Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
349
Head Loss and Critical Reynolds Numbers for Flow in Ascending Equiangular Spiral Tube Coils Shaukat Ali" and Anwar Haider Zaidl Department of Chemical Engineering, University of Roorkee, Roorkee, India
Generalized pressure drop correlations have been obtained for steady Newtonian flow in increasing pitch spiral tube coils called ascending equiangular spirals. The characterizing geometrical group has been obtained. The occurrence of the first critical Reynolds number is quite sharp but the transition from the second critical Reynolds number zone to fully turbulent flow zone occurs smoothly. The first critical Reynolds number has been correlated and seems to depend upon the curvature ratio, dlD,,, where d is the inside diameter of the tubing and D,, is the sum of the maximum and minimum radii of the coils. A comparison has been made for the pressure drop performance of ascending equiangular spiral coil with an Archimedian spiral coil and it is found that the ascending equiangular spiral possesses lesser resistance to flow at high Reynolds numbers.
Introduction Most of the earlier hydrodynamic studies of coil flows have been done for helical coils. Relatively, much less work has been reported for spiral coil flows. The spiral and helical coil flow differ with each other on two accounts. Coil curvature remains constant throughout the length of helical coils giving rise to fully developed downstream flow. This is not the case for flow in spiral coils where curvature keeps on varying continuously along its length and hence giving rise to continuously varying intensity of secondary flow along the length. Thus the flow is never fully developed in spiral coils. Secondly, there exists a need to specify two critical Reynolds numbers distinguishing the flow regime for spiral coils instead of one needed for helical coils and straight tubes. As the Reynolds number is slowly increased, in a spiral coil completely filled with laminar flow, the turbulent flow zone first appears at the outer end a t the approach of the first critical Reynolds number and then stretches itself toward the inner end of the coil until it completely fills the coil with turbulent flow at the second critical Reynolds number. The hydrodynamics of coil flows has been discussed in detail by Ali (1974). Published work on the hydrodynamics of spiral coil flows has been limited to spirals of constant pitch called Archimedian spirals. Based on the pressure drop data of Noble et al. (1952), Kubair and Kuloor (1966) obtained correlations. However, these correlations failed to correlate their own data and they instead proposed the following equations for the Fanning friction factor fa,
= 12.74 [ d2/LDav] 0'3N~c0'5
Extending their proposed correlation for helices, Kubair and Kuloor have also proposed the following modified form of the equation to predict the critical Reynolds number for spiral coil flows NRe,crit = 12730(d/D,v)02
For defining the coil geometry uniquely, the tube inside diameter, d, plus three of the following geometrical variables, the minimum coil radius, R,,,, the maximum coil radius, R, the number of turns, n, the lengths of the coil, L , and the pitch of the coil, p , are necessary. None of the above three correlations proposed by Kubair and Kuloor accounts for all four required geometrical variables. Moreover, these equations are based on the arithmetic mean of the diameters of the first and the last turns of the spiral without specifying pitch and hence there are many possible spirals having the same length and average diameter but different inside and outside radius. It is obvious that the above correlations will not hold for all of these possible coils. Ali (1969) and Ali and Seshadri (1971) avoided the mistake of putting the pressure tapping within the coiled length of the spiral as done by Kubair and Kuloor and accounted for the effect of variation of all the required geometrical parameters to come to the correlations: laminar flow
Eu.GaS = 49NR;O6'
(1)
Eu-G,, = O.65NRe4"
+
(4)
NRe > 10000
(5)
where Eu = P / ( 2 p P ) is the Euler number and the geometrical group Gas is given by
and = 0.0791NRe-0.25 0.1025(d/D,v)0.9
NRe < 6000
turbulent flow
300 < NRe < 7000
fa,
(3)
(2)
for turbulent flow Reynolds numbers, where fa, is the Fanning friction factor for Archimedian spiral coils, d is inside diameter of the coil tube, D,, is average coil diameter, NRe is the Reynolds number of flow, and L is the length of the coil.
