, No. 1 (January
2016).
3. K. Johanson, "Successfully Dealing
with Erratic Flow Rates," Powder
Pointers, Vol. 3, No. A (2009).
4. T. A. Royal, and J. W. Carson, "Fine
Powder Flow Phenomena in Bins,
Hoppers, and Processing Vessels,"
presented at Bulk 2000: Bulk Handling
Towards the Year 2000, London, 1991.
For further reading
Find more information on this topic
in articles listed under "Agglomera-
tion" in the article archive on PBE's
website, www.powderbulk.com.
Greg Mehos, PE (greg@mehos.
net, 978-799-7311) is a chemical
engineering consultant specializing
in bulk solids handling, storage, and
processing and an adjunct professor
at the University of Rhode Island. He
received his BS and PhD in chemi-
cal engineering from the University
of Colorado and his master's from
the University of Delaware. He's a
Fellow of the American Institute of
Chemical Engineers.
Greg Mehos & Associates
Westford, MA
978-799-7311
www.mehos.net
where R
H
is the hydraulic radius
of the cylinder, k is the Janssen
coefficient (typically 0.4 to 0.6), ϕ′ is
the wall friction angle, and h is the
height of bulk material inside the
cylinder.
Combining Equations 13 and 14
yields a quadratic
2(m+1)tanϴ′
v
2
o
+
1
1−
ρ
bo
v
o
−1
Bg K
o
ρ
bmp
16
The maximum steady solids
discharge rate is the product of the
powder's velocity, its bulk density
at the outlet, and the cross-sectional
area of the outlet
ṁ
s
= ρ
bo
A
o
v
o
17
To ensure a steady solids dis-
charge rate and allow agglomerates
with maximum strength and con-
sistent properties, the hopper that
feeds the agglomeration equipment
should have an outlet large enough
to prevent an arch and allow the
required throughput. PBE
References
1. G. Mehos and C. Kozicki, "Consider
Wet Agglomeration to Improve Pow-
u
g
= −
K dP
ρ
b
g dz
13
Using Darcy's law and apply-
ing continuity to the gas phase, the
relationship between the air and
solids flowrates can be derived
dP
=
v
o
ρ
2
bo
g 1
−
1
dz
o
K
o
ρ
bmp
ρ
bo
14
where K
o
is the powder's permea-
bility at the hopper outlet, ρ
bo
is its
bulk density at the outlet, and ρ
bmp
is
its bulk density at a location inside
the hopper where the pressure
gradient is equal to zero (that is, the
gas pressure is at a minimum). A
value of ρ
bmp
equal to the powder's
bulk density at the solids stress
at the cylinder-hopper junction is
often used for design purposes,
3
as Figure 3 shows that the maxi-
mum solids stress in the cylinder
is approximately equal to the stress
where the gas pressure gradient is
zero.
4
The solids stress, σ
1max
, at the
cylinder-hopper junction can be cal-
culated using the Janssen equation
σ
1max
=
ρ
b
gR
H
1 − exp
−ktanϕ′
h
ktanϕ′ R
H
15
FIGURE 3
Consolidating pressure, bulk density, and gas pressure profiles for
coarse (high permeability) and fine (low permeability) powders
High permeability
Low permeability
Consolidating
pressure
Bulk
density
Gas
pressure
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