Chapter XVI Longitudinal Fins and Twisted Tapes within Ducts Longitudinal fins within ducts are widely used in compact heat exchanger applications [6]. Such ducts are also referred to as internally finned tubes. Laminar fully developed convection in tubes with longitudinal rectangular and triangular fins has been studied analytically, employing some simplified idealizations. In this chapter, the "fin efficiency" is considered as 100% for allfingeometries, except for the twisted tapes. For the flow area evalua tion, thefinthickness is considered as zero for all longitudinal thin fins. These idealizations are needed to make the problem amenable to mathe matical analysis. Additionally, the fluid properties are treated as constant. Fully developed and developing laminar flow forced convention problems have been analyzed for the circular tube having a twisted tape. Only fully developed laminar fluid flow and heat transfer problems have been inves tigated for all other geometries considered. The friction factorReynolds number product and Nusselt numbers for internally finned tubes, designated as (/Re) d and Nu bcd , are defined using a freeflowarea and a hydraulic diameter identical to those of the corresponding finless tube. For example, fd and Red of (/Re) d and NubCfd are computed from Λ = Δ ρ * ( ^  ρ ) = (^)(^c2Dh)finless
Re d = ^  H μ
XT
NUb =
=
 ^^ö
=
(543)
^ Y ^
\μΙ\Α e/finless
«"Aitale» " ^h,tinless _
q' Ί
2
(D\ / ~ n
l
4^^jl^' 366
(542)
icAA\
(544)
A.
THIN FINS WITHIN A CIRCULAR D U C T
367
where the formula for Nu bc d is intended for the axially constant heat flux boundary conditions. The reason for this choice is that finned and finless tubes would occupy the same space in a heat exchanger; hence one can compare how much better the finned tube performs relative to the finless tube. For longitudinal fins within a tube, (/Re) d and N u b c d based on the finless tube geometrical dimensions, and / Re and Nu bc based on the actual geometrical dimensions, are related to each other as follows : (/Re) d = ( / R e ) f ^ i ^ Y i ^ h A V^hjinned/
Nu bc , d = N u b c (^ î ! s ! î » Y ^ Ι Η Η Λ νΛι,Πηηεα/
(545)
\^c,finned/ (546)
\^c,finless/
Note that the flow area ratios are generally different from unity; only for the zero thickness will the flow area ratios be unity. A. Longitudinal Thin Fins within a Circular Duct Hu and Chang [129,442] analyzed fully developed laminar flow through a circular tube having n longitudinal rectangular fins equally spaced along the wall, as shown in the insert in Fig. 109. For the velocity problem, they considered the usual noslip boundary condition all along the tube wall and the fin surface. For the temperature problem, they employed the (H2) boundary condition, namely, constant heat flux axially as well as peripherally along the tube wall and fins. The fin efficiency was treated as 100%, and the fin thickness was treated as zero for the flow area calculations and velocity profile determination. Hu and Chang obtained the solutions to the differential momentum and energy equations in the forms of integral repre sentations by means of Green's function. The integral representation for the velocity involves a boundary velocity from the tube center to the fin tip, which is defined by an integral equation. The integral equation for the boundary velocity was solved numerically. The (/Re) d factors and Nu H2 , d of Hu and Chang [129,442] are presented in Tables 127 and 128 and Figs. 109 and 110. Note that (/Re) d /16 and NuH2>d/4.364 are used as the ordinates of Figs. 109 and 110 in order to provide a ready indication of the increase in (/Re) d and Nu H2 , d for the finned geometries relative to the unfinned duct. The actual hydraulic diameter Dh of the finned tube is related to the inside diameter of the tube and the number of fins of the height / as follows (for the fin thickness treated as zero):
D
— = J^i
(547)
368
XVI.
LONGITUDINAL FINS WITHIN
DUCTS
TABLE 127 LONGITUDINAL THIN FINS WITHIN A CIRCULAR D U C T : ( / R e ) d FOR FULLY DEVELOPED LAMINAR FLOW (fRe) d
n
from Hu [443]
I* = .2
0.4
0.6
0.7
2 8 12 16 20
17.28 21.22
20.83 42.87
27.42 1 0 1 . 10
31.89 139.55
22 24 28
25.99
1 32 n
1 4 8 12 16 20 24
0.79
0.795
35.68 161 .03 434.40 607.72
69.57
219.54
91.65
3(fRe) 7 2 . 3 7d
348.86
701.75 1221 .0
30.4 3 I*
0.4
= .2
29.04 47.02 64.22 77.86 88.04 95.57
19.13 22.39 25.55 28.20 30.33 32.01
from 7 7 3 . 6Masliyah 9
35.98 162.03 286.66 439.37 616.52
0.8
0.9
36.64 164.84
40.54 172.70
448.43 6 3 2 . 11
732.60 712.76 838.23 813.67 1062.7 1025.6 1298.7 1251 .6
[516]
0.6
0.7
0.8
50.31 110.81 179.77 246.08 305.47 356.04
59.31 137.35 233.09 336.15 439.13 537.09
73.43 170.4 7 303.16 469.92 668.23 894.15
48 1.12
1546.8 1
0.9
1.0
491.04 708.29 965.09
77.26 175.96 314.99 495.24 716.79 980.77
100
50
20
(fRe)d 16
.0 5
2 I
0
4
8
12
16 Π
20
24
28
32
FIG. 109. Longitudinal thin fins within a circular duct: (/Re) d /16 for fully developed laminar flow (based on the results of Table 127).
