Pump. Heterogeneous Slurries Type A Warman
Pump. Heterogeneous Slurries Type A Warman
Pump. Heterogeneous Slurries Type A Warman
+
|
|
.
|
\
|
(
b b
b c
c
b
b c
c
c
b c
c
c
p Q
Q Q
C Q
Q
Q Q
C Q
Q
Q Q
C Q
C Q
t Q s Q r P
P
+ + =
2
s Q r
Q
P
P
+ =
c
c
2
0 =
c
c
Q
P
P
s Q r
Q
P
C
C
P
+ =
c
c
2
=
c
c
C
P
Q
P
=
c
c
C
P
Q
P
C
C
w
Q C
Q
P
=
c
c
2
=
c
c
C
w
Q
P
=
c
c
C
w
Q
P
=
c
c
c
c
C
P
C
w
Q
P
Q
P
t Q s Q r P
P
+ + =
2
0 OK
=
c
c
c
c
C
P
C
w
Q
P
Q
P
Rev. cjc 30.01.2014
1
A simple way to solve the problem is to
provide a "Cc value" taken from the
Point "B" of the resistance curve experience, which is known to originate
results, very similar to those obtained in
Experience data indicates that the pulp the real life.
resistance curve has its minimum at Warman uses the value
approximately a value C
c,W
= C
c,Warman
C
B
= 0.7 C
c,W
= 1.43
where Using this value, the two curves
Q
B
= C
B
* Q
L intersect at point "C", with very similar
C
B
= 0.7
slopes (no tangency).
Q
L
= 0.009
m
3
/s The flow corresponding to this point is
Q
B
= 0.006
m
3
/s
Q
c,W
= C
c,W
* Q
L
C
c,W
= 1.42857203
The pressure loss corresponding to this Q
L
= 0.009
m
3
/s
point is Q
c,W
= 0.0128
m
3
/s
AP
B
= AP
A
AP
A
= 21.21 mca The pressure loss corresponding to this
AP
B
= 21.21 mca point is
Ap
c,W
= C * Q
c,W
^2
Thus, point "B" is defined by C = 262505
Q
B
= 0.006
m
3
/s
Q
c,W
= 0.0128
m
3
/s
AP
B
= 21.21 mca Ap
c,W
= 43.3 mca
When enforced the requirement
Point "C" of the resistance and water of tangency (indicated by Warman),
curves point "C" scrolls as shown
Experince data indicates that these in the following table
two curves merge from a point "C" to Q H
be determined. m
3
/s
m ca
The flow at point "C" will be "A" 0.009 21.21
Q
C
= C
C
* Q
L
"B" 0.006 21.21
with "Cc" a value to be determined. "C" 0.0128 43.29
2
Point "C"
Point "C" is defined as the point where
both curves have a common point and
where they ar tangent as well.
6.- Theory of Warman's empirical method
Pumping heterogeneous slurries
Flow at point "C"
Flow at point "C" is defined as
Q
C
= C
C
* Q
L
where C
C
is a value determined by the
conditions impose to the system.
