JOURNAL OF

Journal of Petroleum Science and Engineering 16 (1996) 203-208 ELSEVIER

Explicit pressure drop-flow rate relation for laminar axial flow of power-law fluids in concentric annuli

J. David, P. Filip *

Institute of Hydrodynamics, Acad. Sci. Czech Rep., Podbabskci 13, I66 12 Prague 6, Czech Republic

Received 23 October 1995; accepted 4 June 1996

Abstract

This paper is an examination of laminar axial flow of power-law fluids in concentric annuli. The objective is to derive an accurate expression relating pressure drop to flow rate over the whole range of entry parameters: aspect ratio of inner-to-outer diameters, flow behavior index, and consistency parameter of the rheological model. Accuracy of the expression enables determination of the influence of individual parameters to the functional dependence of the explicit relation of pressure drop vs. flow rate.

Keywords: boreholes; channel geometry; rheology; laminar flow; steady flow

1. Introduction

The determination of pressure drop-flow rate re- lation plays an important role in analyzing flow conditions for various geometric arrangements. Un- like laminar Newtonian flow, where complexity is almost exclusively due to geometric conditions of the given problem, for laminar non-Newtonian flow this complexity is intensified by nonlinear depen- dence between shear stress and shear rate. To obtain an adequately simple and a relatively accurate pres- sure drop-flow rate relation, balance between geo- metric and rheological entries must be achieved; it is not possible to expect a simple solution simultane- ously for a complicated geometric arrangement and a strongly nonlinear rheological model. Nevertheless,

* Corresponding author.

there is still a series of hitherto unsolved problems, even for relatively simpler geometries and constitu- tive equations. Recently Aadnoy and Ravnoy (1994) solved a relationship for pressure drop vs. flow rate for flow of Bingham and Collins-Graves fluids in a circular tube. The usefulness of simpler rheological models was shown in Khataniar et al. (1994) who used a power-law model.

The objective of this paper is to derive a simple, explicit, relation between volumetric flow rate and pressure drop for axial flow of power-law fluids in concentric annuli (Fig. I). The crucial point for

solving this problem exactly consists of determining

a parameter h that represents the location of maxi-

mum velocity. The parameter A depends both on

geometric and rheological parameters. Fredrickson

and Bird (1958) derived an integral equation for A.

Nevertheless its applicability is not so easy. This is

confirmed by Luo and Peden (1990) who substituted

into their calculation the value of A corresponding to

0920-4105/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PZI SO920-4105(96)00039-3

204 J. Dahl, P. Filip / Journd of Petrole~trn Science and Engirwerir~~ 16 (1996) 203-208

fqll? I

d Fig. 1. Radii of an annulus.

a Newtonian case as no explicit relation for A is available. The present procedure uses quasisimilarity nature (verified in David and Filip, 1994) of the whole problem instead of using a parameter such as A. This violates exactness of the solution, but the deviation of the approximate pressure drop-flow rate relation obtained by this approach from the exact solution is almost negligible in the broad region

described by the entry parameters K, n.

2. Problem formulation

A power-law fluid generated by the constitutive equation is as follows:

‘T= -klj/[‘J- ‘j/ (1)

Further analysis of this model is based on the pressure drop-flow rate relation:

rR’n PR ‘/I’ Q tube= - -

i i 1 + 3n 2k (2)

Eq. (2) is valid for flow in a circular tube (Bird et al., 1965). The equation derived for flow in a con- centric annulus (Hanks and Larsen, 1979) is:

Qimcx,,, rR’n PR ‘/”

i-i 1+3n 2k

(1 _*2y_ Kl~l/iy*?_ KZ)‘+‘/l~]

_Qr”be[(l _h’)‘+‘~‘l_KI~‘/ir(hl-K1)‘+“ir]

(3)

While Eq. (2) is simple and the unknowns k and y1 are determined by measurements of viscosity, Eq. (3) in addition to k and n. contains a parameter, A = A( K, n), that is given by the integral equation of Fredrickson and Bird (1958):

This equation ensures continuity of the axial ve- locity profile at the point of zero shear stress.

Vaughn (1963) proved that for all aspect ratios of K the following relationship is exact:

lim A( K,II) = fi;

1 -K' lim A(K,n)= ___.

II --t 0 I,’ I hi/K’ ’

l+K

lim A(K,~z)=- 2

(5) li * x

Substituting these limiting values into Eq. (3) and introducing the following notation

nor’n = Qtube( 1 - K)'+ I”‘( 1 + K) (6)

one obtains from Eqs. (2) and (3):

i (7)

lim Q,,,, = i II’=

(8)

In these relationships the right-hand sides are independent of A.

The advantage of the normalized dimensionless flow rate Q,,,, over the dimensional flow rate

Qann,exact is its independence from pressure drop, outer diameter and consistency index. For this rea- son, the normalization using Qlube is inevitable.