Hu and Chang found the optimum value of Nu H2 , d to be 86.82 for l/a = 0.795 and n = 22. For this fin geometry, (/Re) d is 712.76. Thus the ratio of Nu H2 , d to (/Re) d is 0.116 for this socalled optimum geometry. This ratio for the finless circular tube is 0.273 (4.364/16). Clearly, only when the pressure drop
369
A. THIN FINS WITHIN A CIRCULAR DUCT TABLE 128 LONGITUDINAL THIN FINS WITHIN A CIRCULAR D U C T : Nu H1
d
AND Nu H2>d FOR
FULLY DEVELOPED LAMINAR FLOW NUH2
n 2 8 12 16 20 22 24 28  32
n 4 8 12 16 20 24
,d
fr
°m
Hu [443]
I* = .2
0.4
0.6
0.7
0.79
4*25 4.27
4.32 4.67
4.88 8.66
5.38 16· 79
6*11 29*49 72*66 81*89
4*12
4.04
7.29
3*84
3.39
N U 4.10
21*65
84*11 55.76
I* = .2
0.4
4*58 4.74 4.77 4.74 4.68 4.62
6.05 6.98 6.65 6.09 5.64 5.32
40 I
ι
Hl,d
from 8*62Masliyah
0.7
0.6 11*82 21· 10 20.52 16*22 12·73 10.41
1
ι

0.8
15·34 34·27 40*92 34*45 26*07 19*83
1

19.30 42.58 72.2 7 106.50 138*35 156*92


l
0.795
0.8
6*16 30*10 53*65 73*48 83*60
71.06 80.41
86*82 85*00 75*32 62*43
84.02 83.70 78.06 67.05
6.23 30.65
[516]
6.93 27.26
31.85
25.15
0.9
1.0
105*00 147.20 195.40
19.08 40 .68 68.80 103.40 144.60 192.40
1 J'
0.9

1
1 Γ
FIG. 110. Longitudinal thin fins within a circular duct: Nug l i d and Nu£2,d for fully developed laminar flow (based on the results of Table 128).
370
XVI.
LONGITUDINAL FINS WITHIN DUCTS
is of no concern does the foregoing finned tube geometry offer optimum heat transfer (in the case of fin efficiency approaching 100%). Hu and Chang also examined the effects of viscous dissipation and thermal energy generation within the fluid on the basis of the same total heat transfer rate without these effects. They found that the effect of viscous dissipation is insignificant in all cases. However, a superimposed rate of thermal energy generation appreciably decreases the Nusselt number. For Sa/qw" > 2.4, their optimum number of fins is reduced from 22 to 16. As described on p. 375, Masliyah and Nandakumar [128] analyzed the (Hi) boundary condition for the longitudinal triangular fins. When the angle 2φ approaches zero (Fig. 117), the longitudinal triangular fin geometry reduces to the longitudinal rectangular fin geometry analyzed by Hu and Chang. Hence (/Re) d and N u H M for this geometry, obtained from Masliyah [516], are presented in Tables 127 and 128 for the case of 2φ = 0°. (/Re) d /16 and Nu H1 d/4.364 are presented in Figs. 109 and 110. Some interesting contrasts with the (H2) solutions are evident. The comparison in Fig. 109 shows that (/Re) d of Hu and Masliyah agree within about 5% for /* > 0.7 and within about 11% for /* < 0.7. The differences are due to different numerical methods employed. B. Longitudinal Thin Fins within Square and Hexagonal Ducts Chen [133] extended the study of Hu [129] by considering longitudinal fins within a duct of arbitrary cross section. By the use of exact or approximate conformai mapping functions, a circle can be mapped onto the desired arbitrary cross section. Knowing the solutions for the circular tube, the solutions for the arbitrary duct geometries can be obtained with appropriate transformations. Chen employed the method of conformai mapping and Green's functions to obtain numerical solutions for square and hexagonal ducts with longitudinal fins. These ducts have rounded corners and nonstraight sides as shown in Fig. 111. If the square duct of Fig. 11 la is approxi mated as having straight sides with rounded corners (radius = 0.40a), the
(a)
(b)
FIG. 111. Longitudinal thin fins within square and hexagonal ducts.
