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ARNOLD G. FREDRICKSON and
R.
BYRON BIRD
Department of Chemical Engineering, University of Wisconsin, Madison, Wis.
Non-Newtonian Flow in Annuli
Extrusion of molten plastics and
Row
of drilling muds in annular
space are typical problems to
which
th is
study can be applied
T o A K E deductions from flow data
concerning t he range of applicab ilit y of
various empirical non-Newtonian flow
models, it is necessary to have good ex-
perimental flow data and accurate solu-
tions to t he equations of motion in various
geometrical arrangements. Experimen-
tal data for a few fluids and analytical
solutions are available for axial flow in
tubes and for tangential flow in cylindri-
cal annuli
(3, 7, 77,
72). Thi s discussion
is concerned with the analytical solution
of the equation of motion for the steady-
state axial flow of an incompressible,
non-Newtonian fluid in a long cylindrical
annulus. This problem is of importance
in connection with heat transfer to and
from fluids flowing in annular spaces,
flow of molten plastics in extrusion ap-
paratus, and flow of drilling muds in
annular spaces.
Th e equations describing the flow of a
compressible, isothermal fluid are equa-
tions of continuity and motion (2,
4
:
dr/dt + (V.YV) = 0
(1
1
-VP
V.7) + Yg ( 2 )
Assumption of isothermal flow implies
not only th at there is no impressed tem-
peratur e field (6) but that the viscous dis-
sipation term 7:Vu) n the energy bal-
ance equation is negligible (7, 2).
In
the developments which follow, the
flow between two coaxial cylinders (Fig-
ure 1) is considered. Th e following as-
sumptions are made:
The fluid is incompressible
y
= con-
stant).
T he flow is in steady-state-Le., time-
independent.
Th e flow is laminar.
Th e cylinders are sufficiently long that
end effects may be neglected.
For the specific system under con-
sideration Equations 1 and
2
may be
written in cylindrical coordinates and
combined a nd simplified to
:
r[bulbt + ( U . V ) V l =
in whichpo and p L are the static pressures
a t z
=
0 and z = L, respectively, and gs
is the component of gravitational ac-
celeration g in the direction of flow. P
designates the sum of forces per unit
volume on the right side of Equation 1.
This first-order differential equation,
valid over the entire annular region for
an y kind of fluid, may be integrated to
give :
4)
in which
X
is the constant of integration.
The radial distance r = XR represents
that position at which r,, = 0. Equation
4
is
taken as the starting point for the
derivations for the Bingham plastic and
the power law models.
Solution f o r Bingham Plast ic Model
Van Olphen
(70)
has presented an
approximate solution; Mori and Oto-
take
(9)
have given the complete solution,
but their analytical and graphical re-
sults are in erro r; Laird's solution 5)
is
correct. Howe ver, Laird's expression
for the volume throughput is more
complicated than that given here, and
he has not presented his results in terms
of
a
dimensionless correlation for general
use.
For this model the local shear stress,
T, ,
s related to the local shear rate,
dv,/dr, according to the formula :
wherein + is used when momentum is
being transported in the +r direction
and - when transport is in the -rdi-
rection. Th e meaning of TO and pa is
given in Figure 2 , where the Bingham
model is compared with the Newtonian
model.
The introduction
of
the following di-
mensionless variables is useful :
T = 2r,,/PR
To = 2ro/PR
= dimensionless shear stress
= dimensionless limiting
=
dimensionless velocity
shear stress
9
=
( 2 ~ 0 / P R ~ ) p ~
= dimensionless radial distance
p = r / R
Th e equations describing the system are
X+ and 1- epresent the bounds on the
plug flow region. Clearly they a re those
values of
p
for which TI = TO
Figure 1 . Shear stress distribution for
axial annular flow corresponding to
Equation 4 and characteristic velocity
distributions for power law fluids
Equations 22 and 23) and Bingham
plastic fluid Equations 1 1 , 12, and 13)
Actually it is convenient to express all the
final results not in terms of X but rather
in terms of either A+ or
X ;
et
us
choose
k+
to
be
consistent with Mori and
Ototake (9). From Equation
9
it follows
that:
Xa
=
X+(X+ To)
= A+ To)
(10)
Hence
X
is just the geomet ric mean of
A+ and X .
Figure 2. Shear stress vs. shear late
for several types of fluids
N.
