22
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
Flow of Non-Newtonian Fluids through Helical Coils B. A. Mujawar and M. Raja Rao“ Department of Chemical Engineering, indian institute of Technology, Bombay-400 076, india
Studies on isothermal frictional pressure drop were carried out for flow of water and several pseudoplastic polymer solutions in helical coils of curvature ratios 0.0695, 0.0476, 0.0198, and 0.0100. The criterion for laminar flow in coiled tubes was established on the basis of a new dimensionless number, M, deduced from a knowledge of the effect of coil curvature ratio on the flow curves. Suitable correlations for friction factors in coiled tubes are proposed.
Introduction Non-Newtonian fluids are often encountered in processing industries such as rayon, plastics, foods, dye-stuffs, and pharmaceuticals. Helical coils are frequently used for transferring heat in mixing tanks and reactor vessels as well as in process heat exchangers. Curved flow passages can be used to improve mass transfer rates, such as in membrane blood oxygenerators, in kidney dialysis devices, and in reverse osmosis units. In many technological processes, the process fluids are often pumped through pipes which are sometimes curved. In some cases, the use of curved tubes is necessiated because of geometrical restrictions. To analyze the phenomena of heat and mass transfer in helical coils, it is necessary to know the hydrodynamics of the flow in such geometries. Review of Literature A critical review of both theoretical and experimental studies on the flow of non-Newtonian fluids through helical coils has been reported recently by Mashelkar and Devarajan (1976a,b).However, the present review attempts to highlight some of the shortcomings of the previous published work. Mashelkar and Devarajan (1976a) recently solved the problem of laminar flow of a power law type of non-Newtonian fluid in a coiled tube in the modified Dean number, De’ range of 70 to 400, using a boundary layer approximation. A convenient form obtained theoretically and tested experimentally (197610) is f c = [9.069 - 9.438(n’)
+ 4.374(n’)2](~/R)@.~(De’)-o 768f0m2n’
(1)
applicable for (70 IDe’ 5 400), (0.01 I( u / R ) 5 0.135), and (0.35 In’ I1). The results of f c predicted by eq 1for Newtonian fluids (n’ = 1)are lower than those of White’s (1929) equation. It is not clear from their report whether they had provided any calming section in their experimental work. The power-law model is applicable for laminar flow in straight circular tubes and it should be modified for coiled tube flow. Further, eq 1does not cover the entire range of laminar flow in coiled tubes. Rajasekharan et al. (1970) presented experimental data on the flow of aqueous solutions of 0.5 and 1.0% CMC and 1.0% sodium silicate through coiled tubes of curvature ratios 0.037, 0.056, 0.074, and 0.097 and obtained the following correlation
(k)
= 1.23 (De’)@
(2)
The most serious drawback of the above correlation is that it does not reduce to the correlation for Newtonian fluids given by the equation
(k)
= 0.37(De)0,36
(3)
Equation 3 was misquoted by these authors by removing the 2 in the denominator. The correct form of Prandtl’s (1928, 1952) equation is (4a) i.e. = 0.288(De)0.37
(4b)
f s
Further, the coefficient and exponent of the modified Dean number De’ in eq 2 have taken constant values, which seems to be incorrect according to eq 1.From eq 1,it is quite evident that the coiled tube friction factor is a function of Reynolds number, coil curvature ratio, and flow behavior index, n’. Gupta and Mishra (1975) studied the flow of aqueous CMC solutions in the concentration range of 0.602 to 2.181% using only two coils of curvature ratios of 0.0395 and 0.076, and obtained the following correlation (fc/fs) =
1
+ 0.026(De’)0,6i5
(5)
The same objections regarding constancy of coefficient and exponent raised against eq 2 are also applicable to eq 5. The above literature survey reveals the following points. (1) A new flow model is required to be developed to better understand the complex flow in helical coils. (2) Since neither the Reynolds number nor the Dean number could characterize the hydrodynamics of flow through helical coils, a new dimensionless parameter is required to be established. (3) Most of the theoretical work has been carried out for a low range of Dean number, which does not cover the entire laminar flow range in coiled tubes. (4) The criterion for laminar flow of non-Newtonian fluids through helical coils must be established. (5) It would be interesting to find out the lower limit of the coil curvature ratio below which the coiled tube would behave like a straight tube. The present work was undertaken to resolve the above points.
Experimental Work The experimental work was carried out with the object of studying the isothermal frictional pressure drop a t 30 “C for the flow of Newtonian and non-Newtonian liquids through straight and coiled tubes. A schematic diagram of the experimental setup is shown in Figure 1. The geometrical details of the test sections are listed in Table I. The coils were constructed by winding a 1.21 cm i.d. polythane tube tightly over the mandrels of required diameters fabricated from mild steel sheets and were held in position by means of spacers. A wall thickness of 3.2 mm was chosen
0019-7882/78/1117-0022$01.00/00 1978 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
23
LEGEND No.(
Unit
1 Water t a n k 2 Solution t a h k
3 Centrifugal pump
From cold water line
Figure 1. Flow diagram of the experimental setup. Table I. Geometrical Details of the Test Sections Helix diameter of coil, 2R
Coil curvature ratio, (a/R)
Pitch, cm,
Sr. no.
