WILLIAM M. LAIRD1 Gulf Research & Development Co., Pittsburgh 30, Pa.
Slurry and Suspension Brans Basic Flow Studies on Bingham Plastic Fluids
A simplified flow equation, applicable to many engineering problems, gives an approximate picture of annular flow behavior without laborious calculations
T H E flow of non-Nekvtonian fluids through noncircular conduits is a problem frequently encountered in industries dealing with the transporting of industrial wastes, slurries, and suspensions of all kinds. Of particular interest to the oil industry is the flow of the drilling fluid in the annular space formed by the drill pipe and well casing or bore hole. illthough the nature of non-Sewtonian liquids remains control-ersial. it has been established that Buckingham‘s analysis of a Bingham body (2) applies with sufficient accuracy in many cases. I n particular, it has been demonstrated that oil well drilling fluids may be considered as Bingham bodies in the range of velocities encountered in practice (3, 4 ) . Since the flow of Bingham bodies in turbulent flow in circular pipes may be calculated with conventional equations and friction factor charts (3, 4,8), turbulent flow in noncircular sections may be predicted by using a “mean hydraulic radius” defined as cross-sectional areajwetted perimeter for noncircular sections ( I , 8 ) . However, it is generally recognized that mean hydraulic radius may not be used in laminar flow calculations for a Newtonian liquid, and it is reasonable to assume that This limitation also applies to non-Newtonian liquids. Therefore, the laminar flow of a Bingham body in an annular section is considered in detail.
shear force associated with 7 0 and p must be in the same direction. h-ow consider a hypothetical cylindrical surface of radius, r , in a Bingham body flowing in a n annulus (Figure 1). Take positive velocities and forces acting in the direction of flow. I n addition, the following boundary conditions are assumed for this analysis:
1. There is no “slip” a t the annulus walls, or u (R1) = v (R3) = 0. 2. T h e definition of a Bingham body implies “plug flow” where the shear StreSS resisting the pressure (AF) is insufficient to overcome the yieId stress ( T O ) or v ( u ) = v ( b ) = uo. 3. From the same physical reasoning,
b
+ >
\
+
1 Present address, Department of Mathematics, University of Pittsburgh, Pittsburgh 13, Pa.
138
the shear force acting beyond the imaginary surface is -2nlr
( + TO
y
2)
-
. The
change in this force, as r is increased to dr, must equal the change in pressure force across the distance (&) thus:
+
r
fluid beyond the surface is 2arl
A Bingham body is defined to be a fluid whose shear stress-rate of shear 1 dv I p - (see Nomenrelation is T = T O ldr~ clature). T h e absolute value of the du/dr term is necessary because the
Ll‘hen R1< r < a. the shear stress a t the imaginary surface is T = T O f p dz! 5,
\$‘hen b < < R P ,the shear stress at the imaginary surface is T O - pdc/dr. T h e negative sign on the du/dr term is necessary to allow the two shear stress terms to be additive, since dv/dr is negative for all r > 6. As before, the shear force acting on the
n
Development of Theory
which presupposes plug f l o ~ ,we conclude that the shear stress associated with the viscosity ( p ) must reduce to zero a t the boundary and inside the plug, or d 4 a ) - d 4 b ) - 0, dr dr
dr -
+
d
[ 27ilr
(ri
-
[.(
--ro
p
g)]
or dr
INDUSTRIAL AND ENGINEERING CHEMISTRY
TO--
Equating the differential shear dr ’ force to the differential pressure force as r is increased to r dr:
g?)
d
Figure 1. Annular velocity profile for non-Newtonian liquid
(
+
.$)I
=
=
-
y
Integrating Equations 1 and 2 yields
A z In r
+
To7
1
+C
R2
>r >b
(4)
where A I , Az, B, and C are constants of integration. To obtain B and C, apply boundary condition 1-that is u ( R i ) = u (Rz) = 0
(5)
g(b2
rdR1
161
- a)])
(12)
(ao
thus
=
[$ (Rz2 -
b b2)
TO(&
+ A I n - Rz -
-
611)
~(u+b)(Rz24
+ Rz) (RzP - Hi2) 2 ( R i + Rz) ( b 2 - 2 ) + 4 + b ) (bz - + 4
R12)
andatv = b
- a2)2 -
+
' (19)
(Ri
7 0
a')]
(Q
(13)
*
Rz
- 70) Rs
(11)
2~ M
LR2
[$(Rz2r
- 7 3 ) + YA In Rz
-
+ 7rvo(b2 - a 2 )
where ___ d'(R1) and d ' ! are obtained by dr d7
rO(R2r - r2) dr
differentiating Equations 3 and 4 and letting 7 = R1 and Rz,respectively. Thus, 0 = A I - Az or AI = A, = A
Integrating, replacing uo and A by Equations 12 and 14, and simplifying, gives
1
Adding Equations 23 and 24 results in the relation
'pub __ 21
=
A
(25)
It is interesting to note that subtracting Equation 23 from 24 gives the same result as Equation 16, thus showing that the VOL. 49, NO. 1
JANUARY 1957
139
conditions implied by Equation 15 are included in the boundary condition described by Equation 22. W e may reduce Equation 25 to a relation between a and b only by substituting in Equation 14; thus
where r~ is given by Equation 30. As a final check for Kewtonian liquids To = 0 and Equation 30 becomes *[(Rs4
Q
8d
- R14) -
In R2/R1
which is identical with Lamb’s equation for annular flow.