Newtonian, with slope
p
B.
P.
D.
In annular flow rpSs negative
for r / R
< X and
positive for
r / R
>
A
see Figure
1 )
Bingham plastic, slope po and i ntercepts
1 r p
Pseudoplastic, n <
1
in Equation 21
Dilatant, with n >
1
in Equation
21
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NO. 3
MARCH
1958 347
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Table 1
Values
for
Bingham
Flow
TO
0.1 0 . 2 0 . 3 0.4 0 .5 0 . 6 0.7 0.8 0 .9
.01
0
0.01
0.03
0.05
0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.01
0.03
0.05
0.07
0.1
0.2
0.3
0.4
Q.5
0.6
0.7
0.8
0.9
1.0
0
0.01
0.03
0.05
0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.01
0.03
0.05
0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000
0.3295
0.3776
0.4080
0.4326
0.4637
0.5461
0.6147
0.6770
0.7355
0.7915
0.8455
0.8981
0.9495
1.0000
0.1000
0.3709
0.4195
0.4507
0.4757
0.5075
0.5916
0.6614
0.7246
0.7838
0.8404
0.8960
0.9481
1.0000
0.3000
0.4727
0.5194
0.5501
0.5751
0.6070
0.6922
0.7630
0.8272
0.8873
0.9447
1.0000
0.4000
0.5335
0.5775
0.6072
0.6316
0.6629
0.7474
0.8181
0.8823
0.9425
1.0000
0.5000
0.6006
0.6410
0.6691
0.6925
0.7229
0.8059
0.8760
0.9399
1.0000
0.6000
0.6734
0.7095
0.7356
0.7577
0.7868
0.8676
0.9367
1.0000
0.7000
0.7512
0.7826
0.8064
0.8269
0.8544
0.9323
1.0000
0.2000
0.4185
0.4667
0.4980
0.5232
0.5552
0.6402
0.7108
0.7746
0.8344
0.8915
0.9465
1.0000
0.8000
0.8334
0.8600
0.8811
0.8999
0.9256
1.0000
0.9000
0.9195
0.9411
0.9595
0.9764
1.0000
0.005000
0.003306
0.001807
0.0008701
0.0002977
0
90
0.1250
0.08914
0.07486
0.06523
0.05749
0.04787
0.02502
0.01069
0.002639
0
0.5000
0.3252
0.2902
0.2675
0.2497
0.2273
0.1671
0.1271
0.09204
0.06330
0.04044
0.02214
0.009983
0.002480
0
0.4050
0.2687
0.2375
0.2175
0.2016
0.1817
0.1315
0.09393
0.06413
0.04066
0.02272
0.01007
0.002541
0
0.3200
0.2165
0.1896
0.1721
0.1582
0.1409
0.09743
0.06559
0.04123
0.02292
0.0101 1
0.002535
0
0.2450
0.1689
0.1463
0.1315
0.1196
0.1049
0.06821
0.04226
0.02330
0.01024
0.002555
0
0.1800
0.1263
0.1080
0.09577
0.08596
0.07374
0.04405
0.02394
0.01041
0.002572
0
0.08000
0.05776
0.04734
0.04018
0.03441
0.02731
0.01122
0.002686
0
0.04500
0.03269
0.02579
0.02088
0.01701
0.01232
0.002840
0
0.02000
0.01437
0.01042
0.007669
0.00533 1
0.003122
0
Q B
0.4752 0.3541
0.3638 0.2700
0.3192 0.2324
0.2876 0.2057
0.2611 0 1833
0.2268 0.1545
0.1372 0 . O m 2
0.07353 0.03486
0.03085 0.008273
0.007177
0
0
__
0.06990 0.01875
0.05133 0.01252
0.03784 0.006939
0.02820 0.003382
0.02053 0.001178
0.01174 0
0
0.8667
0.6745
0.6123
0.5685
0.5315
0.4825
0.3480
0.2401
0.1541
0.08893
0.04349
0.01597
0.003030
0
0.7339
0.5677
0.5109
0.4709
0.4371
0.3928
0.2726
0.1788
0.1067
0.05473
0.02 151
0.004579
0
0.6028
0.4636
0.4126
0.3767
0.3464
0.3069
0.2015
0.1224
0.06460
0.02648
0.005929
0
1.0000
0.7829
0.7153
0.6679
0 6276
0.5743
0 4258
0.3041
0.2043
0.1260
0.0686
0.03066
0.009608
0.001298
0
...
