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Other Doubly Connected Ducts
Chapter XIV Other Doubly Connected Ducts In this chapter, we describe the remaining solutions for doubly connected duct geometries, other than those discussed in Chapters XII and XIII. Only the fully developed laminar fluid flow and heat transfer problems have been investigated for the five geometries that follow. A. Confocal Elliptical Ducts 1. FLUID FLOW
Fully developed laminar flow through confocal ellipses, Fig. 97, (also referred to as concentric ellipses) has been analyzed by Piercy et al. [74], Sastry [80,89], Shivakumar [489], and Topakoglu and Arnas [490]. The conformai mapping method was used by the first three investigators to arrive at the flow characteristics. Closed-form solutions to the momentum and energy equations were obtained by Topakoglu and Arnas by the use of elliptical coordinates. While Piercy et al. employed ζ' = c cosh ζ as the mapping function, Sastry [89] employed z' = c(£ + l/ζ). The equations for the velocity profile confocal : a^ - b02 = a,2 - bj2 r
*
=
_bj_ b
a
* _ b0
o '
FIG. 97. A confocal elliptical duct. 341
a
o
342
XIV.
OTHER DOUBLY CONNECTED DUCTS
and volumetric flow rate of these investigators are so highly complex that numerical values were not computed. Shivakumar [489] employed z' = c(C + λ/ζ) as a mapping function, and obtained numerical results for seven different confocal elliptical geometries. The fRe factors, calculated from Shivakumar's volumetric flow rate — δπβ/^, are in excellent agreement with those of Table 113. Topakoglu and Arnas [490] obtained closed-form solutions to the mo mentum and energy equations written in an elliptical coordinate system. They presented equations for the velocity profile, coordinates for the points of maximum velocity, and the mass flow rate through the cross section. Note that in the present terminology the definition of Reynolds number employed by Topakoglu and Arnas is —ροχ(α0 + ο0)3/(32μ). The following expression for fRe is derived from the equations of Topakoglu and Arnas:
(522)
'--d£i*7
where
41ηω
(1-ω2)2(ΐ-^-| \ or
, 22)) (1l + 3—? '—s = 7- ( 1l -- a ω T ) 2 V (a0 + b0) 4 \ ω2
(524) YYl
(a0 + b0)
2\
2 {1 + m z )Ê! + ( 1 + —2 )ω£, [
m=
(523)
{ï^)
(525)
(526)
aj + bi
ω =a +K 0 a*r* + [l - a * 2 ( l - r * ) 2 ] 1 / 2 1 +a*
(527)
El and Εω are the complete integrals of the second kind, which are evaluated for the arguments (1 — b02/a02) and (1 — b2/a2\ respectively, where
H'-£)/K) The / Re factors, calculated by the present authors, are presented in Table 113 and Fig. 98.
343
A. CONFOCAL ELLIPTICAL DUCTS TABLE 113 CONFOCAL ELLIPTICAL DUCTS: / R e
FOR FULLY DEVELOPED LAMINAR FLOW
[FROM EQ. (522)] fRe r*
a*=0.2
0.40
0.60
0.80
0.90
0.95
0.02 0.05 0.10 0.20
19.419 19.433 19.452 19.478
19.468 19.534 19.622 19.759
20.291 20.454 20.662 20.965
21.766 22.071 22.388 22.750
22.436 22.825 23.151 23.454
22*620 23*049 23*366 23*643
0.30 0.40 0.50 0.60
19.495 19.507 19.516 19.525
19.871 19.973 2 0 . 0 72 20.171
21.201 21.404 21.585 21.749
22.974 23*135 23.257 23*353
23.610 23.708 23.773 23.819
23*777 23.855 23*903 23*934
0.70 0.80 0.90 0.95 0.98
19.534 19.544 19.555 19.561 19.565
20.268 20.365 20.460 20 · 50 6 20.534
21.896 22.029 22.148 22.203 22.234
23*429 23*490 23.539 23.560 23.572
23*85 1 23*874 23*890 23*896 23*900
23.953 23.966 23.973 23.975 23.976
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
r* FIG.
Confocal elliptical ducts: / R e for fully developed laminar flow (from Table 113).
