This article reports the results of a numerical computation of the length and total pressure drop in the entrance region of a circular tube with laminar flows of pseudoplastic and dilatant fluids at high Reynolds numbers (i.e., approximately 400 or higher). The analysis utilizes equations for the apparent viscosity that span the entire shear rate regime, from the zero to the infinite shear rate Newtonian regions, including the power law and the two transition regions. Solutions are thus reported for all shear rates that may exist in the flow field, and a shear rate parameter is identified that quantifies the shear rate region where the system is operating. The entrance lengths and total pressure drops were found to be bound by the Newtonian and power law values, the former being approached when the system is operating in either the zero or the infinite shear rate Newtonian regions. The latter are approached when the shear rates are predominantly in the power law region but only if, in addition, the zero and infinite shear rate Newtonian viscosities differ sufficiently, by approximately four orders of magnitude or more. For all other cases, namely, when more modest differences in the limiting Newtonian viscosities exist, or when the system is operating in the low- or high-shear rate transition regions, then intermediate results are obtained. Entrance length and total pressure drop values are provided in both graphical form, and in tabular and correlation equation form, for convenient access.

References

References
1.
Skelland
,
A. H. P.
,
1967
,
Non-Newtonian Flow and Heat Transfer
,
Wiley
,
New York
, pp.
5
7
.
2.
Dunleavy
,
J. E.
, Jr.
, and
Middleman
,
S.
,
1966
, “
Correlation of Shear Behavior of Solutions of Polyisobutylene
,”
Trans. Soc. Rheol.
,
10
(
1
), pp.
157
168
.
3.
Capobianchi
,
M.
, and
Irvine
,
T. F.
, Jr
.,
1992
, “
Predictions of Pressure Drop and Heat Transfer in Concentric Annular Ducts With Modified Power Law Fluids
,”
Heat Mass Transfer
,
27
(
4
), pp.
209
215
.
4.
Capobianchi
,
M.
,
2008
, “
Pressure Drop Predictions for Laminar Flows of Extended Modified Power Law Fluids in Rectangular Ducts
,”
Int. J. Heat Mass Transfer
,
51
(5–6), pp.
1393
1401
.
5.
Cross
,
M. M.
,
1965
, “
Rheology of Non-Newtonian Fluids: A New Flow Equation for Pseudoplastic Systems
,”
J. Colloid Sci.
,
20
(
5
), pp.
417
437
.
6.
Shah
,
R. K.
, and
London
,
A. L.
,
1978
,
Laminar Flow Forced Convection in Ducts, A Source Book for Compact Heat Exchanger Analytical Data
(Advances in Heat Transfer (Supplement 1)),
T. F.
Irvine
, Jr.
and
J. P.
Hartnett
, eds.,
Academic Press
,
New York
, pp.
85
99
.
7.
Collins
,
M.
, and
Schowalter
,
W. R.
,
1963
, “
Behavior of Non-Newtonian Fluids in the Entry Region of a Pipe
,”
AIChE J.
,
9
(
6
), pp.
804
809
.
8.
Fargie
,
D.
, and
Martin
,
B. W.
,
1971
, “
Developing Laminar Flow in a Pipe of Circular Cross-Section
,”
Proc. R. Soc. London A
,
321
(
1547
), pp.
461
476
.
9.
Bogue
,
D. C.
,
1959
, “
Entrance Effects and Prediction of Turbulence in Non-Newtonian Flow
,”
Ind. Eng. Chem.
,
51
(
7
), pp.
874
878
.
10.
Chebbi
,
R.
,
2002
, “
Laminar Flow of Power-Law Fluids in the Entrance Region of a Pipe
,”
Chem. Eng. Sci.
,
57
(
21
), pp.
4435
4443
.
11.
Mashelkar
,
R. A.
,
1974
, “
Hydrodynamic Entrance-Region Flow of Pseudoplastic Fluids
,”
Proc. Inst. Mech. Eng.
,
188
(
1974
), pp.
683
689
.
12.
Matras
,
Z.
, and
Nowak
,
Z.
,
1983
, “
Laminar Entry Length Problem for Power Law Fluids
,”
Acta Mech.
,
48
(1), pp.
81
90
.
13.
Sparrow
,
E. M.
,
Lin
,
S. H.
, and
Lundgren
,
T. S.
,
1964
, “
Flow Development in the Hydrodynamic Entrance Region of Tubes and Ducts
,”
Phys. Fluids
,
7
(
3
), pp.
338
347
.
14.
Langhaar
,
H. L.
,
1942
, “
Steady Flow in the Transition Length of a Straight Tube
,”
ASME J. Appl. Mech.
,
9
(
2
), pp.
A55
A58
.
15.
Durst
,
F.
,
Ray
,
S.
,
Ünsal
,
B.
, and
Bayoumi
,
O. A.
,
2005
, “
The Development Lengths of Laminar Pipe and Channel Flows
,”
ASME J. Fluids Eng.
,
127
(
6
), pp.
1154
1160
.
16.
Christiansen
,
E. B.
, and
Lemmon
,
H. E.
,
1965
, “
Entrance Region Flow
,”
AIChE J.
,
11
(
6
), pp.
995
999
.
17.
Vrentas
,
J. S.
,
Duda
,
J. L.
, and
Bargeron
,
K. G.
,
1966
, “
Effect of Axial Diffusion of Vorticity on Flow Development in Circular Conduits: Part I. Numerical Solutions
,”
AIChE J.
,
12
(
5
), pp.
837
844
.
18.
Hornbeck
,
R. W.
,
1964
, “
Laminar Flow in the Entrance Region of a Pipe
,”
Appl. Sci. Res., Sect. A
,
13
(
1
), pp.
224
232
.
19.
Poole
,
R. J.
, and
Ridley
,
B. S.
,
2007
, “
Development-Length Requirements for Fully Developed Laminar Pipe Flow of Inelastic Non-Newtonian Liquids
,”
ASME J. Fluids Eng.
,
129
(
10
), pp.
1281
1287
.
20.
Mehrotra
,
A. K.
, and
Patience
,
G. S.
,
1990
, “
Unified Entry Length for Newtonian and Power-Law Fluids in Laminar Pipe Flow
,”
Can. J. Chem. Eng.
,
68
(
4
), pp.
529
533
.
21.
Soto
,
R. J.
, and
Shah
, V
. L.
,
1976
, “
Entrance Flow of a Yield-Power Law Fluid
,”
Appl. Sci. Res.
,
32
(
1
), pp.
73
85
.
22.
Bejan
,
A.
,
1984
,
Convection Heat Transfer
,
Wiley
,
New York
, pp.
28
39
.
23.
Brewster
,
R. A.
, and
Irvine
,
T. F.
, Jr
.,
1987
, “
Similitude Consideration in Laminar Flow of Modified Power Law Fluids in Circular Ducts
,”
Heat Mass Transfer
,
21
(
2–3
), pp.
83
86
.
24.
Patankar
,
S. V.
,
1980
,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere Publishing Company
,
New York
, pp.
11
153
.
25.
Brewster
,
R. A.
,
2013
, “
Pressure Drop Predictions for Laminar Fully-Developed Flows of Purely-Viscous Non-Newtonian Fluids in Circular Ducts
,”
ASME J. Fluids Eng.
,
135
(
10
), p.
101106
.
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