...
e . .
I
...
...
...
...
...
...
...
...
...
...
0.2432
0.1845
0.1546
0.1331
0.1153
0.09267
0.03882
0.009272
0
0.1467
0.1103
0.08859
0,07284
0.05995
0.04398
0.01034
0
QBITO
0.7082
0,5400
0.4648
0.4104
0.3666
0.3090
0.1642
0.06972
0.01655
0
2.009
1.545
1.388
1.256
1.155
1.023
0.6717
0.4080
0.2153
0.08827
0.01976
0
1.188
0,9095
0.7980
0.7190
0.6528
0.5670
0.3430
0.1839
0.07712
0,01794
0
8.667
6.745
3.670
2.839
2.555
2.354
2.186
1.964
1.363
0.8940
0.5335
0.2736
0.1076
0.02190
0
0.4053
0.3075
0.2577
0.2218
0.1922
0.1544
0.06470
0.01545
0
0.08738 0.02083
0.06416 0.01391
0.04730 0.007710
0.03525 0.003758
0.02566 0.001309
0.01469 0
0
0.2096
0.1576
0.1266
0.1040
0.08564
0.06283
0.01477
0
6.123
5.685
5.315
4.825
3.480
2.401
1.541
0.8893
0.4349
0.1597
0.0303
0
...
Table II. Values of X
for Power Law
Model
K
8
0
0 . 5 =
1
2
3
4
5
6
7
8
9
10
m
a Obtain
0.01
0.1
0.5050
0.5500
0.4000
0.5000
0.3295
0.4637
0.2318
0.4192
0.1817 0.3932
0.1640
0.3787
0.1503
0.3712
0.1413 0.3606
0.1350
0.3550
0.1304
0.3506
0.1268
0.3470
0.1237 0.3442
0.1000
0.3162
ed by interpolation and chec
0 . 2
0.6000
0.5710
0.5461
0.5189
0.5030
0.4927
0.4856
0.4804
0.4764
0.4733
0.4707
0.4687
0.4472
:ked
b y num
0.3
0.6500
0.6300
0.6147
0.5970
0.5866
0.5797
0.5749
0.5713
0.5686
0.5664
0.5646
0.5632
0.5477
erica1 integral
0 . 4
0.7000
0.6875
0.6770
0.6655
0.6587
0.6541
0.6509
0.6486
0.6467
0.6453
0.6441
0.6325
ion.
b Obi
0.6429
0 .5 0.6
0.7500 0.8000
0.7420
0.7960
0.7355
0.7915
0.7280 0.7872
0.7239
0.7847
0.7211 0.7830
0.7191
0.7818
0.7175
0.7809
0.7164
0.7801
0.7154
0.7794
0.7147
0.
7tBb
0.7141b
0.7784b
0.7071 0.7746
;ained from a plot of X
u s .
0.7
0.8500
0.8470
0.8455
0.8433
0.8420
0.8411
0.8405
0.8401
0.8397b
0. 3945
0.8390b
0,8389'
0.8367
l/ s by interp
0 .8
0.9000
0.8990
0.8981
0.8972
0.8967
0.8962
0.8960
0.8958*
0.8957*
0 895S5
0.8959
0.8953b
0.8944
Nolation.