Pressure at point "C"
Point "C" is determined as follow
3
c) Tangente de la curva de e) Curvas de resistencia se cortan
resistencia de la pulpa en punto "c" en "C"
Pulp resistance curve
The pressure derivativeis
(Ec. c1)
Evaluating at point "C" Water resistance curve
(Ec. b2)
(Ec. c2)
d) Condition of common tangent at Equating equations c1 y c2
point "c"
2
Q C p =
t Q s Q r p
c c c
+ + =
2
t Q s Q r p + + =
2
2
c c
Q C p =
t Q s Q r Q C
c c c
+ + =
2 2
t Q s Q r P
P
+ + =
2
s Q r
Q
P
P
+ =
c
c
2
s Q r
Q
P
C
C
P
+ =
c
c
2
C
P
C
w
Q
P
Q
P
c
c
=
c
c
(Ec. c)
f) Pulp resistance curve
goes through point "b"
Curva de resistencia de la pulpa
(Ec. b)
(Ec. d)
4
.(e)
(g)
(f)
Equating equations (g) y (f)
(h)
(f)
(g) (h)
c c
Q r Q C s = 2 2
) ( 2 r C Q s
c
=
t Q s Q r p
b b b
+ + =
2
t Q s Q r p + + =
2
t Q s Q r Q C
c c c
+ + =
2 2
c c c
Q s Q r Q C t =
2 2
( ) t Q Q r Q r p
b b b b
+ + = 2
2
( )
c b c c
Q Q r Q r Q C t = 2
2 2
t Q r Q r p
b b b
+ =
2 2
2
t Q r p
b b
+ =
2
( )
c b c
Q Q r r C Q t + = 2
2
( )
2 2
2
b b c b c
Q r p Q Q r r C Q + = +
( ) 0 2
2 2
= +
b b b c c
p Q r Q r Q r C Q
C
P
C
w
Q
P
Q
P
c
c
=
c
c
s Q r Q C
C C
+ = 2 2
.(e)
(i)
5
Results for the pulp resistance
curve
r = 514,716
s = -6,477
t = 42
Flows
Q
A
= 0.009 m/s
Q
B
= 0.006 m/s
Q
C
= 0.0128 m/s
b c c
Q r r Q C Q = 2 2 2 ( )
b c c
Q Q r C Q =
( )
b c
c
Q Q
C Q
r
=
( ) ( ) ( )
0 2
2 2
=
(
+
|
|
.
|
\
|
(
b b
b c
c
b
b c
c
c
b c
c
c
p Q
Q Q
C Q
Q
Q Q
C Q
Q
Q Q
C Q
C Q
Water and slurry curves. Durand limit velocity
According Weir empirical method [10]
For validity range, see Note 1
Mc Ewans and Cave correction factor
F
L
= Pipe_Fl_McElvain_d50_Cv(d50,Cv)
Input data d
50
= 145 m
Following data is required to draw the
curves
C
v
= 20.1 %
dn 4 in
sch = STD - Limit deposition velocity according JRI [3]
d
50
= 145 mm d
50
= 145 m
C
v
= 20.1 % C
v
= 20.1 %
S
S
= 2.65
-
dn 4 in
s
lining = 9 mm Ss = 2.65 -
Q =
0.0104
m/s
s
lining
= 9 mm
AP = 28.56 mwc
v
L
= Slurry_Limit_Deposition_Velocity_Lining_JRI_Imp_d50_Cv_dn_Ss_Slining
v
max
= 4 m/s v
L
= 1.61 m/s
Inside diameter with lining Slurry flow velocity
di = d - 2 * s
lining
v = Q
P
/ A
m
3
/s
where s
lining
is the sum of the lining pipe Q
P
= 0.0104
m
2
and the deposition film
A = 0.0056 m/s
Inside pipe diameter
v = 1.87
d = Pipe_Imp_CS_Dint_dn_sch
d = 102.3 mm
s
lining
= 9 mm
di = 84.26 mm Limiting flow rate
di = 0.08426 m Q
L
= v
L
* A
v
L
= 1.612 m/s
Area of pipe section A = 0.0056
m
2
A = (t/4)*di^2 Q
L
= 0.0090
m
3
/s
di = 0.08426 m
A = 0.005576129 m Note 1. The method is valid for the Weir defined type-A fluids. See sheet Ref. 10 & 11
Water system curve constant From pages 3 to 5, sheet 6, [3]
The system curve for water is the pulp curve is
for H
estat.tot
= 0
H = r*Q^2 + s*Q + t
AP = C * Q^2 with
Velocity enough to avoid settlement
7.- Weir method for slurry pump selection for A-type fluids
OK. d50 within range
OK. Cv within range
OK. v < vmax
( )
(
=
b c
c
Q Q
C Q
r
thus, the water system constant is
C = AP / Q^2
From input data, the water pressure drop is r = C * Qc / (Qc - Qb)
Ap = 28.56 mwc C = 262,505
and the flow rate (Q
P
= Q
w
) is Qc = 0.0128410
Q
P
= 0.0104
m
3
/s
Qb = 0.006
then r = 514,716
C = 262,505 -
From page 4 of Sheet 6, the flow rate at point "c" is s = - 2 * r * Qb
calculated from equatio (i)
[3]
r = 514716
(Ec. i) Qb = 0.006
s = -6477
This equation is solved with the function
Function Qc_C_Qb_Pb(C,Qb,Pb) t = Qc^2*(C - r) - s*Qc
Using an auxiliar variable Qc = 0.0128410
z = (Qc*C)/(Qc-Qb) C = 262505
The equation to be solved is r = 514716
Zero = Qc^2*(C-z) +2*Qc*Qb*(z) - Qb^2*(z)-Pb s = -6477
The equation is solved using the function t = 41.6
Qc = Qc_C_Qb_Pb(C,Qb,Pb)
C = 262,505 Thus, the parabola constants for the
Q
b
= 0.0063
m
3
/h
slurry curve are:
P
b
= 21.2 mca r = 514,716
Qc = 0.0128
m
3
/h
s= -6,477
t = 41.6
( ) ( ) ( )
0 2
2 2
=
(
+
|
|
.