2. HEAT TRANSFER
Sastry [80] analyzed the (Hi) temperature problem for a region bounded by confocal ellipses by employing the method of complex variables. He used the boundary condition for the velocity problem as u = 0 on the outer ellipse, u = u0 Φ 0 on the inner ellipse. Hence, his results for the velocity and temperature problems are not relevant to stationary ducts. Topakoglu and Arnas [490] analyzed the fundamental problem of the fifth kind f for confocal ellipses. The solution to the energy equation was f One wall at axially constant q\ the other wall insulated; the temperatures along the periph eries uniform at a section but different (see Table 4).
344
XIV. OTHER DOUBLY CONNECTED DUCTS
obtained in closed form by the use of elliptical coordinates. Based on their equations, the numerical results for the fundamental solutions of the fifth kind were determined by the present authors and are presented in Table 114. The other quantities of interest are [βίΒ5/-^] = ηβ&-0ίο5)]
(529)
Nul?» = „,.,' „,(5 gÎ 5)
(530)
1 Nui 5 » = 0(5) _ 0(5)
(531)
Nu05) = NuS05) = 0
(532)
0< - 0 ·>
where P* is the ratio of inner to outer ellipse perimeters. TABLE 114 CONFOCAL ELLIPTICAL DUCTS : FUNDAMENTAL SOLUTIONS OF THE FIFTH KIND FOR FULLY DEVELOPED LAMINAR FLOW (CALCULATED BY THE PRESENT AUTHORS BASED ON THE WORK OF TOPAKOGLU AND ARNAS [490])f θ(5).θ(5)
θ (5) -θ (5)
io
H»i?> 11
0.02 0.05 0 . 10 0.20 0 .30 0.40
.0084733 .0084179 .0083411 .0082406 .00820 38 .008 22 33
5.0556 5.0571 5.0566 5.0 4 8 1 5.0327 5.0126
4.7885 4 . 7 9 14 4.7957 4.8031 4.8097 4.8160
.93277 .93313 .93421 .93782 .94292 .94915
.018465 .018190 .017833 .017434 . 0 1 7 3 70 .017561
5.1361 5. 1449 5. 1466 5.1210 5.0760 5.0240
4.3924 4.4062 4.4271 4.4649 4.5010 4.5370
.79690 .79852 .80321 .81765 .83636 .85764
0.50 0.60 0.70 0.80 0.90 0.95
.008 2 9 2 5 .0084058 .0085579 .0087441 .0089584 .0089174
4.9895 4.9645 4.9 3 8 3 4.9U6 4.8849 4.8679
4.8224 4.8289 4.8358 4.8430 4.8505 4.8507
.95628 .96411 .97251 .98135 .99054 .99524
.0 1 7 9 4 6 .018473 .019101 .019796 .020531 .020920
4.971 1 4.9206 4.8740 4.8319 4.7946 4.7779
4.5736 4.6108 4.6485 4.6864 4.7242 4 .7433
.88046 .90413 .92820 .95236 .97635 .98823
0.02 0.05 0.10 0.20 0.30 0.40
.040344 .039484 .038432 .037328 .037073 .037311
5.4154 5.4462 5.4553 5.4003 5.3156 5.232 3
4.1682 4.2063 4.2620 4.3593 4.4480 4.5322
.62794 .63250 .64482 .67905 .71917 .76138
.078352 .075656 •072488 .068620 .066252 .064555
6.0394 6.1164 6 . 1 0 19 5.9296 5.7568 5.6206
4.3262 4.40 38 4.5023 4.6477 4.7622 4.8599
.42663 .43907 .46833 .53686 .60658 .67360
0.50 0.60 0 .70 0.80 0 .90 0.95
.037818 .038453 .039129 .039790 .040407 .040696
5.159 7 5.0994 5.0506 5.0116 4.9806 4.9677
4.6128 4.6897 4.7627 4.8315 4.8960 4.9267
.80391 •84585 .88668 .92610 .96392 •98218
.063199 .062035 .060992 .060036 .059152 .058731
5.5169 5.4378 5.3 766 5.3285 5.2901 5.2737
4.9459 5.0224 5.09 1 1 5.1530 5.2088 5.2345
.73703 .79675 .85282 .90534 .95439 .97762
0 .02 0.05 0.10 0.20 0 .30 0.40
.099392 .094830 .089767 .083280 .078792 .075289