0.9
0.9500
0.9497
0.9495
0.9493
0.9492
0.9491b
0.9491*
0. 9490b
0.9490b
0.948gb
0.948gb
0. 948g5
0.9487
348
INDUSTRIAL AND ENGINEERING CHEMISTRY
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NON-NEWTONIAN
FLOW
Table 1 1 1
Values of s
+
2)Qp/ l
) +
= T(s,
K )
and
v,,,/v,,
for Power Law Model
0.01 0 . 1 0.2 0.3 0.4 0 . 5 0.6 0.7 0.8 0.9
0.5050 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500
0.5312
0.5606 0.6062
0.6539
0.7024 0.7516
0.8009
0.8504 0.9002 0.9500
0.5397
0.5641
0.6082
0.6552
0.7032 0.7521 0.8011
0.8506 0.9003
0.9501
0.5566 0.5710 0.6122 0.6577 0.7048 0.7531 0.8018 0.8509 0.9004 0.9501
0.6051
0.5908 0.6237 0.6649 0.7094 0.7560 0.8034
0.8517
0.9008
0.95OZb
0.6929
0.6270 0.6445 0.6781
0.7179 0.7611 0.8064
0.8533 0.9015b 0.9504b
0.7468
0.6547 0.6612
0.6882
0.7246 0.7651
0. SO81
0.8546 0. 9022b 0. 9506b
0.7819 0.6755 0.6736 0.6966
0.7297 0.7685 0.8107 0.8551 0.9027b O.950Sb
0.8064
0.6924 0.6838 0.7030
0.7342 0.7711 O.812Sb
0.856Sb
0.9032b 0.951ob
0.8246
0.7046 0.6919 0.7084
0.7372 0.7732 0.8144b
0.8577b
0.9036b O.951lb
0.8388
0.7150 0.6987
0.7128 0.7401 0.7751 0.8160b
0.8585b
0.9041b 0.9513b
0.8502
0.7235 0.7042 0.7164
0.7418 0.7760 0.816Sb
0.8594b 0.9045h 0.9515b
0.8595
0.7306 0.7089
0.7195 0.7446
0 .
7770b 0.8176*
0.8602b 0.9050b
0.9517b
0.8673
0.7367 0.7130
0.7222b
0.7462b 0.777Sb 0.8184b 0.8611b 0.9054b 0.951Qb
K
S
f
2)/ '/(1 - ) ' =
y(S,
K )
S
0
1/4a
1/3
1 2
1
2
3
4
5
6
7
8
9
10
0 2.000 2.000
2.000
2.000
2.000 2.000
1/4a 1.869
1.803 1.815 1.810
1.806 1.803
1/3 1.835 1.787 1,769 1.762
1.757 1.754
1/2 1.778 1.714 1.693
1.682 1.676
1.672
1 1.662
1.567 1.538
1.523
1.514 1.508
2 1.540 1.419 1.380
1.361 1.350
1.344
3 1.442 1.342 1.297 1.278 1.267 1.261
4 1.365 1.280 1.246 1.227 1.217 1.212
5 1.310 1.240 1.210 1.194 1.186 1.180
6 1.270
1.210 1.183
1.168
1.152 1.146
7 1.239 1.187 1.162 1.146
1. 129' 1. 127c
8 1.215 1.168 1.145 1.130 1.114c 1.113c
9 1.192 1.154 1.132
1.117 1.102c
1.101
10 1.176
1.140 1.120 1.106c 1.093c
1.092c
m 1.000
1.000 1.000 1.000
1.000 1.000
* Obtained by parabolic interpolation using values for
8
= 0 , 1, 2.
Obtained by graphical interpolation
of
T s,
K ) / T (O ,
) , using the fact that lim T(s ,
K )
K + 1
=
1.
2 000
1.802
1.752
1.670
1.505
1.340
1.257
1.208
1. 177c
1. 145c
1. 126c
1.1120
1.1000
1.091c
1.000
2.000
1.802
1.752
1.669
1.504
1.3370
1. 254c
1.205
1.
174c
1. 144c
1.126O
1.111c
1.lOOC
1.091c
1.000
2 000
1.800
1.751
1.668
1.502
1. 336c
1.
252c
1. 203c
1. 17lC
1. 143'
1.125'
1.111c
11OOG
1.0910
1.000
2.000
1.800
1.750
1.667
1.501.
1,334=
1.2518
1.201s
1.
16QC
1. 143c
1. 125c
1.111'
1 low
1
0910
1,000
Obtained by graphical interpolation of '=using th e fact th at
vmax /z ~Bv-
1
a s s + m
+ 2 I
Combination of Equations 7 and
8
and
integration over p gives th e following ex-
pressions for the velocity distribution :
1
L = -T T o p ) - 3 p 2 2 )
+
X 2 h P ;
K
5
P 5
h
11)
Go
=
(h) ++ ; A-
5 P 5
X
(12)
1
++ =
To(1
) + 2 1 021 +
Xalnp;
A+
5
p
5
1 13)
where use has been made of the boundary
conditions that 4 = 0 at p = K and
p =
1.
The determining equation for
A,
is just t he statement that the velocity
4- A- ) be the same as q5+
A,)
:
To
- ) ~I- 2To(l A + ) = 0
(14)
From this equation X+ has been deter-
mined as a function
of K
and TO nd is
plotted in Figure
3.
