- Email: [email protected]
Chute profile for maximum exit velocity in gravity flow of granular material
J. agric. Engng Rex (1970) 15 (3) 292-294
Chute
Profile
for Maximum
Exit
of Granular
Velocity
in Gravity
Flow
Material
W. H. CHARLTON*; A. W. ROBERTS* An analysis of the required profile shape of gravity mum exit velocity is presented. The dynamic equations are developed and utilizing energy considerations, an Maximizing exit velocity involves minimizing losses; required chute curvature are drawn. 1.
flow discharge chutes to achieve maxifor a grain element in the flowing stream expression for exit velocity is obtained. on this basis conclusions concerning
Introduction
It sometimes may occur, when granular materials flow in chutes under gravity, that a maximum possible exit velocity is desired. This case has been known to arise in connection with the loading of rail cars with wheat. A rigorous analysis is a complex problem but Roberts ‘3 2 has shown that under fast flow conditions, with grain contact on the chute bottom and side walls, the average particle behaves approximately as though it moves subject to a constant Coulomb frictional force. Using such a model, the conditions for maximum exit velocity can be examined. 2.
Dynamic analysis
Fig. 1 shows the co-ordinates with respect to which the particle gravitational force acts in the direction of the positive y axis.
dynamics
are formulated.
The
Fig. I
Assuming
the coefficient
of friction,
p, is constant,
mj; +N(cos mi+N(p where N is the normal
8+p
the equations
of motion
sin 0)=mg
cos O-sin
O)=O
force and d2y i;S-@
Using the abbreviation
*Division
of Engineering.
Wollongong
University
College, Wollongong,
292
New South Wales, 2500 Australia
are: . . (1) . . . (2)
W.
14. CHARLTON;
A.
W.
293
ROBERTS
1 cos e=t/
[l +(y’)“]
?’ sin (3= &Y’)2] and hence
-t(Y’)“l P-Y’ .
-me41
N=
Substituting
(3) into Eqn (1) . . . (4)
With respect to the describing Then,
co-ordinates,
the chute profile can be written
4v ’ ;iT”
as Y-f(x).
z-_$.t
and d2y --#=Y where
Substituting
=Y’~+sy”~2
(5)
d2.r Y”Zdxz. Eqn (5) into Eqn (4) and re-arranging ~=
-(P-Y’) h-Y”X2) .
. ..
1MY')" When Eqn (6) is substituted
(6)
into Eqn (3) N_m(g-Y”i2) [l +(Y’)“l*
. . (7)
A convenient method of establishing a constraint equation for the motion of the particle is to write an energy balance equation. Hence, assuming the particle starts from the co-ordinate system origin with initial velocity I’,,, +m(v2--r,2)=mgY-frictional The frictional
losses.
loss in going from the origin to a general Loss =
x, Y uNds s o, o
..I (8)
point (x, y) is
where ds is an increment
--s
of arc length
X
PNv’P+W21dx.
. ..
0
Substituting
(9)
Eqn (7) into Eqn (9). X
Loss =
s
dx
pm(g--y”22)
,
0
i.e.
X
Loss =pmgx-pm
y”i”
I
0
dx
.
(10)
294
PROFILE
The constraint
SHAPt.
OF
GRAIN
(‘HUTI’?
Eqn (8) can now be written
3. Comments on chute curvature Examination of Eqn (11) permits certain deductions to be made concerning the required chute profile to achieve maximum exit velocity. For the discussion to be meaningful it is necessary to consider the possible chute profiles which will permit discharge of the material at a given point located by the co-ordinates x and y as shown in Fig. 2. The initial velocity 13,and friction coefficient u are considered the same for the various chute profiles examined.
:F;
;KX
(0)
lb)
Fig. 2
The term within the square brackets and depends only on the terminal point entirely on chute curvature and it is this of the integral term is dependent on y”; y” is zero. These considerations lead to
of Eqn (11) is independent of of the chute. The integral term term, therefore, which requires the term is zero in the case of the first deduction.
chute shape or curvature in Eqn (11) is dependent consideration. The sign a straight chute in which
(1) A continuously ‘concave’ chute [Fig. 2(a)] will yield an exit velocity lower than the corresponding straight chute between the same end points. For exit velocities higher than that obtained with a straight chute, the integral term of Eqn (11) must be positive. Thus for practical chute profiles, a second deduction can be stated. (2) For maximum exit velocity, a practical chute starting with a straight or ‘concave ’ section must be followed by a ‘convex ’ section [Fig. 2(b)]. Thus (2) asserts that a straight chute section, followed by a ‘convex ’ section will give a higher exit velocity than a continuous straight chute through the same end points. The above pass through greater than to achieve a
deductions are subject to a proviso that the normal force on the particle does not zero. Further, for a straight chute, and as can readily be shown, the slope must be u, i.e. Fig. 1, tan 8 must be greater than u. This latter requirement is necessary stable fast flow condition without the danger of flow stoppages. REFERENCES
Roberts, A. W. The dynamics ojgranular material jfow through curved chutes. Mech. and Chem. Eng. Trans. Instn of Engrs, Auk, Vol. MC3, No. 2, Nov. 1967 2 Roberts, A. W. An investigation of the gravity flow of non-cohesive granular materials through discharge chutes. Trans. A.S.M.E., J. Engng Jnd. Vol. 91, Series B, No. 2, May 1969
’
Chute
Profile
for Maximum
Exit
of Granular
Velocity
in Gravity
Flow
Material
W. H. CHARLTON*; A. W. ROBERTS* An analysis of the required profile shape of gravity mum exit velocity is presented. The dynamic equations are developed and utilizing energy considerations, an Maximizing exit velocity involves minimizing losses; required chute curvature are drawn. 1.