They also noticed the presence of two critical Reynolds numbers instead of one and correlated them as NRe,crit I = 2100[1 + 4.9(d/Rmax)o.21(~/Rm,)o.1]
NRe,crit11 = 2100[1 + 6.25(d/R,in)0~'7(p/Rmin)o~1] (8) Srinivasan et al. (1970) have extended their correlation for helices to that for spiral coils considering a spiral to be a two-dimensional helix with coil diameters varying
*Address correspondence to this author at the DRPD Division of Research and Development Centre, Steel Authority of India Limited, Ranchi-834 002, India. 0019-7882/79/1118-0349$01.00/0
(7)
C 2
1979 American Chemical Society
350
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
H
I
Iki
W
Figure 2. Geometry and Configuration of an ascending equiangular spiral tube coil.
Figure 1. (a) Archimedian spiral (R = a0). (b) Ascending equiangular spiral ( R = Kerns).
from point to point. This leads to the following correlations: laminar flow 0.62(n20.7- n10,7)2,0 fas
[ 500
;)"il"i
=
(9) C-CML T-TANKS P-PUMP R-ROTAMETER M-Hg & CCld MANOMETERS
NRe(
< NRe(2p/d)O" < 20000; 7.3 < (2p/d) < 15.5
transition flow
),y5
0.52(n20.7s- n10,75) fa,
[ 20000
$
=
(10)
NRe(
< NR,(2p/d)0.5< 40000; 7.3 < (2p/d) < 15.5
turbulent flow 0.0074(n,0.9- n1°'9)1'5 fa,
=
[ ;) "1
(11)
o.2
40000 < N ~ , ( 2 p / d ) O< . ~15000; 7.3
< (2p/d) < 15.5
where n, = [ L / ( v n )- nl/2
n2 = [ L / ( v n )+ nl/2 (12) It is pointed out that the dimensionless number NRe( 2 ~ / d ) Ois. ~not a characteristic for flow in spirals as the dimensionless number NR,(D/d)"j is for flow in helices. for helices falls on a single A plot off, against N&(D/C~)O.~ line. This is not the case for a plot of fa, against NRe( 2 ~ / d ) Ofor . ~ spirals. The two critical Reynolds numbers have been correlated by extending the correlation for helical coils as
+ 11 = 2100[1 + 12id/(2Rmin)Io~51
NRe,crit I
NRe,crit
= 2loo[l
(13) (14)
Figure 3. Experimental setup.
and thus do not depend upon the pitch of the spiral. Apart from the Archimedian spirals there are many other categories of regular spirals. For them, the pitch does not remain constant but varies along the length of the spiral. In order to study the effect of variation of pitch on the hydrodynamics of spiral flow, the present investigation deals with Newtonian fluid flow through ascending equiangular spiral tube coils. The shape of Archimedian spiral and ascending equiangular spiral curves are shown in Figure 1. The ascending equiangular spirals are true spiral curves belonging to the family of curves called logarithmic spirals described by the polar equation
R
=A
+ Kerns
(15)
with A = 0 and positive values of 0, eq 15 describes ascending equiangular spiral curves. Zaidi (1977) has discussed the geometrical properties of these curves in detail. Figure 2 shows geometry and configuration of ascending equiangular spiral tube coil. Experimental Section The experimental setup is shown in Figure 3. Coils were made by winding thick-walled flexible polyethylene tubing in the shape of ascending equiangular spirals on a wooden plank and fastening them by means of tin clips and nails. Coils were provided at both ends with tangential straight lengths so that initial entry disturbances could be decreased in the coil and exit secondary flows could be decreased in the straight length before the manometer
Ind. Eng. Chem. Process Des. Dev., Vol. 18,No. 2, 1979
Table I.