The volume rate of
flow
for the Bing-
ham plastic is obtained by integrating
the velocity distribut ion in Equations
11,
12, and 13 over the annular region and
simplifying the result with the help of
Equation 1 4
1
Q = 2rR2 v,pdp
Q/ rR4P/8pO)s plotted in Figure 4 s a
function of
K
and
TO.
This graph
enables one to compu te easily the volume
rate of flow for a given pressure drop
when the dimensions of R and K and po
and
TO
re known.
A
plot of
C l B / T o
(Figure 5 is useful for calculating the
This expression holds for
TO> 1
K ) ;
there is no flow if
TO5 (1
) . The
dimensionless volume rate of flow OB =
Figure
3.
X ,
and , for Bing-
ham plastic flow
in an annulus
10
0.8
06
2
0 4
a2
00
VOL.
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NO. 3 MARCH 1958
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YO
4
Figure
4.
for Bingham plastic
10
flow through an annulus
09 09
08 08
07
07
06
Figure 6.
X
for power law flow
b 06
through an annulus
-
0 5
50s
-
I-0
P O
04
03
03
0 2
0 2
01
01
00
00
00
1
02 a3 04 05
06
07 08 09 i o
K
pressure drop when the volume rate of Bingham Flow in Circular Tubes Solut ion
for
the Power l a w Mode l
flow is known. Th e use of Figures 4 and
5
is discussed in Exam ples
1
and
2.
The For this model the local shear stress
TR4P 4 1 depends on the local shear rate as follows:
numerical results for the Bingham
flow
(21
1
For high rates of
flow
when the plug
flow region is small compared with the
dimensions of the annulus, Equation 1 5
in which m and n are the rheological con-
stants, obtainable by viscometric tech-
ma y be simplified by using the Newtonian
niques
(7,
I I ,
12).
Neivtonian fluids are
a special case
of
Equation 21 with
rn
=
p
expression for
X
(see Equation
18)
and
making the further assumption that
A,
+ X- F
2X.
Then:
Equations
21
and
4
may be combined
and integrated to give the velocity dis-
( K = 0
X-
=
0, X+
= To)
Q = __ [ I
-
T~+ W ] ( 1 7 )
calculations are presented in Table I. PO
3
Newtonian Flow in Circular
Annulus
( T , = O , P ~ = ~ , X + ~ = X _ ~ = ( ~
21n (1;;;)
Q
= __ [(l - i*
(1
18)
R4P
8 P
In 1 / K ) a n d n = 1.
Newtonian Flow in
a
tribution:
(15a) Very Thin
lit
~1~ R ( P R / 2 m ) s ) d p ; ti
5
p
5 X (22)
5 P 5 1 (23)
(1
4) M
( 1 +
3) To
Q e[ n
(1/.)
This expression differs slightly from that
(To
=
0,
/
=
p ,
K w
1)
given by Laird 5, Equation 30).
Eauation 15 are: Q = 1 1 9 )
l1P
Five important limiting cases
of
?rR4P v, = R ( P R ) / 2 m ) e
UP
Newtonian Flow in Circular Tubes
in which s = l / n ; in the integrations,
and p =
1
have been used. Clcarly both
Equations 22 and 23 must give the same
value of velocity at p =
X :
To
=
0
=
0,
Po
=
P )
Bingham in a V e r y
Thin
the boundary conditions
u, = 0
at
p =
K
Iar
Slit ( K
= 1
r R 4 P
GPO
I -
) 3
[ I -
16)
r R 4 P
Q = -
8~
LA p) dp =
L1
P - ? ) d p (24)
This is the determining equation for A,
which is a function of K and s.
Finally the volume rate of flow is ob-
tained by substituting Equations 22 and
23 into the first line of Equa tion 15. The
order of integration may then be inter-
changed and one integration performed
to give:
Q = T ~ 3 ~ ~ / 2 m ) h I ~2 6 + l p - a d
P
(25)
L1
which can easily be integrated once
X
has
been determined from Equation 24.