B. THIN FINS WITHIN SQUARE AND HEXAGONAL DUCTS
371
hydraulic diameter of this duct with 8 and 16 fins is given by oh 2a
7.72531 9.30150+ 16/*'
for
n= 8
(548)
7.72531 10.6198 + 32/*'
for
n = 16
(549)
The effect on the hydraulic diameter of rounding of the corners of the hexagonal duct is negligible. Thus, using sharp corners and straight sides approximations, the hydraulic diameter of the hexagonal duct (Fig. 111b) with 12 fins is given by 2a
6.92820 8.78461 + 2 4 / * '
for
12
(550)
Chen [133] presented / R e and Nu H2 , d for these duct geometries. By employing the relationship of Eq. (545), the ( / Re)d factors were calculated by the present authors as presented in Table 129 along with Nu H2 , d . TABLE 129 LONGITUDINAL THIN FINS WITHIN SQUARE AND HEXAGONAL DUCTS : ( / Re) d AND Nu H2 , d FOR FULLY DEVELOPED LAMINAR FLOW (FROM CHEN [133])
square duct n = = 8 l* 0.2 0.4 0.6 0.7 0.8 0.9
(fRe)d 25.59 48.00 105.4 138. 6 157.5 161.4
NU
square duct n = 18
H2,d
4.0 1 4.32 8.34 17.21 28.30 28.04
(fRe)d 43.75 70.52 140.2 175.5 188.4 184.6
Nu
H2,d
hexagona 1 duct n = 12
3.74 4,57 13.51 51 .83 76.0 34.78
3 0 . 14 67.18 1 75.1 247.2 291.7 302.7
Nu
H2,d
3.73 3.96 8.17 23.28 49 .53 35.90
Gangal and Aggarwala [131] analyzed fully developed laminar combined free and forcedconvection flow through a square duct having four equal internal fins as shown in Fig. 112. They considered constant heat flux axially with peripheral constant wall and fin temperatures. The fin efficiency was considered as 100%. Thefinthickness was idealized as zero in the calculations of the flow area, and the velocity and temperature distributions. They
H
1
a
2a FIG. 112. A square duct with four equal longitudinal thin fins.
372
XVI.
LONGITUDINAL FINS WITHIN D U C T S TABLE
no
LONGITUDINAL FOUR THIN FINS WITHIN A SQUARE D U C T : ( / R e ) d AND Nu H1 d FOR FULLY DEVELOPED LAMINAR FLOW (FROM GANGAL AND AGGARWALA [130])
!
(fRe)
a 0 0.125 0.250 0.375 0.500 0.625 0.750 1
d 14.261 15.285 18.281 23.630
NU
31.877 42.527 52.341 56.919
7.309 11.096 14.025 14.431
Hl,d 3.609 3.721 4.160 5.172
employed their method in [130] to analyze the square duct problem and obtained numerical values for 2(/Re) d and Nu H i, d for /* varying from 0 to 1. Their (/Re) d and N u H l d are represented in Table 130. C. Longitudinal Thin Fins from Opposite Walls within Rectangular Ducts Aggarwala and Gangal [130] analyzed fully developed laminar combined free and forcedconvection flow through a rectangular duct having internal fins as shown in Fig. 113. They considered constant heat flux axially with peripheral constant wall and fin temperatures. The fin efficiency was con sidered as 100%; the fin thickness was idealized as zero in the calculations of the flow area, and the velocity and temperature distributions. The coupled momentum and energy equations were combined in a nonhomogeneous Helmholtz wave equation in the complex domain by introducing a complex function. This complex function is directly related to the velocity and temperature distributions. The solution of the equation gives rise to dual series equations, which are reduced to Fredholm integral equations of the second kind in the complex domain. These latter equations were solved numerically. Aggarwala and Gangal presented 2(/Re) d and Nu H1?d for (1) a square duct with two longitudinal fins, (2) a square duct with four longitudinal
Î 1 M H h*2—[~f K
 1—
■
I
2a
H
FIG. 113. A rectangular duct with four longitudinal thin fins from opposite walls.
C.