Inside diameter of tube, 2a
1
1.21
2 3 4 5
1.21 1.21
1.21
17.4 25.4 61.0 121.0
0.0695 0.0476 0.0198 0.0100
1.9 2.6 5.7 13.6
1.21
m
0
P
-
for the polythene tube to ensure that the tube remains circular in cross section when wound around the mandrels. The maximum curvature ratio of the coil was restricted to 0.0695, as it was observed that the tubes tend to become slowly elliptical in cross section a t still higher curvature ratios. The first and the last two turns of the coiied tubes were provided respectively as upstream and downstream calming sections in accordance with the equation of Austin and Seader (1974),given by YD
= 49 [De(a/R)I1/3
(6)
The test solutions used were dilute aqueous polymeric solutions of sodium alginate (SA) and sodium carboxy methyl cellulose (SCMC) in the concentration range of 0.3 to 1.0% (w/w). These dilute solutions were found to be nonviscoelastic as per the test suggested by Suryanarayanan et al. (1976). Dilute sulfuric acid was used to clean the inside surface of the polythene tube after using each dilute aqueous polymer solution and then the tube was flushed with water for about 3 h. The test solution was admitted into the test section at the desired flow rate a t 30 f 1 "C. After attaining steady-state conditions, pressure drop was measured by means of a U-tube manometer, and the test liquid flow rate was obtained by weighing the effluent liquid collected in a definite time interval. This procedure was repeated for different flow rates with each of the test liquids. Results and Discussion As a part of the standardization of the experimental setup, experiments were carried out in laminar and turbulent flow
Angle of elevation,
Number of turns, n
Test section length, cm, L
3
8 6 4 2
437 480 766 770
PO 3 3
3
-
-
262
of water in four helical coils. The data points in laminar flow were well correlated by the equation
(k)
= 0.26(De)0."6
(7)
over the range (35 IDe I 2200); (0.0100 5 ( a l R )5 0.0695). The coefficient 0.26 in our eq 7 agreed closely with 0.288 of Prandtl's (1928) eq 4b. Further, an attempt was made in Figure 2 to compare our eq 7 with the results available in the literature. It is quite clear that the proposed eq 7, based on the present work, gives excellent agreement with the earlier correlations. Excellent agreement between the results of present work with water and Ito's turbulent flow correlation
fcm = O . O ~ ~ [ R ~ ( U / ~ ) ~ ] - ~ ) . ~ "(8) was observed. (i) Flow Curves. It would be quite interesting to know the nature of the flow curve for flow of Newtonian fluids through helical coils, using the usual defining relation of wall shear stress, T~ = ( d U ) / 4 L and average shear rate, 7 = ( 8 V / d ) .The relationship, 7 = ( 8 V / d ) is strictly applicable for flow of Newtonian fluids in a straight tube of circular cross section. Due to the complexity of defining the average shear rate for coiled tubes for flow of Newtonian fluids, the straight circular tube relationship for average shear rate was used so that it serves as a basis of comparison. The variation of wall shear stress, T ~ with , ( 8 V l d )and ( a / R ) is shown for a typical aqueous solution of 0.50/0sodium aliginate in Figure 3. The data points of laminar flow in a straight tube were well fitted by a straight line whose slope and intercept give a flow behavior index, n,' and consistency index
24
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
- Present 9 White + Prandtl -
A
m X
0
D
work,(fc/fs)
i
0.26
It0
Gupta 8 Mishra Mashelkar P D e v a r a j a n Tarbell e Samuels Truesdell & Adler Mori E NaKayama Akiyama & Chong
I
X
5x16~ 102
30
t 0.11 1
'
'
'
1
' '
'$,'I
-
"
10
""1
lo3
relationship between sw,,(8V/d),and ( a / R )for laminar flow might be represented by
Figure 2. Comparison of laminar flow friction factors iii coiled tubes of present work with the results of previous investigators for Newtonian fluids.
Kd. The flow curves for coiled tubes are separated from that
Tw,
ofthe straight tube and the enhancement in pressure drop was found to be a function of ( a / R )and the nature of the fluid. A cross plot of rw,vs. ( a / R )at fixed values of (8Vld)in laminar flow is shown in the right-hand portion of Figure 3. Figure 4 shows a correlation plot of T ~ ( 8~V /, d ) ,and ( a / R ) .Similar correlation plots were made for other test fluids (not shown). (ii) Flow Model. From these experimental flow curves for Newtonian and non-Newtonian fluids in coiled tubes, the
=
d AP (a) K,' =
8V
a
(yyC' (E)
LEGEND
A - A
8 - 8
0,154
It
16'
I
I
I
I
I
I
#,,I
I
IO'
I o3
8V/d,
s-'
me'
(9)
for (0.010 I( a / R )5 0.0695). The constants Kc', n,', and m,' so determined in coiled tubes are listed in Table IT. By definition, the friction factor is given by
LEGEND
10
loL
Figure 4. Correlation between 7wc,(8V/d),and ( a / R )for laminar flow of 0.5%aqueous sodium alginate.
I
10 2
De
lo3
Wid, 5-1
I
-
I 1o4
Figure 3. Variation of wall shear stress T~~ with (8Vld) and ( a / R )for flow 0.5% aqueous sodium alginate a t 30 "C.
E