The integrals appearing in the general
results in Equations 24 and 25 may
b
350
INDUSTRIAL AN D ENGINEERING CHEMISTRY
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8/18/2019 scambio termico
5/6
WON-NEWTONIAN FLOW
0
0 1
0 2 0 3 0 4
05 06 0 1 0 8 0 9
I O
K
easily be integrated when
s
is a positive
integer (pseudoplastic materials), by ex-
pandin g the integrands in a binomial ex-
pansion and interchanging the order of
summation and integration. Equation
24, which determines
X,
then becomes:
k A s g hac+ A s f 1 = 0
s
= 1,2,3. . .(26)
i = O
i # T
+ l
in which
(;) (-1)ifl
1
1
+
(-1)8 K 8 - - 2 i t 1
,I =
s - 2 i + 1
A = 2 2 (:) (-')' [seven] (29)
i = O s 2i + 1
Equations 26 to 29 are polynomial
equations which can be solved to get
X
as
a
function of
s
and
K ,
although this
method breaks down for high values of
s
and K . Values of X so computed were
used to prepare Table
I1
and Figure
6.
The limiting values of h are h =
(1
+ ) /
2 a t
s
= 0 and X = t
s
= co.
The latter may be shown by expanding
the int egrand of the left side of Equatio n
24 in a Taylor series about p = K and
the right side about p = 1, and taking
the first term in both expansions.
The expression for the throughput rate
given in Equation 25 may be expanded
in a similar fashion to get:
Figure 7.
vmnx/vav for
power law
flow through
an annulus
4
IO
0 9
o n
-
0 7
b
Figure 8.
T(s, K )
for
power law
flow through
an annulus
K
Power Law Flow in
a
Very
Thin
Annular Slit
=
R3(PR/2rn)8 n p ( s , K ) (30)
Applications
in which
1
(31)
- l )a+ lKI- -2 i+8
s - 1
( -1) In (1 / ~) [sodd]
(32)
[seven] (33)
i = o 2i
- s + l
T(s, K ) , defined by 'T (s, E ) = (s + 2 )
QP/ l- ) ' + ~ ,
is tabulated in Table
111
as a
function of s and
X .
One can
easily compute the throughput for any
pressure drop once the dimensions of
R
and
K
and m and
s
are known. Table 111
may also be used to deduce the rheo-
logical constants from annular flow data
(Examples 3 and
4).
The various limiting cases of Equation
30 may be tabulated :
Newtonian Flow
in
a Circular Tube
(s
=
1, m = p, K = 0)-Equation 16
Power-Law Flow
in a
Circular Tu be
S
1, K =
0 )
Example
1.
Calculation of Pressure
Drop for Annular Flow
of
a Bingham
Plastic Material.
A
mud having a den-
sity of 1.69 grams per cc.
flows
at 5 feet
per second average velocity through an
annulus made from 0.5-inch sta ndard pipe
(outside diameter = 0.840 inch =
0.0700
foot) and 2-inch standard pipe (inside
diameter = 2.067 inches
=
0.1726 foot).
The Bingham plastic constants for this solu-
tion are 7 = 0.554 pound/ per square foot
and po
=
0.000582 pound, second per
square foot. Compute the pressure drop
per unit length required.
SOLUTION.From the dimensions of the
annulus, K
=
0.840/2.067
=
0.406 and
R = 0.1726/2 = 0.0863 foot. The
volume throughput is given by:
Q = nR2(1 ')U,,.
~ ( 0 . 0 8 6 3 ) ~l 0.406)2]
= 0.09775 cubic foot per second.
The quantity &/To = 4fi0Q/nR3~os then:
O B I T O=
4(0.000582) (0.09775)/
~ ( 0 . 0 8 6 3 ) ~0.554) = 0.204
From Figure 5, TO= 0.295. Thus , the
pressure drop per u nit length:
P = 270/ToR = 2(0.554)/(0.295) (0.0863)
= 43.5 poundf per square foot per foot
= 0.30 pound/ per square inch per foot
Newtonian Flow
in a
Circular
An
Example 2. Deduction
of
Bingham
Plastic Rheological Constants from An-
nulus (s
= = kh * o)-Equation nu la r Flow Data,
A
Bingham plastic
18
material flows through the annulus de-
Newtonian Flow in
a
Verv Thin An- scribed in Example 1. The following
data are obtained; For v,, = 5 feet p&
second, P = 16.8 pound, square foot per
foot; for 0, 10 feet per second, P = 28.3
nula r Slit
(s
= 1,
m = p , K =
1)-Equation 19
VOL. 50, NO. 3 MARCH 1958 351
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8/18/2019 scambio termico
6/6
pound, per square foot per foot. Calcu-
late T Oand P O or the substance.