THIN FINS FROM OPPOSITE WALLS IN RECTANGULAR DUCTS
373
fins, and (3) finless rectangular ducts with 0.1 < a* < 1 [426]. Their / R e and Nu H 1 for finless rectangular ducts for a Raleigh number of 1 are within 0.24 and 0.03% of the exact values of Table 42. (/Re) d and Nu H1?d for the square duct with two and four internal fins for a Rayleigh number of 1 are presented in Table 131. Partial results from this table for fins with equal heights (^ = l2) are shown in Fig. 114. Gangal [426] obtained (/Re) d and NuHi,d for two fins with lt = l2 = a as 34.997 and 9.278, in comparison to the TABLE 131 LONGITUDINAL THIN FINS FROM OPPOSITE WALLS WITHIN A SQUARE D U C T : ( / Re) d AND Nu H l t d FOR FULLY DEVELOPED LAMINAR FLOW (FROM AGGARWALA AND GANGAL [131,426])
!± a
h. a
0.0 0.0 0*0 0.0 0*0
0.0 0.2 0.4 0.6 0.8
0*2 0.2 0.2 0.2 0.4
Four
Two f i n s (fRe)d
NU
(fRe)d
Hl,d
fins NU
Hl,d
14.261 14.853 16.329 18.506 21.32 5
3.609 3.683 3.932 4.395 5.151
14.261 15.265 17.737 21 . 4 9 7 26.797
3.609 1 3.684 3.934 4.398 5.232
0.2 0.4 0.6 0.8 0.4
15.483 17.055 19.368 22.340 18.860
3.766 4.044 4.557 5.385 4.407
16.388 19.166 23.408 29.385 22.719
3.777 4.081 4.646 5.674 4.549
0.4 0.4 0.6 0.6 0.8
0.6 0.8 0.6 0.8 0.8
21.472 24.741 24.466 27.917 31.366
5.058 6.067 5.926 7 . 108 8.288
28.132 35.652 35.313 44.579 55.398
5.430 7.069 6.970 9.752 14.136
1.0
1.0
34.983
9.277
68.359
19.179
a: yl
—j
L
/
/
NU
HI,CQ_
(fRe):
u
O.Ol j 0.0
vIfReL me Hi,d" 3 . 6 0 8 ' d "14.227 I i I i I i I i_ 0.8 I.0 0.2 0.4 0.6
l/Q FIG. 114. Longitudinal thin fins from opposite walls within a square duct: ( / R e ) J and Nugi
d
for fully developed laminar flow (based on the results of Table 131).
374
XVI.
LONGITUDINAL FINS WITHIN DUCTS
more accurate values of 34.983 and 9.277 based on Table 42. Similar results for four fins with lx = l2 = a by Gangal are 68.365 and 19.179, in comparison to the values of 68.359 and 19.179 based on Table 42. D. Longitudinal Thin VShaped Fins within a Circular Duct Kun [132] analyzed fully developed laminar flow through a circular duct having longitudinal Vshaped thin fins, as shown in the insert in Figs. 115
0.4 0.6 0.8 0.9 */*max FIG. 115. Longitudinal thin Vshaped fins within a circular duct: (/Re) d /16 for fully developed laminar flow (based on the results of Table 132). 4.0
3.5 ω 3.0 XJ
3
2.5
X
2.0 1.5 1.0
a7
S
0.4 0.8 0.9 J/'max FIG. 116. Longitudinal thin Vshaped fins within a circular duct: Nu H1 d/4.364 for fully developed laminar flow (based on the results of Table 132).
E. TRIANGULAR FINS WITHIN A CIRCULAR DUCT
375
TABLE 132 LONGITUDINAL THIN VSHAPED FINS WITHIN A CIRCULAR DUCT : ( / R e ) d AND Nu H1
d
FOR FULLY DEVELOPED LAMINAR FLOW
(FROM KUN [517]) Φ
I &
max
(fRe)d
NU
Hl,d
30
0.20 0.40 0.60 0.80
16.353 17.059 17.776 18.203
4.359 4.386 4.400 4.403
40
0.20 0.40 0.60 0.80
16.799 18.567 20.497 21.731
4.385 4.472 4.535 4.521
50
0.20 0.40 0.60 0.80
17.444 2 1 . 0 34 25.735 29.046
4.426 4.681 4.895 4.887
60
0.20 0.40 0.60 0.80
18.154 24.351 34.527 43.552
4.496 5.071 5.994 6.376
65
0.40 0.60 0.80
26.022 39.903 54.368
5.341 7.209 8.70 3
Φ
I
max
(fRe)d
NU
Hl,d
67.5
0.40 0.60 0.80
26.724 42.620 60.129
5.501 7.990 10.885
70
0.40 0.60 0.80
27.328 45.009 65.356
5.647 8.856 13.981
72.5
0.40 0.60 0.80
27.810 46.607 69.095
5.768 9.771 17.218
75
0.40 0.60 0.80
28.072 47.342 70.942
5.872 10.469 18.010
77.5
0.60 0.80
47.078 69.702
10.787 16.353
80
0.60 0.80
46.070 65.779
10.477 13.967
and 116. He employed the (Hi) boundary condition for the heat transfer problem. He idealized the fins as 100% efficient, and considered the fin thickness as zero for the calculations of the flow area and the velocity and temperature distributions. Employing Green's function, Kun first presented the velocity and tempera ture distributions in terms of integral equations, and subsequently obtained the solutions numerically. He presented graphically Nu H1 as a function of the dimensionless fin height l/a with the fin angle φ as a parameter. His (/Re) d and Nu H l j d are presented in Table 132. The ratios (/Re) d /16 and Nu H1 d/4.364 are presented in Figs. 115 and 116, where 'max = 2a sin ( Dh
la 1 + (21/πα)
(551) (552)
E. Longitudinal Triangular Fins within a Circular Duct Masliyah and Nandakumar [127,128] analyzed fully developed laminar flow through a circular tube having n triangular fins equally spaced along the wall, as shown in Fig. 117. The flow area and wetted perimeter for this
376
XVI.