AP
~( b - a)
from Equation 16, 21 so Equation 26 becomes but rO =
Cancelling out
AP
-
21
Now consider a numerical example in which pressure versus flow is plotted using the complete equation (Equation 20). This will be compared with a similar plot using the simplified equation (Equation 30). Finally, a typical velocity profile is plotted from Equations 8, 9, and 25. Similar plots are shown for a Newtonian liquid for comparison. Take the following values for annular dimensions and fluid properties :
R1= 4 inches
= 0.3333 ft., inner radius of annulus R2 = 6 inches = 0.5000 ft., outer radius of annulus 1 = 1000 ft., length of conduit fi = 20 cp. = 20 x 0.2089 x 10-:3 lb.sec./sq.ft., viscosity of fluid T~ = 0.15 Ib.jsq.ft., yield stress of fluid
and combining terms
Equation 28 may be checked by considering the limiting case of a Newtonian liquid ( 6 ) where to = 0 , and a = b = Y O . Thus, for a Newtonian liquid, Equation 28 becomes
Numerical Example
These values for fluid properties correspond to an aqueous Bentonite suspension with properties similar to those of a typical oil well drilling mud. Inserting the proper values into Equation 20 gives Q
Equation 29 is exactly the result obdu - = 0 and solving for 1 dr for the annular flow of a Newtonian liq.lid as established by Lamb (6). T o solve Equation 20, we must have expressions for a and b in terms of variable Ab and the other flow parameters. This could be done by solving Equations 16 and 28 simultaneously. This would be difficult to do analytically but may be done numerically for specific
=
+ 376.00[a3 + b3 - 0.162001 + + 282.00 ( b + ( b 2 - + 65.280 - 470.00(b2
0.9399A$[0.05016 - (64 - ad)]
In 1.5
a1)
[-0.01813Ap
a)
a2)
- a2)]
(32)
N E W T O N I A N LIQUID: V I S C O S I T Y = 2 0 C P . B I N G H A M BODY: V I S C O S I T Y = 20CP. Y I E L D S T R E S S = . I ~ L B S . /F T . ~
tained by letting
problems
by
assuming
values of
a,6
0
2
Equation 20 to get Q. A numerical example of this procedure is included here. However, for engineering calculations, we may make the same simplification that Bingham 12) does for pipe flonnamely, that a t reasonably high pressures, the dimensions of the plug become negligible. T o accomplish this, assume that b = a or ( b - a) = 0. These simplifications must be applied to Equation 19 because subsequent equations involved simplification with (6 - a) = 2170
-.
Ab
16
LL
14-
solving for the corresponding values of
Ap, a, and b and then substituting in
18-
3
W
5
12
LT
IO 8 6-
Under these conditions, Equation
19 becomes Figure
740
INDUSTRIAL AND ENGINEERING CHEMISTRY
2.
Laminar
pressure
drop
Now assume various values of
2b
-O 2 r
ANNULAR DIMENSIONS
I
I
and
solve Equation 28 for a and 0. For an assumed value of
n --
b
= 0.7000, Equation
28 becomes 0.2208 b2 - 0.2500 b f 0.06945 = 0 (33)
This could be solved by the quadratic formula. However, the Newton-Raphson iterative process was resorted to for ease of computation (7). Solving for a and b and substituting in Equation 16 gives = 0.3422 ft. b = 0.4888 ft. A@ = 2046 lb./sq. ft.
a
,300
Figure 3.
and, finally, substituting these values in Equation 32 gives i
Q
=
0.44 cu. ft./sec. at Afi
z. b
those obtained by the approximate formula, the appropriate values are inserted in Equation 30 which becomes =
500 R2
body and Newtonian liquid
Q = 4.50 cu. ft./sec.
2046 Ib./sq. ft.
To compare these results with
Qapprox
Velocity profile-Bingham
.450
=
This process is repeated for other values of
,400 RADIUS- FT.