SOLUTION.By using Equation 15
we
can calculate the ratio of 0,
( 5
feet per
second) to C ~ 10 feet per second)
:
Q B ( ~
t./sec.)
Q ~ ( 1 0t./sec.)
Q ( 5 ft./sec.) P(10 ft. /sec.)
Q(l0 ft./sec.) P(5 ft./sec.)
Similarly, from Equation 6, we find:
Ta(5 ft./sec.) P(10 ft./sec.)
To(1O ft. /sec.) ~ ( 5t. /sec.)
-
’$ = 1.685
I n order to calculate th e value of
TO
t
either flow rate, a trial-and-error method
is
used. This can be done by assuming a
value of
TO
t one velocity from which one
can get
To
at the other velocity. Values
of fie corresponding to these two TO alues
may be found from Fi ure 4. T h e ratio
of
ne (5
feet per second7 to Q B (IO feet per
second) is then computed; the procedure
is repeated until this ratio is as near to 0.843
as is desired. Such a trial- and-error pro-
cedure takes the form
:
To
To(l0 Q B 5 Ft./Sec.)/
(5 Ft./Sec.) Ft./Sec.)
Re(10 Ft./Sec.)
0.100 0.060 0.868
0 . 2 0 0
0.119
0.730
0.138 0.082
0.837
Hence, Equation 6 gives TO =
ToPR. 2
=
(0.138)(16.8)(0.0863)/2
=
0.10 poundf
per square foot. Th en from Equation 15
Q
= 0.09775 =
~ ( 0 . 0 8 6 3 ) ~16.8) (0.1 33)/8~0
whence it is found tha t
po
= 0.0005 pound?
second per square foot.
Example 3. Calculation of Pressure
Drop for
Power-Law Flow through
an Annulus.
A
0.67% aqueous solution
of carboxy methyl cellulose flows at 5 feet
per second average velocity through the
annulus described in Example
1.
The
power law constants for this solution 8) are
s = 1.398 and rn = 0.00635 pound/
(second)O.7’6 per sq uare foot. Compute
the pressure drop per un it length required.
SOLUTION. s in Example
1,
K = 0.406
and R = 0.0863 foot. The volume
throughput is
Q = .rrR’(I - K~)U,\..
~(0.0863)’ l - (0.406)’] (5)
= 0.09775 cubic foot per second
For the values of K and s given above, we
find by interpolation from Table I11 that
th e dimensionless function T s,
K )
is 0.7155.
Hence the dimensionless throughput is:
fi p =
(1 - ) ’ + ’ ( S
+
2 ) - ’ T S ,
K )
(1 0.406)’.’9*+2 1.398 + 2)-’
(0.7155)
= 0.0359
From Equation 30
0.09775 = ~ ( 0 . 0 8 6 3 ) ~0.0863 P /
(2) (0.00635))’,[email protected])
whence
P =
25.5 pound, per cubic
foot
or
0.177 pound, per square inch per foot.
Deduction
of
Power Law
Constants from Flow through an
An-
nulus. A polymer solution known to be
of
the power-law type flows through the
annulus described in Example
1.
The
following dat a are obtained : at Q =
0.09775 cubic foot per second,
P
= 326
Example
4.
pound, per square foot per foot and at
Q
=
0.19550 cubic foot per second, P = 460
pound, per square foot per foot. It is de-
sired to calculate the power-law constants
of this fluid.
SOLUTION. rom Equation 30 we have
0.09775 =
?iR3
(326R/2m)s Q p ( s , 0.406)
0.19550 = .rrR3((460R/2m)’Qp(~,.406)
Division of these two expressions gives the n:
(112) = (326/460)s
when s = 2.0 (or n = 0.50). Interpolating
from Table 111, we f i nd T(2.0 , 0.406)
=
0.7205, whence Q p = 0.0224. Hence
Equation 30 gives (for Q = 0.09775 cubic
foot per second).
0.09775 = ~10.0 863) ’ (326) (0.0863)/
(2m)]*.O 0.0224)
from which = 0.30 pound, (second)0,50
per square foot.
Acknowledgment
The authors are greatly indebted to
the computing staff of the University of
Wisconsin Naval Research Laboratory,
under the direction
of
Elaine Gessert,
for assistance with the computational
work. Th ey wish to thank J. 0 Hirsch-
felder for making these arrangements
possible.
