Environmental Engineering Reference
In-Depth Information
ʲ
ʲ
=
valid for a wide range of values of
, including the cases
0 (vertical orifices)
ʲ
=
ʲ
∗
(the critical negative inclination when the flow is arrested). More-
over, it must be valid for the vertical case and it will be a sine-like function to take
into account the experimental behavior given in the previous section. Therefore, we
propose that
and
sin
(ʲ
+
ʱ
−
ʸ
r
)
sin
m
ʲ
=
c
m
0
(ʲ
+
ʱ
−
ʸ
r
)
˙
,
(7)
(ʱ
−
ʸ
r
)
which yields
m
ʲ
=
,
ʲ
=−
(ʱ
−
ʸ
r
),
0
if
(8)
when the flow is arrested, and
m
ʲ
=
0
=
cm
0
(ʱ
−
ʸ
r
) ,
if
ʲ
=
0
,
(9)
which is a general correlation valid for the vertical cases (Medina et al.
2013
,
2014
).
In order to show that Eq. (
7
) gives a correct correlation for circular orifices, we
display in Fig.
14
the experimental flow rates,
m
ʲ
expt
, versus the theoretically pre-
dicted,
m
ʲ
T
, given by Eq. (
7
). From Fig.
14
we see that straight lines fit very well the
experimental data obtained when the bin is gradually tilted.
In our experiments with sand we found that the value of
c
was
c
=
.
±
.
0
070
0
002
=
.
=
.
ʱ
=
.
=
if
D
0
9 cm and
w
0
9 cm, i.e., with a wall angle
0
57 rad; if
D
1cm
and
w
=
0
.
9 cm, then
ʱ
=
0
.
83 rad and we have that
c
=
0
.
083
±
0
.
002. For sugar
c
=
0
.
066
±
0
.
002 if
D
=
0
.
9 cm and
w
=
0
.
9cm;if
D
=
1 cm and
w
=
0
.
9cmwe
have that
c
=
0
.
083
±
0
.
002 and finally,
c
=
0
.
127
±
0
.
002 if
D
=
2cm,
w
=
0
.
9cm
and
15.
Actually, we found that
m
ʲ
ʱ
=
1
.
as given by Eq. (
7
) reaches a maximum value and then
decreases. So, this theoretical formula is not valid in a region close to
ʲ
ₒ
ˀ/
2. To
estimate such a value, we compute
dm
ʲ
/
d
ʲ
.Ityieldsthat
ʲ
+
ʱ
−
ʸ
r
=−
tan [
ʲ
+
ʱ
−
ʸ
r
]
.
(10)
Consequently, to get the maximum value
ʲ
m
we need to solve the transcendental
equation
x
=−
tanx
, where
x
=
ʲ
m
+
ʱ
−
ʸ
r
. Whence the general solution of
Eq. (
10
)is
ʲ
m
=
2
.
028
−
(ʱ
−
ʸ
r
),
(11)
where the value 2.028 is correct up to a numerical precision of 10
−
13
. Thus, for
each couple of values (
ʲ
m
, where the maximum occurs. This
criterion was used to build the plots in Fig.
15
. Finally, we note that this formula is
worst if
ʱ, ʲ
) we get a value of
is large. Incidentally, we have plotted in Fig.
16
m
ʲ
T
, given by Eq. (
7
), as
a function of
ʱ
ʲ
in order to show that this later plot is similar to the plots in Fig.
14
.
Search WWH ::
Custom Search