LONGITUDINAL FINS WITHIN
DUCTS
FIG. 117. A circular duct with n = 6 triangular longitudinal fins.
finned geometry are given by Achmed = ™2  n\a2 φ  a(a  I)sin φ] tinned = 2πα + 2πΓ  Ιηφα
(553) (554)
where /' = [a2 + (a
I)2  2a(a  /)cos(/>] 1/2
(555)
Then the hydraulic diameter can be calculated from its definition, Dhifinned = 4(^cAP)finned ·
For the velocity problem, they considered the usual noslip boundary condition all along the tube wall and the fin surface. For the temperature problem, they considered the (HI) boundary condition, namely, constant heat flux axially and uniform temperature along the tube wall and fins. The fin efficiency was considered as 100%. Note that the fin thickness was considered finite, as shown in Fig. 117. They divided the smallest symmetrical region of the tube into triangular elements and employed a finite element method to obtain solutions for the velocity and temperature problems. Nandakumar and Masliyah [127] presented a solution to the velocity problem for the limited range of /*, φ, and n. Their (/Re) d factors are presented in Table 133 and Fig. 118. Masliyah and Nandakumar [128] further extended their work in order to cover a wider range of parameters for / Re and also to obtain the (Hi) heat
E. TRIANGULAR FINS WITHIN A CIRCULAR DUCT
377
TABLE 133 LONGITUDINAL TRIANGULAR FINS WITHIN A CIRCULAR D U C T : ( / R e ) d FOR FULLY DEVELOPED LAMINAR FLOW (FROM MASLIYAH [516])
(fRe) d 2φ=12°
(fRe) d f o r 2φ = 3° n 4 8 12 16 20
1 24
A*=.l
ί,*=.2 £*=.4
l*=.6
i*=.7
18.27 19.36 20.36 21.25 22.02
19.20 22.56 25.78 28.50 30.61 32.26
51.73 118.4 198.3 277.4 348.0 407.3
69.56 174.2 326.3 523.5 759.4 1021.8
18.11 19.11 20.03 20.85 21.55
19.29 29.74 22.77 49.38 26.09 68.25 28.86 82.79 31.02 93.25 32.65 100.6
29.35 48.16 66.13 80.36 90.66 98.15
£*=.l
l*=.8 77.24 194.2 376.7 638.1 990.2 1440.5
(fRe) d f o r 2φ = 6° 4 8 12 16 20 24
53.25 126.9 219.2 311.8 392.7 456.8
18.27 19.32 20.27 Z*=.2
71.91 193.0 388.5 665.5 1013.3 1402.2
81.41 223.4 479.4 903.8 1560.9 2502.2
23.34 27.06 29.8 2 31.97 33.53
80 60 40 30 20
(fRe)d 16 I0
Mf>r""\ 0
■ i 4
8
i
i I2
i
i 16
i
i 20
i I 24
n FIG. 118. Longitudinal triangular fins within a circular duct: (/Re) d /16 for fully developed laminar flow (based on the results of Tables 127 and 133).