|
\
|
(
b b
b c
c
b
b c
c
c
b c
c
c
p Q
Q Q
C Q
Q
Q Q
C Q
Q
Q Q
C Q
C Q
( )
(
=
b c
c
Q Q
C Q
r
b
Q r s = 2
c c
Q s r C Q t = ) (
2
40
50
60
70
F
r
i
c
t
i
o
n
h
e
a
d
l
o
s
s
H
f
[
m
]
Figure 6.2.- Heterogeneous slurry
Pulp resistance curve [mpc]
H = r*Q^2 + s*Q + t
C = 262505 C = 262505
Limiting flowrate
Q
a
= Q
L
Q Ap Q Ap Ap
m
3
/s
m fc
m
3
/s
m fc m fc
0 0.00 0.0090 0.00 0.00
0.001 0.26 0.0090 10.00 0.00
0.002 1.05 0.0090 20.00 20.00
0.003 2.36 0.0090 24
0.004 4.20 0.0090 28
0.005 6.56 0.0090
0.006 9.45 0.0090
0.007 12.86 0.0090
0.008 16.80 0.0090
0.009 21.26
0.01 26.25
0.012 37.80
0.014 51.45
0.016 67.20
C = 262505
r = 514716
s= -6477
t = 42
Limiting flow rate
Q
L
= v
L
* A
Water curve QL * 1.1
Ap = C * Q
0
10
20
30
40
0.00 0.01
F
r
i
c
t
i
o
n
h
e
a
d
l
o
s
s
H
f
[
m
]
Flowrate Q [m
B
A
Minimum pressure
v
L
= 1.61 m/s
A = 0.01
m
2
Q
L
= 0.00899
m
3
/s
1.1 * Q
L
= 0.0099 m/s
Subroutine for the pressure loss, for the actual flow in metres of pulp column
For Q < 1.1 * Q
L
,
deposition will occur Q
L
= 0.0090
m
3
/s
For 1.1 * Q
L
< Q <= Q
c
Pulp curve is to be used P = r * Q^2 + s * Q + t
For Q > Q
c
Water curve is to be used P = C * Q^2
Flow "1.1 * Q
L
"
1.1 * Q
L
= 0.0099
m
3
/s
Q >= 1.1 *QL
Flow at point "c"
Q
c
= 0.0128
m
3
/s
Actual flow
Q = 0.0104
m
3
/s
Slurry curve
P =
r * Q^2 + s * Q + t, C *Q^2
r = 514,716
s = -6,477
t = 42
C = 262,505
P = SI( Q<=Q
c
, r * Q^2 + s * Q + t, C *Q^2)
P = 30.02 mpc
Flow enough to avoid deposition
1.1*QL < Q < Qc. Flow to be calculated
using the slurry curve
Ratio Q
P
/Q
L
Q
B
= C
B
* Q
L
whith C
B
= 0.7
Q
P
= 0.0104
m
3
/s Thus, with
Q
L
= 0.0090 m
3
/s
C
B
= 0.7
Q /Q
L
= 1.16 Q
L
=
0.0090
the value of the limit flow is
Q
L
= 0.0090
and Q
B
= 0.006
Points "A" y "B"
Pressure in point "B"
Flowe rate and pressure in point "A" According "Weir" asumption, it should be
Point "A" is defined as the point in the
P
B
= P
A
Water curtve, where the flow rate
coincides with the limiting flow rate. If the total static height is not zero, the
v
L
= Slurry_Limit_Deposition_Velocity_Lining_JRI_Imp_d50_Cv_dn_Ss_Slining
Thus parabola equation shall be
Q
A
= Q
L
P = H
stat_tot
+ C * Q^2
.