The problem now consists of deriving a suffi- ciently accurate approximation of Q,,,,,( ~,n> by means of K and n in their sufficiently broad region as the pressure drop-flow rate relationship is given by Eqs. (2) and (6).

J. David, P. Filip / Journal of Petroleum Science and Engineering 16 (19961203-208 205

3. Problem solution

The first approximating step reflects the behavior

of Qnorm for n + 0. Inasmuch as no analytical solu- tion is known, the numerical calculation is approxi- mated by:

Q,,,,( K,O) = i [ 1 + (3Q,,,,( K,l) - 2)0.~~“‘] (9)

the deviation is shown in Fig. 2. Furthermore, Eq. (6) provides that:

Qnorm(0~~) = 1 (10) The opposite case (i.e. K + 1) which is character-

ized by the limiting passage of the diameter of the inner cylinder to the diameter of the outer cylinder can be derived from equation 4.3-37 in Bird et al. (1987). Their approximate relation for flow rate in a concentric annulus is:

rrR3n = 2(1 +2n) (1 - K)2+1'n(l + K)

(11) which represents, in fact, the flow rate between parallel plates given in McKelvey (1962, p.102).

0 numerical calculation

rel.(9)

Fig. 2. Inaccuracy of Q,,,,( K,O) - relation (9).

n

1

1

0

Qnor,,&rO) i c - Fig. 3. Boundary behavior of Qnor,,,(~,n) for a pseudoplastic case

n<l.

Limitations of Eq. (11) in application to an annular geometry are summarized in David and Filip (1995). Substituting Eq. (11) into Eq. (6) and using Eq. (2) we obtain:

1+3n Qnorm(l~~) = 2c1+2nj (12)

The pseudoplastic case, 0 < n < 1, is illustrated in Fig. 3. With respect to the quasisimilarity nature (verified in David and Filip, 1994) of the pressure drop-flow rate relation the approximate solution for Q,,,,< ~,rz) for 0 < n < 1 is proposed in the form:

Q,,,,( K,n> = 42 - 3QmdW1 Qnom( ~70)

+ 3[2Q,,,,<l>n) - Wnorrn(~~l)

(13)

where Q,,,,< K,O), Q,,,,< ~~1) and Q,,,,<l,n) are expressed by Eqs. (91, (7) and (121, respectively. From which it follows that:

Qnorrn(~~~) = &-I(' -~)Qnorm<4

+ 3~Q,,,nd ~41 ( 14) Finally using Eqs. (21, (31, (6) and (14), the flow

rate vs. pressure drop relationship is given by:

%-R3n Q an” ,appr

= l+(l - K)2+1'n(l + K)

xQnorm~~4(~)1'n (15)

206 J. Dal,id, P. Filip/Journal of Petroleum Science and Engineering 16 (1996) 203-208

P

Fig. 4. Deviation (in %) of the approximate flow rate from the

exact one - a pseudoplastic case.

Eq. (15) is based on simple algebraic functions. Its deviation from the exact values:

y _ ,OO ~Qann.nppr - Qann.ed d-

Qannxxict ('6)

does not exceed 2.15% for 0.025 < K < 1 in the whole pseudoplastic region 0 < n < 1. For 0.5 < K <

1 the deviation is even less than 0.4%; for 0.6 < K <

1, less than 0.16%. These results seem to be satisfac- tory, and are illustrated in Fig. 4.

The situation for the dilatant case II > 1 is illus- trated in Fig. 5. There is a singularity at the point

3 (O,n> for n + x as Q,,,.,(K,Y~) + 3 for K > 0, IZ +

A.(7) Ot

0 Qnor&J) ' Ic

Fig. 5. Boundary behavior of Q,,,,,, (K,IZ) for a dilatant case IZ 1 1.

Fig. 6. Deviation (in %;) of the approximate tlowrate from the

exact one - a dilatant case.

x and Q,,,,,(O,n) + 1 for IZ + x. This locally uni- form convergence is sufficient for the problem stud- ied because only positive values of K are considered. Analogously to Eq. (13) the following relation for

Q,,,,,.,,, ' II > I, is proposed:

Q,,,,,,,( K,n) = 313 - 4Qnm( 1 .n>l QmA K,’ )

+ 4[3Q,,A 1 ,?I> - 21 Qr,o,,,,( K.~) (17)

Using Eqs. (8) and (12):

Q,,,,,n,( ~317) = & I7 - 1 Q,,w,,,( ~1') + 2

and the flow rate vs. pressure drop relationship is given again by Eq. (15). For the dilatant case (n > I >, the deviation does not exceed 1.5% for 0.025 < K < 1

and 0. I % for 0.4 < K < 1 (see Fig. 6). Example demonstrating normalization procedure

is presented in Appendix A.

