Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979 fST,
turbulent flow friction factor in the straight pipe
137
Dean, W. R., Phil. Mag. S.T., 4, 208 (1927). Dean, W. R., Phil. Mag. S . T . , 5, 673 (1928). Detra, R. W., Mitteiiungen aus dem InstRut fuer Aerodynamik Herausgegeben Var, Prof. J. Ackert, No. 20, 1953. Eustice, J., Proc. R. Soc., A84, 107 (1910). Grindley, J. H., Gibson, A. H.,Proc. R. Soc., A80, 114 (1908). Gupta, S. N., Ph.D. Thesis, Banaras Hindu University, 1974. Hawthorne, W. R., R o c . R. Soc., 374 (1951). Ito, H., J. Basic Eng., Trans. A.S.M.E., BID, 123 (1959). Ito, H.,ZAMM, 49, 653 (1969). Keulegan, G. H.,Beij, K. H., J . Res. Nat. Bur. Stand., 18, 89 (1937). Miyagi, C., J. Jpn. Soc. Mech. Eng., 35, 1121 (1932). Thomas, G. B., Jr., "Calculus", Addison-Wesley Publishing Co., Inc. (London), 2nd ed, p 464, 1964. Tomita, K., J. Jpn. Soc. Mech. Eng., 35, 293 (1932). Truesdell, L. C., Jr., Adler, R. J., AIChE J., 16, 1010 (1970). White, C. M., R o c . R . SOC., 123A, 645 (1929).
L , length of the straight pipe or the coil NRe,Reynolds number, D g U / p ND, Dean number, NR,(D,/D,)'/2 ND,, modified Dean number, N ~ , ( D t / 2 R , ) ' ~ 2
p , pitch of the helical coil, distance between center lines of
two adjacent wraps U ,pressure difference
r,, radius of the coil R,, radius of curvature of the coil, see eq 4 U , average fluid velocit,y p , viscosity p , density Literature Cited Adler. ZAMM. 14, 257 (1934). Cummings, G. H.,Aeronautical Research Council Reports and Memoranda No, 2880, 1955.
Heceiued /or r e v i t u December 1, 1977 Accepted June 29, 1978
Momentum Transfer in Curved Pipes. 2. Non-Newtonian Fluids P. Mishra' and S. N. Gupta Departments of Chemical Engineering and Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-22 1005. India
Experimental data for non-Newtonian fluids flowing through helical coils are presented. An appropriate viscosity term prevailing at the wall shear stress is defined and is used in the Reynolds number to obtain a correlation between friction factor and Dean number for helical coils. The use of differential viscosity in turbulent flow Reynolds number is found to give a single valued correlation for both the Newtonian and non-Newtonian fluids.
Introduction In spite of the increasing importance of non-Newtonian fluids in industries, little effort has been made to study the behavior of such fluids flowing through curved pipes. Rajshekharan et al. (1964, 1966) obtained diametrical pressure drop data for helical coils of different geometries for non-Newtonian fluids and correlated the results in terms of flow rate and pressure drop. Later (1970) they presented the data on non-Newtonian fluid flow through helical and spiral coils in the form of a plot of friction factor vs. Reynolds number. No correlation is available so far in the literature for laminar and turbulent flow of nonlNewtonian fluids through curved pipes. With little data available it is not possible to present any correlation in terms of friction coefficient as a function of Dean number, the well known dimensionless variable. It is, therefore, felt necessary to obtain more experimental data for non-Newtonian fluids flowing through helical coils. Theoretical Background (i) Laminar Flow. The shear stress-shear rate relationship, velocity profile equation, and friction factorReynolds number relationships for the laminar flow of non-Newtonian fluids (Ostwald power law type) flowing through straight tubes of round cross section as given by Skelland (1967) are summarized below: (a) the constitutive equation of a power law fluid 7
= -K(
(b) the generalized power law 7,
where
K ' = K 3( n + 7 1 ) ; n = n' (c) the velocity profile
and the velocity gradient a t the wall (4) (d) the laminar friction factor fSL
= 16/NrRe
where D,R
NRL =
U(2-n)p
K'8n-l
and
$)n
27,
D,AP
piY
2pCr-L
f=-=-
* Author to whom all correspondenceshould be addressed at the Department of Chemical Engineering. 0019-7882/79/1118-0137$01.00/0
= K'(Y)n'
(7)
The prediction of the pressure drop for laminar flow of non-Newtonian fluids through straight pipe systems poses C
1978
American Chemical Society
138
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979 __-
1
tM .-.. . .. ..... ...~. .. .. .... .. .. ...