Q
L
=
0.0090 m/s
Q
A
=
0.0090 m/s
Total static height
Since the water curve is a parabola EL
suc
= 0.0
and in this case the total static E
lpump
= 0.0
height has been defined as zero, the EL
disch
= 0.0
parabola equation is
H
stat.tot
= EL
desc
- El
suc
. P = C * Q^2 H
stat.tot
= 0.000
For the point "A"
P
A
= C * Q
A
^2
Weight concentration
Cw = Ss*Cv / ( 1 + Ss*Cv - Cv )
Flow rate in point "B" S
s
= 2.65
Experience data indicates that the pulp C
v
= 0.20
resistance curve has its lowest point at Cw = 0.40
a flow rate defined by a value "CB",
Cw = 40
Note 1. The method is valid for the Weir defined type-A fluids. See sheet Ref. 10 & 11
From pages 3 to 5, sheet 6, [3] Pressure at point "A" The ratio Q
c
/ Q
L
is
Q
A
= Q
L
Q
c
/ Q
L
=
Q
A
= 0.0090
m
3
/s
Q
c
=
P
A
= C * Q
a
^2 Q
L
=
C = 262,505 Q
c
/ Q
L
=
7.- Weir method for slurry pump selection for A-type fluids
No settling, but increase diameter if possible
Q
A
= 0.0090
m
3
/s
P
A
= 21.2 m Warman indicates a value
Q
c
/ Q
L
=
C * Qc / (Qc - Qb) Pressure at point "B" as an approximate value
P
B
= P
A
=
The reason being that up a value of 1.3,
P
A
= 21.2 m ca
both water and slurry curves are
P
B
= 21.2 m ca practically the same (see Fig. 6.2)
The constant "C
c
" For this application, the ratio Q / Q
L
is
Relating the flow at point "C" to the Q / QL =
limiting flow "Q
L
" with the pameter "C
C
" Q =
Q
c
= C
c
* Q
L
QL =
C
c
= Q
c
/ Q
L
Q / QL =
Q
c
= 0.0128
m
3
/s
Q
L
= 0.009
m
3
/s
According Weir, at flows over "Qc" the
C
c
= 1.43 slurry curve and the water curve should
be similar. Thus, for Q > Q
C
, the pressure
Qc^2*(C - r) - s*Qc Pressure at point "C" can be calcualted using the water curve,
P
C
=
r*Q^2 + s*Q + t expressed in metres of slurry column and
Q
c
= 0.0128
m
3
/s no correction is required.
r = 514716
s= -6477 On the other hand, the flow should not
t = 42 be lower than the limiting value "1.1*QL"
P
C
=
43.29 m to ensure no deposition will occcur.