Nomenclature
9
2
L
m,
P
Po
PL
P
0
r
R
S
t
T
TO
V
VZ
2
Y
K
x
= external body force per unit
= dummy index used in sum-
= length
of
annu lar region
= parameters in power law
model (Equation 21)
=
static pressure
= static pressure at entrance
= static pressure at exit to
mass
mations
to annulus
(z
=
0)
annulus ( z
= L )
=
Po P d / L
+
Ygz
= volume rate of flow through
annulus
= radial coordinate, measured
from common axis of cyl-
inders forming annulus
= radius of outer cylinder of
annulus
=
reciprocal of
= time
= dimensionless shear-stress for
Bingham flow (Equation 6)
= dimensionless limiting shear-
stress
for
Bingham flow
(Equation
6)
= velocity vector
= z-component of velocity vec-
tor
=
axial coordinate, measured
from entrance of annulus
= mass density of fluid
= ratio of radius of inner cyl-
inder to that of outer cvl-
P = Xewt onian viscosity
= “plastic viscositv”
of
the
ABingliarn plastic (Equ a-
tion 5)
= expansion coefficients in ex-
pression for volume rate of
flow of a power-law fluid
through an annulus (de-
fined in Equations 31, 32,
and 33)
= 3.1416
= r,’R = dimensionless radial.
coordinate
= she ar stress tensor
= limiting shear stress of Bing-
ham fluid
=
rt-component of shear stress
tensor
= function defined just after
Equation 33
= dimensionless velociry for
Bingham plastic (Equation
6)
40
=
dimensionless maximum ve-
locity for Bingham plastic
(Equation 12)
+-, ++ = dimensionless velocity for
Bingham plastic outside
plug flow region (defined
in Equations 11 and
13)
OB: Q p = dimensionless
flow
rates for
Bingham and power-law
models, respectively
A = “del” or “nabla” operator
literature Cited
(1) Bird,
R.
B.,
SPE
Journal 11, 35-40
(1955).
(2) Bird, R. B., “Theory of Diffusion,”
Chau. in “Advances in Chemical
Engineering,” vol. 1, pp. 155-239,
Academic Press,
New
York, 1956.
(3) Christiansen,
E.
B., Ryan, N. W.,
Stevens, Lt’.
E. ,
A.I.Ch.E. Journal
1, 544-9 (1955).
(4) Hirschfelder, J .
O.,
Curtiss, C. F.,
Bird,
R.
B., “Molecular Theory
of
Gases and Liquids,” Chap. 11,
Wiley, New York, 1954.
(5) Laird,
h’. M., ~ N D .
ENG.CHEM. 49,
138-41 (1957).
Lyche, B. k. ird, R. B., Chem. Eng.
Metzner, A . B., “Yon-Newtonian
Technology
:
Fluid Mechanics,
Mixing, and Heat Transfer,” in
“Advances in Chemical Engineer-
ing,” vol. I pp. 77-153, Academic
Press, New York, 1956.
Metzner, A. B., Reed, J. C., A.I .Ch.E.
Journal
1
434-41 (1955).
Mori,
Y.,
Ototake, N . , Chem. Eng.
J u p u n ) 17, 224-9 (1953).
Olphen, H. van, J . Znst.
Petroleum
36 ,
223-34 (1950).
Philippoff,
W.,
Viskositat der
Kolloide,” Steinkopff, 1942; Ed-
wards Brothers, Ann Arbor, Mich.,
Sci.
6, 35-41 (1956).
1944.
(12 ) Reiner, M., “Deformation and Flow,”
H.
K. Lewis and Co., London,
1949.
RECEIVEDor review February 13, 1957
h C E P T E D Jun e 10, 1957
ind er Division of Industri al and Engineering
= value of dimensionless rad ial Chemistry, Symposium
on
Fluid
Me-
coordinate for which chanics in Chemical Engineering, Purdue
University, Lafayette, Ind., December
1956. Presented in part , Society of Rheol-
hear stress is zero
ogy, Pittsburgh, Pa., November 1956.
x = limits of plug
flow
region in
Bin gha m flow Work supported by fellowship from Na-
AS, ASZ = coefficients defined in Equ a- tional Science Foundation and grant from
tions 27, 28, a n d
29
Wisconsin Alumni Research Foundation.
352 INDUSTRIAL AN D ENGINEERING CHEMISTRY