,350
RI
0.002434A) - 6.538
Q = 0.00243 A)
(36)
Equation 37 is a straight line through the origin and is shown on Figure 2. T h e Newtonian velocity profile is also computed from Lamb’s velocity expression (6):
(34)
Acknowledgment
T h e author wishes to express appreciation to N. D. Coggeshall and J. G. Jewel1 of the Gulf Research and Development Co. for their helpful suggestions and criticism. Appreciation is expressed to Blaine B. Wescott, Director, for permission to publish this paper.
L
Figure 2 is a Cartesian plot of the pressure versus rate of flow values calculated as described. Note that the curve representing Equation 20 is asymptotic to the straight line representing Equation 30. T h e question arises concerning the behavior a t zero rate of flow. T h e exact value of the pressure a t this point is determined from Equation 16 by letting a = R1, b = R B . T h e slope may be determined by differentiating Equation 20 : dQ
271.
Z f i = T
[(&4
- R14) 161
(b4
- a4)] I 161
p
(37) Values for velocity are calculated from this equation for the same flow rate of 4.50 cu. ft./sec. as was done for the Bingham body. The Newtonian curve of Figure 2 represents a pressure of 4471.0 pounds per square foot. Conclusion
I n conclusion, a flow equation for a 2~ a R2
[ - (R22 -
In -
- 2)(Rz2- R1’) -~ (5’ - a*) 81
A velocity profile for a given flow may %
4
be plotted by evaluating Equations 8 and 9. This is done for Q = 4.50 cu. ft./sec. (Figure 3). T h e slope of the velocity profile must be zero a t a and b from physical considerations and may be verified by differentiating Equations 8 and 9 and substituting the proper values for a, b, and A. T h a t the slope must be greater than zero a t the annulus wall is obvious from the physical reasoning that there is shear stress a t the walls providing there is no slip. This can also be verified by differentiation. Finally, it is interesting to compare these results with those for a Newtonian liquid having the same viscosity. Solving Equation 3 1 for the same pipe dimensions as for the previous example and for a viscosity of 20 cp., gives
+
bRi
(b2
Equation 35 becomes zero for a = RI, b = Rz.
R1s)2
161
161
(35)
Bingham body in an annular conduit has been developed. This equation is complicated and requires numerical solution. However, certain simplifying assumptions may be made resulting in a simplified flow equation (Equation 30) which is applicable to many engineering problems and will give an approximate picture of the annular flow behavior without requiring laborious calculations. T h e justification of applying Equation 30 to flow problems will be dictated by the accuracy desired and the range of flow rates being investigated. For example, referring to,Figure 2, if the flow rates being investigated lie fairly far out on the curve, the approximate or straight line solution may be satisfactory. However, should interest lie in behavior near the pressure axis, the exact solution should be obtained.
Nomenclature
r TO
= radius, ft. = radius for zero velocity gradient
in Newtonian liquid, ft.
I a, b
= length of flow, ft. = bounding radii of plug flow, ft.
R1
RZ
= inner annular boundary, ft. = outer annular boundary, ft.
A!
= pressure difference along
Apo
=
u
=
uo
=
Q
=
T
TO
= =
p
=
I , Ib/ sq. ft. pressure required for incipient flow, lb./sq. ft. velocity, ft./sec. plug velocity, ft./sec. flow rate, cu. ft./sec. shear stress, lb./sq. ft. yield stress of fluid, lb./sq. ft. viscosity of fluid, lb.sec./sq. ft.
References
(1) Binder, R. C., “Fluid Mechanics,” p. 84, Prentice-Hall, New York, 1943. ( 2 ) Bingham, E. C., “Fluidity and Plasticity,” McGraw-Hill, New York, 1922. (3) Caldwell, D. H., Babbitt, H. E., Trans. Am. Inst. Chem. Engrs. 27, 237-66 (1941 ). (4) Dunn, T. H., Nuss, W. V., Beck, R. V., World Oil 128, 85-94 (Feb. 1,
.,,.
1040’i -,
(5) Fage, A., Proc. Roy. Sac. London 165, 501-29 (1938). (6) Lamb, H., “Hydrodynamics,” p. 586, Cambridge Press, London, 1932. (7) Lovitt, W. V., ”Elementary Theory of Equations,” p. 139, Prentice-Hall, New York, 1939. (8) Moody, L. F., Trans. Am, Sac. Mech. Engrs. 66, 681-8 (1944).
RECEIVED for review January 11, 1956 ACCEPTEDJuly 12, 1956 VOL. 49,
NU. 1
JANUARY 1957
141