Thus, the parabola constants for the
Thus, a correction will be only required
in the range
1.1 * Q
L
< Q < 1.4 * Q
L
c c
Q s r C Q t = ) (
2
Heterogeneous slurry
Qa= QL
DP=C*Q
Qb=Cb*QL
Qc=Cc*QL
C
Water
resistance curve
2
Q C P
w
=
Actual flow rate
Graphic is for static height equal zero
C = 262505
Flow rate at point"B" Actual flow rate
0.0104
QL * 1.1 Q Ap Q Ap
m
3
/s
m fc
m
3
/s
m fc
0.0099 0.0063 0.00 0.0104 0.00
0.0099 0.0063 10.00 0.0104 25.00
0.0099 0.0063 20.00 0.0104 50.00
0.0099 0.0063 0.0104 75.00
0.0099 0.0104 100.00
0.0099 0.0104 125.00
0.0099 0.0104 150.00
0.0099 0.0104 175.00
0.0099 0.0104 200.00
0.0104 225.00
0.0104 250.00
0.0104 275.00
0.0104 800.00
Minimum pressure drop
Ap = Apa
Q P
a
Q Pactual
m
3
/s
mpc
m
3
/s
mpc
0.000 21.2 0.000 30.02
0.010 21.2 0.010 30.02
0.015 21.2 0.015 30.02
0.020 21.2 0.020 30.02
QL * 1.1
Q
b
=C
b
* Q
L
Actual pressure drop
0.01 0.02
Flowrate Q [m
3
/s]
Qb=Cb*QL
Qc=Cc*QL
Slurry resistance
Q
QL*1.1
Pa
Pactual
C
Limiting flowrate Q
L
Flowrate Q
b
= 0.7 * Q
L
Q = 1.1 * Q
L
Q
c
= 1.43 * Q
L
Actual flow rate
Actual pressure
0.025 21.2 0.025 30.02
0.030 21.2 0.030 30.02
0.035 21.2 0.035 30.02
0.040 21.2 0.040 30.02
0.042 21.2 0.042 30.02
0.045 21.2 0.045 30.02
0.050 21.2 0.050 30.02
Slurry_Weir_A_DeltaP_d50_Cw_dn_Ss_Pw_Slining(d50, Cw, Dn, Ss, Q, P, Slining)
Use of function DeltaP
The shown subroutine calculation can be repalced by the function DeltaP
Input data for the function DeltaP The slurry pressure is then,
DeltaP = Slurry_Weir_A_DeltaP_d50_Cw_dn_Ss_Pw_Slining(D368, D369, D370, D371, D372, D373,D374)
. P = P
w
+ DeltaP
d
50
= 145 m with
Cw = 40 % P
w
: the pressure calculated as the fluid
d
n
= 4 in
was water
S
s
= 2.65 - P
w
= 28.56
Q = 0.01043
m
3
/s and
P
w
= 28.56 mwc
DeltaP: the difference between the
s
lining
= 9 mm pressures for slurry and water,
measured in metres of pulp column,
Function result (Note 1) corresponding to the actual flow
DeltaP = 1.47 msc DeltaP = 1.47
Note 1. Correction "DeltaP" required Thus, the slurry pressure is
to be applied only if P = 30.02
1.1 * Q
L
< Q <= 1.4 * Q
L
Checking of ranges:
This condition is equivalent to
1.1 < Q/Q
L
<= 1.4
These ranges correspond to the by Weir
In this application defined type-A fluid
Q = 0.0104 m/s
Q
L
= 0.0090 m/s Note.
Q/Q
L
=
1.16 When using the standard method (sheet 2)
the pressure head is
P = 29.23
Between both methods there is a difference of
AP
difference
= 0.791
2.6
OK. Cv within range
Correction is to be applied
OK. d50 within range
Rev. cjc 30.01.2014
1
and
m/s
m
3
/s
m
3
/s
According "Weir" asumption, it should be
If the total static height is not zero, the
H
stat_tot
+ C * Q^2
m
m
m
m
Ss*Cv / ( 1 + Ss*Cv - Cv )
-
-
-
%
2
The ratio Q
c
/ Q
L
is
0.0128
0.0090
1.43
7.- Weir method for slurry pump selection for A-type fluids
0
10
20
30
40
50
60
70
0.00
F
r
i
c
t
i
o
n
h
e
a
d
l
o
s
s
H
f
[
m
]
Figure 6.2.
B
Warman indicates a value
1.3
as an approximate value
The reason being that up a value of 1.3,
both water and slurry curves are
practically the same (see Fig. 6.2)
For this application, the ratio Q / Q
L
is
0.0104
m
3
/s
0.0090
m
3
/s
1.16
m
3
/s
According Weir, at flows over "Qc" the
slurry curve and the water curve should
be similar. Thus, for Q > Q
C
, the pressure
can be calcualted using the water curve,
expressed in metres of slurry column and
no correction is required.
On the other hand, the flow should not
be lower than the limiting value "1.1*QL"
to ensure no deposition will occcur.