378
XVI. LONGITUDINAL FINS WITHIN DUCTS
transfer results. Their (/Re) d and Nu H1 d are presented in Tables 133 and 134 and Figs. 118 and 119 [128] The case of 2φ = 0° represents the longitu dinal fins of zero thickness and has already been discussed on p. 370. TABLE 134 LONGITUDINAL TRIANGULAR FINS WITHIN A CIRCULAR DUCT: N u H l d FOR FULLY DEVELOPED LAMINAR FLOW (FROM MASLIYAH [516])
a* 0.2
n
0.4 NU
4 8 12 16 20 24
4.58 4.73 4.75 4.71 4.64 4.58
4 8 12 16 20 24
4.58 4.71 4.73 4.67 4.60 4.54
H1, d
6.03 6.85 6.42 5.82 5.37 5.07 NU
0.6 0.7 for 2φ = 3° 11.72 20.01 17.82 13.15 9.95 8.02
18.02 43.26 64.58 63.21 46.17 30.86
0.8 19.29 43.60 76.17 112.11 131.90 117.50
H1, d for 2φ = 6°
6.00 6.71 6.19 5.57 5.14 4.87
11.63 18.87 15.30 10.64 7.91 6.42
17.82 43.31 58.88 45.28 26.75 16.01
19.29 44.88 80.81 112.45 98.20 56.04
40 30 20
10 8 6 NuH.,d 4.364 4 3 2
I "0
i
I 4
i
1 8
12
16
I L 20 24
n FIG. 119. Longitudinal triangular fins within a circular duct: Nu H l d /4.364 for fully developed laminar flow (based on the results of Tables 128 and 134).
F. CIRCULAR DUCT WITH A TWISTED TAPE
379
F. Circular Duct with a Twisted Tapef A twisted tape is sometimes inserted in a circular tube to establish swirl flow and thereby to increase the heat transfer coefficient on the inside tube surface. This tape, generally a thin metal strip, is twisted about its longitudinal axis as indicated in Fig. 120. In the analytical results presented below, the width of the tape is considered as equal to the internal diameter of the tube. When the tape twist ratio XL( = H/d) approaches infinity, the duct geometry becomes two semicircular straight ducts in parallel.
FIG. 120. A circular tube with a twisted tape. 1. FULLY DEVELOPED FLOW
a. Fluid Flow Date and Singham [99] and Date [100,101] employed a finite difference method and used the socalled upwind difference scheme for the grid points to analyze constant property fully developed laminar flow through a circular tube containing a twisted tape. They found that the flow with a twisted tape is quite different from the flow through a straight duct because of secondary or swirl flow v and w velocity components. The secondary flow velocity is largest in regions close to the twisted tape surface and decreases in the direction of tube surface. The magnitude of the secondary flow velocities increases with an increase in Red and a decrease in XL. Date [100] found it to be between 10 and 35% of the mean axial velocity. The secondary flow increases with the axial distance, reaching its maximum value when the flow is fully developed. Date and Singham [99] showed typical velocity profiles in twisted tape ducts with XL = 5.24, 3.14, and 2.25 for Red = 1200. The velocity profiles across the duct cross section are asymmetric. As the tape twist increases, they become flatter and flatter. At very high twists (low XL, e.g., 2.25), they disintegrate into a pattern that demonstrates two peaks of axial velocity. A similar phenomenon is observed at a constant tape twist ratio XL when Red is increased. +
The authors are grateful to Professors A. E. Bergles and A. W. Date for reviewing this section.
380
XVI.
LONGITUDINAL FINS WITHIN DUCTS
Date and Singham [99] showed that the friction factor depends upon the Reynolds number and the tape twist ratio XL. They were the first investi gators to recognize the parameter Re d /X L , which accounts for the centrifugal force effect. When small, the parameter Re d /X L represents an effect similar to that of the Dean number in curved pipe flow. However, for large values of Re d /X L , (/Re) d depends upon XL in addition to Re d /X L , as shown later. When XL ► oo, the limiting geometry consists of two semicircular ducts with / R e = 15.523. This is only 1.5% lower than the exact value of 15.767 (Table 77). For XL ► oo, Date and Singham correlated their numerical results within ± 5 % by Eqs. (557) and (558). Equation (556) is proposed by the present authors, based on the numerical results of Date and Singham and the theory for the straight semicircular duct: 42.23, (/Re) d = 38.4(Re d /Z L )
005
,
C(Re d /Z L ) 0 · 3 ,
for
(Re d /X L ) < 6.7
(556)
for
6.7 < (Re d /X L ) < 100
(557)
for
(Re d /X L ) > 100
(558)
where C = 8.8201XL  2.1193XL2 + 0.2108XL3  0.0069XL4
(559)
Equation (557) is a corrected version of the equation originally presented by Date and Singham [99]. In the foregoing correlations, it is idealized that the thickness of the twisted tape is zero. However, it may not be negligible if the tube diameter is small. The thickness of the tape reduces the flow area and increases the pressure drop. The foregoing correlations are modified by the present authors, as follows, in order to account for the finite thickness δ of the twisted tape : Î42.23C, for ReJXL < 6.7 (560) (/Re) d = J38.4(Re d /X L ) 005 C, [c(Re d /X L )°' 3 C,
for
6.7 < ReJXL < 100
(561)
for
Re d /X L > 100
(562)
where C is given by Eq. (559) and ζ is given by (Dh2Ac)ô = 0 = ( π y / π + 2  2ô/dV ( π \ ς { (Dh2Ac)^0 \π + 2)\ π4δ/ά J \π  4δ/dj The factor ζ was obtained from the ratio of (/Re) d δΦ0 to (fRé)dtô = 0. These quantities were determined from Eq. (545) with / Re considered as approxi mately constant^ =
f Note that the finite thickness of the tape would change the shape of a semicircular duct to a circular segment duct with the segment angle 2φ < 180°. However, a review of Table 77 reveals that the / R e factors are constant within 1% for the segment angle 2φ varying from 180 to 0°.