True shear viscosity may be defined as
_. ~
I
and the corresponding Reynolds number as N ~ e a= (DtuP/pa)
L-:
._______
L -
8
8
“IO
- - - - - L O G
Figure 1. Laminar and turbulent flow through curced and straight pipes on the capillary shear diagram for non-Newtonian fluids.
no problem provided the necessary rheological data are available for the determination of the rheological constants K‘and n’. These constants are normally determined either from capillary tube or other types of rotational viscometer data. Once the values of K‘and n‘are known, pressure drop through the pipe can readily be calculated from eq 5 and 7. For Newtonian fluids, friction losses for both the laminar and turbulent flow in straight and curved pipes have usually been expressed in terms of friction factor-Reynolds number as shown in part 1. The same general procedure will be followed for non-Newtonian fluids also except that the definition of Reynolds number involves a viscosity term which has various possible rheological definitions. An appropriate viscosity term and its explanation are needed so that its use in the Reynolds number gives a simple and common correlation for Newtonian and non-Newtonian fluids flowing through the curved pipes. In order to achieve the above objective, laminar and turbulent flow curves of a power law fluid ( n < 1) flowing through straight and curved pipes are explained in the capillary flow diagram as shown in Figure 1. This figure has been drawn after having the knowledge of the nature of the flow curves from experimental data of Newtonian and non-Newtonian fluids flowing through straight and curved pipes. ACR represents the capillary flow curves obtained from a capillary tube viscometer. The entire curve is for the laminar flow regime, and the slope gives the flow behavior index n‘. The curve ACD represents the flow through a straight pipe of diameter D,. The portion AC of the curve represents the laminar flow while CD shows the turbulent flow. The curve AEF represents the flow through a curved pipe of diameter equal to that of straight pipe and curvature radius I?,. Let us consider the flow through a curved pipe. If the flow is very small (ND,< IO), fluid particles will remain blind to the curvature and the centrifugal forces will be very small compared to viscous forces. Under this situation the curved pipe flow follows the straight pipe flow curve. If the flow is say at point J, the pipe apparent viscosity (or pseudoshear viscosity) would be the ratio of T, at point P to (8lT/DJ at point Q. ‘Thus 2DtUP
= ___ PI
where TW
F1
=
E7pj
= K ’(8U / Dt)”’-I
(10)
If the flow is increased, the secondary flow develops due to effective centrifugal force, thereby increasing the pressure drop or T , for the same 8cJ/D, in the straight pipe. The laminar flow curve for curved pipes deviates slowly from that for the straight pipes. Let us consider the laminar flow in the curved pipe corresponding to point G. In such a case, if the Reynolds number N K e defined by eq 8 is used it is seen that it contains a viscosity K’(8U/ DJn’-l being evaluated at point U, i.e., from the straight pipe laminar flow curve a t the same value of 8U/D, but at the shear stress T ~ corresponding ‘ to point L. Thus
The true shear rate (duldr), and pseudoshear rate (8U/Dt) are related by eq 4 for straight pipe laminar flow only. The quantity (SU/DJ does not represent even pseudoshear rate for curved pipe flow. Further, it can be observed that the viscosity p l corresponds to 7,’ whereas the actual shear stress prevailing a t the wall is 7 , . In order to avoid discrepancies noted as above in NRe, it is suggested that one should evaluate the viscosity corresponding to point T, that is a t the same wall shear stress T, as in the curved pipe flow rather than a t the same flow parameter (8U/D,). Thus, the Reynolds number becomes (12) where
The use of Reynolds number NReseems to be more appropriate since it is defined with a viscosity term evaluated a t the prevailing wall shear stress. Rheological parameters K’and n’too are evaluated at T,. For laminar flow of non-Newtonian fluids flowing through straight pipe, all three definitions of the Reynolds number NRe‘, NRel, and NKe are seen to be the same. A c c o r h g to the present approach the Dean number is defined as ND, = N R ~ ~ ( D ~ / D ~ ) ” ‘
(14)
and for the non-Newtonian fluids the corrlstion of the friction factor will take the following form f C / f S L = 40VDJ (15) where fSL
=
16/NRez
(16)
(ii) Turbulent Flow. Let us now examine the turbulent flow curves for straight and curved pipes. The breakdown of laminar flow occurs at points C and E for straight and curved pipes, respectively. The transition flow is being delayed in curved pipes as compared to straight pipes. This is because of the suppression of turbulent eddies by secondary flow. As pointed out by Ito (1959), if the curvature is marked, the distribution of velocity in a curved pipe is entirely altered by the secondary flow, the
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
maximum axial velocity being nearer to the outer side wall. The secondary flow takes place chiefly in a sort of boundary layer in the region near the wall normally known as shedding layer. Assuming lI7th power law and logarithmic velocity distributions in the shedding layer for incompressible Newtonian fluids and for smaller D t / D , ratios, Ito (1959) simplified the momentum integral equations for the shedding layer. The friction factor for turbulent flow f, was found to be a function of Reynolds number NRe and the curvature ratio (Dt/D,). For llith power law velocity distribution in the shearing layer, the following expressions for shedding layer thickness and friction coefficient were presented