Thus, a correction will be only required
3
1.1 * Q
L
< Q < 1.4 * Q
L
Qa= QL
DP=C*Q
Qb=Cb*QL
Qc=Cc*QL
Graphic is for static height equal zero
C = 262505 4
Q
c
=C
c
* Q
L
262505
Q Ap
m
3
/s
m fc
0.0128 0.00
0.0128 10.00
0.0128 20.00
0.0128 30.00
0.0128 35.00
0.0128 40.00
0.0128 45.00
Q H
m
3
/s
m cf
0.002 30.69
0.004 23.91
0.006 21.25
0.008 22.71
Flow rate at point"c"
Slurry resistance curve
H = r*Q^2 + s*Q + t
Qb=Cb*QL
Qc=Cc*QL
Slurry resistance
QL*1.1
Pactual
0.010 28.29
0.012 37.98
0.014 51.79
0.016 69.72
cjc.Rev. 1213.11.2012
5
P
w
: the pressure calculated as the fluid
mwc
DeltaP: the difference between the
pressures for slurry and water,
measured in metres of pulp column,
corresponding to the actual flow
msc
msc
These ranges correspond to the by Weir
When using the standard method (sheet 2)
msc
Between both methods there is a difference of
msc
%
OK. Cv within range
OK. d50 within range
0.01 0.02
Flowrate Q [m
3
/s]
Figure 6.2.- Heterogeneous slurry
B
A
C
8.- Depositation limit velocity, according JRI
1. JRI Formula [1] [Eq. b] 2. JRI Formula
For coarse-grained solids with wide d50 >= 200 m For coarse-grained solids with wide
particle size range and small diameter particle size range and large diameter
tubes d, cualquiera tubes
[Ec.1]
v
L
= F
L
* (2* g * d * (S
S
-1))^0.5 v
L
= 1.25 *F
L
* (2* g * d * (S
S
-1))^0.25
Applicacin Applicacin
d
50
= 250
m d50 >= 200 m
d
50
= 150
C
v
= 20.1 % C
v
= 20.1
d
n
= 200 mm d, cualquiera d = 200
S
S
= 2.65 - S
S
= 2.65
Mc Elvain y Cave correction factor Mc Elvain y Cave correction factor
F
L
= Slurry_Fl_McElvain_d50_Cv(E19;E20) F
L
= Slurry_Fl_McElvain_d50_Cv(E19;E20)
d
50
= 250
m
10 <= d
50
<= 3000 d
50
= 150
C
v
= 20.10 % 5% <= C
v
<= 40% C
v
= 20.1
F
L
= 1.089 - F
L
= 0.988
Limit deposition velocity Limit deposition velocity
v
L
= F
L
* (2* g * d * (S
S
-1))^0.5 v
L
= 1.25 *F
L
* (2* g * d * (S
S
-1))^0.25
F
L
= 1.089 - F
L
= 1.005
g = 9.81 m/s
2
g = 9.81
d
n
= 200 d
n
= 200
Sch = STD Sch = STD
d
i
= 202.74 d
i
= 202.74
d
i
= 0.20274 m d
i
= 0.203
S
S
= 2.65 - S
S
= 2.65
v
L
=
2.790 m/s
v
L
=
2.011
v
L
= Slurry_Limit_Deposition_Velocity_JRI_Imp_d50_Cv_dn_Ss Also, there is a function that allows consideration of a
d
50
= 250
m
lining thickness.