flow discharge chutes to achieve maxifor a grain element in the flowing stream expression for exit velocity is obtained. on this basis conclusions concerning
Introduction
It sometimes may occur, when granular materials flow in chutes under gravity, that a maximum possible exit velocity is desired. This case has been known to arise in connection with the loading of rail cars with wheat. A rigorous analysis is a complex problem but Roberts ‘3 2 has shown that under fast flow conditions, with grain contact on the chute bottom and side walls, the average particle behaves approximately as though it moves subject to a constant Coulomb frictional force. Using such a model, the conditions for maximum exit velocity can be examined. 2.
Dynamic analysis
Fig. 1 shows the co-ordinates with respect to which the particle gravitational force acts in the direction of the positive y axis.
dynamics
are formulated.
The
Fig. I
Assuming
the coefficient
of friction,
p, is constant,
mj; +N(cos mi+N(p where N is the normal
8+p
the equations
of motion
sin 0)=mg
cos O-sin
O)=O
force and d2y i;S-@
Using the abbreviation
*Division
of Engineering.
Wollongong
University
College, Wollongong,
292
New South Wales, 2500 Australia
are: . . (1) . . . (2)
W.
14. CHARLTON;
A.
W.
293
ROBERTS
1 cos e=t/
[l +(y’)“]
?’ sin (3= &Y’)2] and hence
-t(Y’)“l P-Y’ .
-me41
N=
Substituting
(3) into Eqn (1) . . . (4)
With respect to the describing Then,
co-ordinates,
the chute profile can be written
4v ’ ;iT”
as Y-f(x).
z-_$.t
and d2y --#=Y where
Substituting
=Y’~+sy”~2
(5)
d2.r Y”Zdxz. Eqn (5) into Eqn (4) and re-arranging ~=
-(P-Y’) h-Y”X2) .
. ..
1MY')" When Eqn (6) is substituted
(6)
into Eqn (3) N_m(g-Y”i2) [l +(Y’)“l*
. . (7)
A convenient method of establishing a constraint equation for the motion of the particle is to write an energy balance equation. Hence, assuming the particle starts from the co-ordinate system origin with initial velocity I’,,, +m(v2--r,2)=mgY-frictional The frictional
losses.
loss in going from the origin to a general Loss =
x, Y uNds s o, o
..I (8)
point (x, y) is
where ds is an increment
--s
of arc length
X
PNv’P+W21dx.
. ..
0
Substituting
(9)
Eqn (7) into Eqn (9). X
Loss =
s
dx
pm(g--y”22)
,
0
i.e.
X
Loss =pmgx-pm
y”i”
I
0
dx
.
(10)
294
PROFILE
The constraint
SHAPt.
OF
GRAIN
(‘HUTI’?
Eqn (8) can now be written
3. Comments on chute curvature Examination of Eqn (11) permits certain deductions to be made concerning the required chute profile to achieve maximum exit velocity. For the discussion to be meaningful it is necessary to consider the possible chute profiles which will permit discharge of the material at a given point located by the co-ordinates x and y as shown in Fig. 2. The initial velocity 13,and friction coefficient u are considered the same for the various chute profiles examined.
:F;
;KX
(0)
lb)
Fig. 2
The term within the square brackets and depends only on the terminal point entirely on chute curvature and it is this of the integral term is dependent on y”; y” is zero. These considerations lead to
of Eqn (11) is independent of of the chute. The integral term term, therefore, which requires the term is zero in the case of the first deduction.
chute shape or curvature in Eqn (11) is dependent consideration. The sign a straight chute in which
(1) A continuously ‘concave’ chute [Fig. 2(a)] will yield an exit velocity lower than the corresponding straight chute between the same end points. For exit velocities higher than that obtained with a straight chute, the integral term of Eqn (11) must be positive. Thus for practical chute profiles, a second deduction can be stated. (2) For maximum exit velocity, a practical chute starting with a straight or ‘concave ’ section must be followed by a ‘convex ’ section [Fig. 2(b)]. Thus (2) asserts that a straight chute section, followed by a ‘convex ’ section will give a higher exit velocity than a continuous straight chute through the same end points. The above pass through greater than to achieve a
deductions are subject to a proviso that the normal force on the particle does not zero. Further, for a straight chute, and as can readily be shown, the slope must be u, i.e. Fig. 1, tan 8 must be greater than u. This latter requirement is necessary stable fast flow condition without the danger of flow stoppages. REFERENCES
Roberts, A. W. The dynamics ojgranular material jfow through curved chutes. Mech. and Chem. Eng. Trans. Instn of Engrs, Auk, Vol. MC3, No. 2, Nov. 1967 2 Roberts, A. W. An investigation of the gravity flow of non-cohesive granular materials through discharge chutes. Trans. A.S.M.E., J. Engng Jnd. Vol. 91, Series B, No. 2, May 1969
’