F. CIRCULAR DUCT WITH A TWISTED TAPE
381
Hong and Bergles [446,518] experimentally determined the friction factors for fully developed twistedtape flow with XL = 3.125 and oo by using ethylene glycol and water as the test fluids. They employed δ/d of 0.045, so that ζ = 1.152. Their resultsf are shown in Fig. 121. For XL = oo, these results are in excellent agreement with the theoretical predictions. For XL = 3.125, fd is lower than the theoretical value (about 17 and 30% at Red = 1000 for ethylene glycol and water, respectively). It is interesting to note that the theoretical increase in the friction factor above the semicircular duct value is only 40% at Red = 1000. As will be discussed later, the increase in the Nusselt number for XL < oo can be very large, depending upon the fluid Prandtl number. 50.0 THEORETICAL
20.0
X L =co, Eq.(560)
10.0
XL = 3.125 Eqs. (561 )~ and (562)
5.0
1.0 0.5 0.2 0.1
0.05
EXPERIMENTAL • XL = ω , Ethylene glycol o x L = 3.125, Ethylene glycol Δ XL = 3.125, Water
0.02
0.01 10° 2
5
m l _l io1 2
I
I I I 1 II
5
io 2 2
5
03 2
FIG. 121. Circular tube with a twisted tape : fd as a function of Red for fully developed laminar flow (from Hong and Bergles [518]).
b. Heat Transfer Date and Singham [99] and Date [100,101] considered a circular tube subjected to axially constant wall heat flux with a peripherally constant wall temperature at any cross section, namely the (Hi) thermal boundary condition. However, the tape was considered as a fin attached to and in perfect contact with the circular tube. Thus, in the heat transfer analysis, f Note that 4 ( / Re)d is 194.6 based on Eq. (560) and ζ = 1.152. Hong and Bergles [446, 518] considered the same flow area Ac with and without the twisted tape, so that ζ = 1.086 after neglecting the last term on the righthand side of Eq. (563). Hence they arrived at 4(f Re)d as 183.6.
382
XVI.
LONGITUDINAL FINS WITHIN DUCTS
the tape was treated as a fin with a peripherally variable heat flux and temperature. The resulting fin efficiency was found to be less than 100%. Because of this, the temperature of the tape differed from the temperature of the circular tube wall at a specified cross section. This boundary condition does not match any described in Chapter II ; thus, in order to avoid confusion, no boundary condition subscript is used for the corresponding Nusselt numbers. Date and Singham showed that the Nusselt number for the circular duct with twisted tapes is dependent upon four groups: Re d /X L , C fin , Pr, and XL, where Cfin = (kmô/k{d) and km is the thermal conductivity of the tape material. Cfin = 0 signifies that the tape is a heat nonconductor, and Cfin = oo signifies that the tape is a perfect heat conductor. A higher Re d /X L , Cfin, or Pr results in higher Nusselt numbers, while XL has a weak influence on Nusselt numbers. All heat transfer results of Date and Singham [99] and Date [100,101] are in error, because their q' relates to the semicircular duct and not the circular tube with a twisted tape. Hence, all of the Nusselt numbers should be multiplied by a factor of 2, irrespective of the fin param eter used [519]. Date and Singham presented temperature profiles at Red = 378, 809, and 1600 for Xh = 5.24, Pr = 1.0, Cfin = 1.85. The temperature profiles across the duct cross section are asymmetric. As the Reynolds number increases, the temperature profiles become flatter, confirming an increase in the Nusselt number. The effects of increasing Pr and decreasing XL at a constant Red are similar. Table 135 and Fig. 122 are based on the graphical results of Date [100] for XL = 2.25 and 5.24. The reason for the choice of the coordinate for the abscissa of Fig. 122 will be clear after the discussion of Eq. (564). For Re d /X L = 0 and Cfin = oo, Date determined Nu d as 10.8 and Hong and Bergles [446,518] determined it as 11.00, in contrast to a more precise value