C
v
= 20.1 %
d
n
= 8 in v
L
= Slurry_Limit_Deposition_Velocity_JRI_Imp_d50_Cv_dn_Ss_Slining
S
S
= 2.65 -
v
L
= 2.790 m/s
) 1 ( 2 =
S L L
S d g F v | |
25 . 0
) 1 ( 2 25 . 1 =
S L L
S d g F v
19
[1] [Eq. c] 3. JRI Formula [1] [Eq. a]
For coarse-grained solids with wide d50 < 200 m For fine-grained solids with narrow d50 < 200 m
particle size range and large diameter particle size range and small diameter
d >= 150 mm tubes d < 150 mm
[Ec.2]
1.25 *F
L
* (2* g * d * (S
S
-1))^0.25 v
L
= 1.1 *F
L
* (2* g * d * (S
S
-1)^0.6)^0.5
Applicacin
m d50 < 200 m
d
50
= 100
m d50 < 200 m
% C
v
= 20.1 %
mm d >= 150 mm d = 100 mm d < 150 mm
- S
S
= 2.65 -
Mc Elvain y Cave correction factor Mc Elvain y Cave correction factor
Slurry_Fl_McElvain_d50_Cv(E19;E20) F
L
= Slurry_Fl_McElvain_d50_Cv(E19;E20)
m
10 <= d
50
<= 3000 d
50
= 100
m
10 <= d
50
<= 3000
% 5% <= C
v
<= 40% C
v
= 20.1 % 5% <= C
v
<= 40%
- F
L
= 0.920 -
Limit deposition velocity
1.25 *F
L
* (2* g * d * (S
S
-1))^0.25 v
L
= 1.1 *F
L
* (2* g * d * (S
S
-1)^0.6)^0.5
- F
L
= 0.920 -
m/s
2
g = 9.81 m/s
2
d
n
= 100
Sch = STD
d
i
= 102.26
m d
i
= 0.102 m
- S
S
= 2.65 -
m/s
v
L
=
1.665 m/s
Also, there is a function that allows consideration of a
v
L
= Slurry_Limit_Deposition_Velocity_JRI_Imp_d50_Cv_dn_Ss_Slining
6 . 0
) 1 ( 2 1 . 1 =
S L L
S d g F v | |
25 . 0
) 1 ( 2 25 . 1 =
S L L
S d g F v
10.- Water un pulp resistance curves
Warman's empirical method
Application range
50 m < d < 300 m
% < C
w
<= 40 %
The graphic represents the case of a
solids mass flow rate
m
s
= 40 ton/h
with four different weight concentrations
C
w 15 %
C
w 20 %
C
w 30 %
C
w 40 %
To each concentration corresponds
a specific pulp mass flow rate and
a pulp volumetric flow rate
The table shows a resume of the
four cases
C
w % 15 20 30 40
m
p ton/h 267 200 133 100
Q
P m
3
/s 0.0672 0.0486 0.0301 0.0209
The curves for cases with water as fluid, are the four parabolas that start from the
origin of coordinates. This is because it is considered the case in which the height
difference between the suction and the discharge is zero.
If this difference were not zero, the origin of these curves would move upwards or
downwards.
The four curves for the case of pulp, are the four parabolas with minimum flow rates
corresponding to the limit deposition flow rate QL for each case.
[1] Slurry pumping manual
Warman International Ltd.
1st edition, 2002
[2] Warman Slurry Pumping Handbook - AU
Warman International Ltd.
Feb. 2000
[3] JRI
'Curso de transporte hidrulico de slidos
'Tecnex. Ingenieros ltda. (JRI)
'Octubre 1993
'pginas 20 y 21
[4] Slurry systems handbook
[5] Warman slurry pumping handbook
http://www.pumpfundamentals.com/slurry/Warman_slurry_pumping.pdf
1.- Ball valves
http://info.jamesbury.com/public/publicdocs/Docs/T120-1.pdf
2.- Butterfly valves Bray 2021
http://www.bray.com/docs/brochures/20.pdf
2.- Butterfly valves Bray 3031
http://www.bray.com/docs/brochures/30.pdf
3.- Knife valves
http://www.sureflowequipment.com/pdf/Knife-Gate-Valves-Catalog-2008-SureFlow.pdf
4.- Globe valves
http://www.fnwvalve.com/FNWValve/assets/images/PDFs/FNW/FlgGGC_tech-Cv.pdf
5.- Pinch valves
http://www.jecwoodland.com/pinch_valve_CV.html
6.- Diaphragm valves
http://www.thevalveshop.com/pdf/saunders9.pdf
6.- Diaphragm valves
http://www.thevalveshop.com/pdf/saunders9.pdf
6.- Diaphragm valves
http://www.thevalveshop.com/pdf/saunders17.pdf
6.- Diaphragm valves
http://www.thevalveshop.com/pdf/saunders17.pdf
7.- Round plug valve
http://www.gaindustries.com/MProducts/Bulletins/EccentricPlugValves/GAI_PV517.pdf
7.- Rectangular plug valve
http://www.gaindustries.com/MProducts/Bulletins/EccentricPlugValves/GAI_PV517.pdf
8.- Check valves
http://www.acuster.com/files/documentos/FC-69-PVV-10_es.pdf
9.- Angle valves
http://www.controlvalves.com/series/125/125_sizing.html
10.- Cone valve
http://www.detroitcontracting.com/documents/Cone-Valve-1.pdf
11.- Strainers
http://www.coltonind.com/file_library/products/33_STR990-1_Pressure%20Drop%20Data%20Y.pdf
12.- Expansion/reduction, fittings
Crane A-46 [1]
[2], 57-59
Cw
100
90
80
70
60
50
40
30
20
10
0 0 10 20 30 40 50 60
Particles essentially
all finer than 50 m
[1], page S6-3
Homogeneous slurries
At sufficiently low concentrations Hf will be close to that for clear water and may be estimated by the empirical method as applied to Category "A" Heterogeneous slurries
At sufficiently high concentrations, The Yield Stress characteristic largely influernces the value of Hf.