Dimensions of Coils and Symbols Used
coil sym- R, no. bo1 cm 1 2 3 4 5
x o
+ A
v
30.0 22.9 22.9 22.9 22.9
Rmh, cm
d , cm
m
5.0 5.0 6.6 5.0 5.0
0.603 0.603 0.603 0.603 0.464
0.0441 0.0441 0.0441 0.0691 0.0441
,
24,
,
I
351
,
I
parameter changed
R,,
---
R,,
m
d
Log
f
N R 1~
Figure 5. Generalized pressure drop correlation. 02
33
c2
I
34
I
I
I
I
35
36
37
38
Log I
I
I
'
I
39
LO
L1
12
I
Figure 6. Prediction of critical Reynolds numbers.
Figure 4. Pressure drop-flow rate data.
taps. Pressure drop data for an equal straight length of the same tubing were obtained and subtracted from the pressure drop data of the coil-straight length combination a t the same flow rate to give the pressure drop data for the coiled length only. A carbon tetrachloride manometer was used for smaller pressure drops and a mercury manometer for larger pressure drops. Water a t ambient temperature was used as the test fluid. The inside diameters of the tubings were accurately measured by means of a traveling microscope by cutting the tube at different lengths and measuring at two mutually perpendicular diameters. These observed values of diameters were further checked by calculated values obtained by taking the weight of distilled water filled in known lengths of tubing. Visual inspection revealed no deformation of the tubings during the winding and experimentation. Flow rate and pressure drop data were obtained for a set of five coils. Geometries of these coils were such that each of the geometrical parameters, R,,,, the maximum radius vector of the outermost turn, R,,,, the minimum radius vector of the innermost turn, m, the parameter specifying the pitch variation, and d , the inside diameter of the tubing, changes once with respect to a particular coil. These four geometrical parameters, R,,,, R,,,, m , and d, are sufficient to uniquely specify the geometry of the coils. Table I gives the dimensions of the coils used in experimentation and symbols used for them for plotting in the subsequent figures. Results and Discussion Pressure Drop Correlation. The graphical method of appropriate shift was used for obtaining the pressure drop correlations. First a log-log plot of the Euler number vs. dimensional group p V / p was obtained from the pressure drop-flow rate data in Figure 4. Neither of the
groups Euler number and p V / p contains any geometrical parameter. When the Euler number axis was shifted by
by multiplying the Euler number by the geometrical group (16)
and the p V / p axis was shifted by log d by multiplying the group p V / k by d , a satisfactory merging was obtained as shown in Figure 5. The data points in Figure 4 show a sharp break at the onset of the first critical Reynolds number. The nature of the plot in Figure 5 further shows that a straight line fit can be obtained for the laminar flow region and the turbulent flow region separately. A least-squares fit results in the correlations: laminar flow
NRe < 5000
with a maximum deviation of 2.8%; turbulent flow
NRe > 8000
with a maximum deviation of 5.2%. Critical Reynolds Numbers. The method of White (1929) and Adler (1934), originally suggested for curved pipes, was used for the location of critical Reynolds
352
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
Table 11. The First Critical Reynolds Number NRe,crit I
coil no.
d , cm
1
0.603 0.603 0.603 0.603 0.464
2 3 4 5
R,
cm 30.0 22.9 22.9 22.9 22.9
D,,, cm
pmax,cm
obsd
eq 3 (KK)
35.0 27.9 29.5 27.9 27.9
7.84 5.10 5.47 8.00 5.10
5960 6750 6450 6600 4650
5650 5912 5847 5912 5612
eq 7 (SA)
eq 1 3 (SNH)
eq20
6060 6240 6270 6431 6002
4626 4992 4992 4992 4636
5717 6636 6384 6636 5586
numbers. This consists of plotting the ratio AP,/AP, vs. N,, where aPc and AP,are the pressure drops at the same Reynolds number in a coil and a straight tube, respectively, of equal length and cross-sectional diameter. The pressure drops for straight tubes, AP,,are calculated from the Blausius formula f, = 0.0791NR;0.25
2100
< NRe