4. Conclusions

Eqs. (14), (15) and ( 15), (17) for pseudoplastic and dilatant power-law fluids, respectively, provide a

J. Dauid, P. Filip/ Journal of Petroleum Science and Engineering 16 (1996) 203-208 201

good approximation in the whole range of an aspect

ratio K and flow behavior index n. The relative simplicity of the resulting expressions makes possi- ble to determine separately the behavior of the pres- sure drop-flow rate relation with respect to the individual parameters. Moreover, in some regions of the parameters K,n where fitness is very close the infinitesimal analysis can be carried out. In all these respects the present results improve the conclusions in David and Fihp (1995) due to the fact that the present analysis takes into account all “boundary” conditions (for limiting values of K and n).

5. Notation

k n

consistency parameter (Eq. (1)) Pa s” flow behavior index (Eq. (1)) dimen-

sionless P

Q R

'd

pressure drop, Pa/m volumetric flow rate, m3/s radius of an outer cylinder (Fig. I), m relative deviation (Eq. (16)) %

Greek letters:

shear rate (Eq. (1)) l/s aspect ratio of inner-to-outer radii (Fig.

1), dimensionless relative radial location of maximum velocity (Fig. l), dimensionless shear stress (Eq. (1)) Pa

Subscripts:

ann,appr

ann,exact

ann(plates)

norm

tube

approximate value for annular geome-

try (Eq. (15)) exact value for annular geometry (Eq.

(3)) approximate value for annular geome- try taken from a parallel plate model

(Eq. (11)) normalized value (Eq. (6)) dimension- less exact value for circular geometry (Eq.

(2))

Acknowledgements

The authors wish to thank Dr. E.C. Donaldson whose comments were helpful in the revision of the manuscript. This work was financially supported by GA CR, Grant No. 106/93/2274.

Appendix A. Example of the use of the developed equations

The entry values from the example in Gray and Darley (1980) (Chapter 5) are taken:

R = 0.2032 m; K = 0.405; n = 0.22; P = 97

Pa/m; from table 5.3 and fig. 5.51 in the cited source it follows that k = 3.1 Pa so.“.

The calculation of volumetric flow rate consists of three successive steps:

(1) application of geometric parameters: From Eqs. (7) and (9) we obtain Q,,,,(O.405,1)

= 0.6755 and Q,,,,(O.405,0) = 0.5487, respectively. (2) application of a rheological parameter (flow

behavior index): From Eq. (14) it follows that Q,,,,(O.405,0.22) =

0.6068. (3) application of kinematic and rheological (con-

sistency parameter) parameters: Volumetric flow rate is given by Eq. (15) its

approximate value is Qann,+,rr = 0.01911 m3/s. The exact value of volumetric flow rate Q,,, exact

is given by Eq. (3) and equals 0.01896 m3/s (using PC for solving the integral equation (4) - in con- trast to the preceding steps where only a pocket calculator is needed). The relative deviation (Eq. (16)) attains 0.77% and is negligible with respect to inaccuracy of the experiments for determination of the values n and k.

References

Aadney, B.S. and Rawfly, J.M., 1994. Improved pressure

drop/flow rate equation for non-Newtonian fluids in laminar flow. J. Pet. Sci. Eng., 11: 261-266.

Bird, R.B., Stewart, W.E. and Lightfoot, E.N.,1965. Transport Phenomena, Ch.2. Wiley, New York, NY, 780 pp.

Bird, R.B., Armstrong, R.C. and Hassager,O., 1987. Dynamics of

208 J. David, P. Filip/Joumal of Petroleum Science and Engineering 16 (1996) 203-208

Polymer Liquids, Vol. 1: Fluid Mechanics, Ch. 4, Par. 4.3.

Wiley, New York, NY, 649 pp.

David, J. and Filip, P., 1994. Quasisimilarity of flow behavior of

power-law fluids in concentric annuli. Fluid Dyn. Res., 14:

63-70.

David, J. and Filip, P., 1995. Relationship of annular and

parallel-plate Poiseuille flows for power-law fluids. Polym.-

Plast. Technol. Eng., 34: 947-960.

Fredrickson, A.G. and Bird, R.B., 1958. Non-Newtonian flow in

annuli. Ind. Eng. Chem., 50: 3477352.

Gray, G.R. and Darley, H.C.H., 1980. Composition and Properties

of Oil Well Drilling Fluids, Ch. 5. Gulf Publishing Co.,

Houston, TX, 509 pp.

Hanks, R.W. and Larsen, K.M., 1979. The flow of power-law

non-Newtonian fluids in concentric annuli. Ind. Eng. Chem.

Fundam., 18: 33-35.

Khataniar, S., Chukwu, G.A. and Xu, H., 1994. Evaluation of

rheological models and application to flow regime determina-

tion. J. Pet. Sci. Eng., 11: 155-164.

Luo, Y. and Peden, J.M., 1990. Flow of non-Newtonian fluids

through eccentric annuli. Sot. Pet. Eng. Production Eng. (Feb.),

pp. 91-96.

McKelvey, J.M., 1962. Polymer Processing. Wiley, New York,

NY, 409 pp. Vaughn, R.D., 1963. Laminar flow of non-Newtonian fluids in

concentric annuli. Sot. Pet. Eng. J., 3: 274-276.