A Heterogeneous slurries Category "A"
Particles essentially all coarser than 50 m and finer then 300 m, and with Cw from Zero to 40%
Head losses can be calculated by means of the empirical method propossed by Warman in [2]
The most economical slurry velocity is a velocity a little in excess of the limit velocity " v
L
"
This is an empirical method, that provide estimates considered to be rasonable accurate for many practical slurry pimping applications.
B Heterogeneous slurries Category "B"
Particles essentially all coarser than 50 m and finer then 300 m, and with Cw greater than 40%
Generally, friction losses for this category are much higher than for Category "A" due largely to the increased friction effect of the more closed-packed solids content upon
the pipe wall. This effect generally increases with increasing Cw and is so greatly influenced by a number of variables, for example, Cw, S
s
, S
Liq
, d
50
and actual sieve analysis
of solids present that is not possible to provide a simple empirical method of estimating slurry Hf.
See [2], 59
C Heterogeneous slurries Category "C"
Particles essentially all coarser than 300 m, and with Cw from Zero to 20%
Generally, friction head losses for Category "C" slurries are also much higher than for Category "A"
See [2], 59
D Heterogeneous slurries Category "D"
Particles essentially all coarser than 300 m, and Cw greater than 20%
Generally, friction head losses for Category "D" slurries are also much higher than for Category "A"
See [2], 59
Homogeneous slurries
Heterogeneous
slurries
Weir slurry clasification
A-type 0 % <= Cw <= 40 %
50 m < d
50
< 300 m
70 80 90 100 110 120 130 140
At sufficiently low concentrations Hf will be close to that for clear water and may be estimated by the empirical method as applied to Category "A" Heterogeneous slurries
At sufficiently high concentrations, The Yield Stress characteristic largely influernces the value of Hf.
Particles essentially all coarser than 50 m and finer then 300 m, and with Cw from Zero to 40%
Head losses can be calculated by means of the empirical method propossed by Warman in [2]
The most economical slurry velocity is a velocity a little in excess of the limit velocity " v
L
"
This is an empirical method, that provide estimates considered to be rasonable accurate for many practical slurry pimping applications.
Particles essentially all coarser than 50 m and finer then 300 m, and with Cw greater than 40%
Generally, friction losses for this category are much higher than for Category "A" due largely to the increased friction effect of the more closed-packed solids content upon
the pipe wall. This effect generally increases with increasing Cw and is so greatly influenced by a number of variables, for example, Cw, S
s
, S
Liq
, d
50
and actual sieve analysis
of solids present that is not possible to provide a simple empirical method of estimating slurry Hf.
Generally, friction head losses for Category "C" slurries are also much higher than for Category "A"
Generally, friction head losses for Category "D" slurries are also much higher than for Category "A"
Heterogeneous
slurries
B
0 % <= Cw <= 40 %
50 m < d
50
< 300 m
150 160 170 180 190 200 210 220
Heterogeneous
slurries
230 240 250 260 270 280 290 300
Heterogeneous
slurries
D
310 320 330 340 350 360 370 380
Heterogeneous
slurries
C
390 400 410 420 430 440 450 460
Heterogeneous
slurries
470 d [m]
Heterogeneous