of 10.95 based on Nu H i of Table 77 for 2φ = 180°. For Re d /X L = 0 and Cfin = 0, Date determined Nu d as 5.188, while Hong and Bergles obtained it as 5.172. A review of Fig. 122 reveals that the influence of Pr and Cfin is quite significant on Nusselt numbers. The influence of Pr on Nusselt numbers of laminar fully developed twistedtape flow is very similar to that for the curved pipe flow [13], i.e., at a given Re d /X L (Dean number in the case of curved pipe flow), the higher fluid Prandtl numbers result in higher Nusselt numbers. Date's analysis shows that Cfin also has a strong effect on Nusselt number, as shown in Fig. 122. Hong and Bergles [446,518] experimentally evaluated the heat transfer performance for a laminar twistedtape flow. The tube was electrically heated but the twisted tape was electrically isolated from the wall. Hence
383
F. CIRCULAR DUCT WITH A TWISTED TAPE TABLE 135
CIRCULAR TUBE WITH A TWISTED TAPE: Nu d AS FUNCTIONS OF Re d /X L , Cfin,AND Pr FOR FULLY DEVELOPED LAMINAR FLOW (FROM THE GRAPHICAL RESULTS OF DATE [100])
Nud Re
X
d
L
C
C.. fin = 1.85
fin = ° Pr=.l
5.19 5.58 7.04 8.28 9.72
7.60 7.70 7.74 7.78 7.82
7.60 7.60 7.60 7.80 8.0 10.8 8.04 9.3 17.2 8.25 1 0 . 7 25.0 8.55 1 2 . 6 3 7 . 0
10.7 11.6 12.3 13.0 14.4
7.86 7.90 7.95 7.98 8.05
8.76 8.90 9.06 9.18 9.40
14.0 15.2 16.2 17.0 18.6
100 150 200 300 400
7.15 15.5 7.68 17.9 8.22 9.56 11·0
8.12 8.40 8.66 9.34 10.1
9.65 10.2 10.8 12.2 13.8
20.0 23.0 25.7 30.8 36.4
100. 131. 161. 220. 280.
500 600
12.5 13.9
11.1 12.3
15.7 17.8
42.8 50.3
350. 420.
0 2 5 10 20
5.19 5.50 5.58 5.72 5.96
30 40 50 60 80
6.18 6.35 6.50 6.66 6.92
Pr=10
Pr=100 Pr=.l
Pr=10
Pr=l
Pr=l
C
46.6 55.5 64.0 72.0 86.0
10.8 11.0 11.5 11.8 12.0 12.2 12.4 12.7 12.9 13.3 13.7 14.2
fin * 
Pr=l Pr=10 Pr=100 10.8 11.0 11.1 11.3 11.9 12.4 13.0 13.5 14.0 14.9 15.8 18.2 20.8 25.8 30.0
10.8 10.8 17.0 4 8 . 8 2 0 . 2 79.0 23.0 105. 25.4 130. 28.0 153. 30.4 176. 35.6 220. 40.2 52.6 65.0 90.0
262. 365. 470. 690.
33.8 37.4
Pr(Re d /X L )'· 78 FIG. 122. Circular tube with a twisted tape: Nu d for fully developed laminar flow [based on the results of Table 135 and the experimental correlation of Eq. (564)].
384
XVI.
LONGITUDINAL FINS WITHIN DUCTS
their results are for the case of Cfin ~ 0 (Cfin = 0.124 for ethylene glycol). They obtained test results for XL = 2.45 and 5.08 by using distilled water and ethylene glycol as the test fluids (Pr ~ 3 to 191). They correlated the test results by the following expression : Nu d = 5.172(1 + 0.005484[Pr(Re d /X L ) 1 · 78 ] 0 · 7 } 0 · 5
(564)
In Fig. 122, this correlation is compared with the results of Date and Singham. Using an abscissa of Pr(Re d /X L ) 1/78 , Hong and Bergles were able to correlate all of their test data for Pr ranging from 3 to 191 within ±20%, while Date's analytical results show a separate influence of Pr on Nu d . The analytical results of Date and the correlation of Hong and Bergles differ substantially. The strong Prandtl number effect for Cfin > 0, predicted by Date, needs experimental verification. An examination of (/Re) d and Nu d for the laminar twistedtape flow reveals that with increasing Re d /Z L there is a dramatic increase in heat transfer with only a slight increase in pressure drop. Both analysis and experiments show this increase in heat transfer. More analyses and experi ments are required to determine more precisely the augmentation charac teristics for laminar twistedtape flow in a circular tube. 2. DEVELOPING FLOWS
Date [100,520] investigated simultaneously developing flow through a circular tube containing a twisted tape. For the heat transfer problem, he considered the tube at constant temperature axially and peripherally. He showed that Ap* and 0m are functions of the following parameters: Ap* = Ap*(Re d ,X L ,x/a) 0m = 0 J R e d , P r , X L , x / a ) The results have been presented graphically by Date.
(565) (566)