6.79 10 6.8 2 7 1 6.6459 6.2544 5.9841 5.7999
4.5920 4.6769 4.7654 4.8791 4.9648 5.0380
.29979 .32361 .37128 .46681 .55461 .634 76
.107724 .102188 .096232 .088414 .082898 .078598
7.7718 7.5342 7.0391 6.4203 6.0796 5.8656
4.7438 4.8133 4.8760 4.9588 5.0268 5.0884
.21908 .25646 .31993 .43272 .53017 .61683
0.50 0 .60 0.70 0 .80 0.90 0.95
.072399 .069942 .067816 .065955 .064318 .063569
5.6687 5 . 5 7 14 5.4970 5.438 5 5 . 3 9 18 5.37 17
5.1033 5.1626 5.2168 5.2665 5.3121 5.33 33
.70833 .77619 .83894 .89701 .95064 .97585
.0 7 5 0 8 7 .072141 .069629 .06 7 4 6 2 .065582 .064735
5.7187 5.6119 5.5311 5.4680 5.4178 5.3964
5.1454 5.1986 5.2483 5.2946 5.3375 5.35 78
.69524 .76690 .83273 .89331 .94900 .97508
r*
mo
Nu<5> OO
P./P l'
O
mo
io
a*=0.20
Nu(5> oo
V*o
a*=0.40
a*=0. 60
a*=0. 80
a*=0.90
f
Hui?> 11
a*=0.95
K2 of Eq. (6.2) of [490] should have(l - ω4) instead of (1 - ω 2 ).
345
A. CONFOCAL ELLIPTICAL DUCTS
As mentioned on p. 292 for concentric annular ducts, additional solutions of practical interest can be obtained by the superposition of the fundamental solutions. The solution for the case of heat flux specified constant on both walls is obtained in the same manner as that for concentric annular ducts (described on p. 294), except that the superscript designating the kind of fundamental solution is changed from (2) to (5). When constant axial heat fluxes q^'i^qJ/Po) and q"{ = q\IPx) are specified on each wall, the corre sponding Nusselt numbers are obtained from Eqs. (519) and (520) with the superscript 2 replaced by 5. The Nusselt numbers for the cases of q^lql = — 0.5 and —2.0 analyzed by Topakoglu and Arnas can be obtained by the use of Eqs. (519) and (520) and the results of Table 114. Similarly, the Nusselt numbers Nup b) and NuJ,5b) for the equal wall temperature case analyzed by Topakoglu and Arnas can also be determined from Eqs. (519) and (520)
TABLE 115 CONFOCAL ELLIPTICAL DUCTS: Nu H1
FOR FULLY DEVELOPED
LAMINAR FLOW [FROM EQ. (533)] NuH1 r* α*=0.2
0.40
0.60
0.80
0.90
0.95
0.02 0·05 0.10 0.20 0.30 0·40
5.1237 5.1248 5.1252 5.1230 5.1185 5.1130
5.1231 5.1311 5.1395 5.1479 5.1541 5.1626
5.4782 5.5121 5.5534 5.6162 5.6770 5.7441
6.5083 6.6178 6.7384 6.8973 7.0218 7.1325
7.1933 7.3679 7.5273 7.6945 7.7961 7.8696
7.4100 7.6216 7.7940 7.9574 8.0427 8.0955
0·50 0*60 0.70 0.80 0.90 0.95
5.1072 5.1016 5.0965 5.0921 5.0885 5.0788
5.1751 5.1922 5.2137 5.2390 5.2676 5.2836
5.8179 5.8966 5.9779 6.0597 6.1404 6.1801
7.2325 7.3224 7.4023 7.4724 7.5336 7.5606
7.9259 7.9699 8.0046 β.0320 8.0536 8.0621
8.1306 8.1546 8.1711 8.1825 8.1901 8.1928
Ό·0|
■ _ ι —■—I—'—I—'—I—■—I—'—I—'—I—'—I—'—I— Γ
FIG. 99. Confocal elliptical ducts: Nu H1 for fully developed laminar flow (from Table 115).
346
XIV.
OTHER DOUBLY CONNECTED DUCTS
(with superscript 5) with q"/q0" obtained from Eq. (518) (with superscript 5). Nu H1 is then computed from the following equation, where h is defined by Eq. (76): Nu<,5b) + P*NuS5b) NuH1= 1 + p«
_„„, (533)
Nu H1 for confocal elliptical ducts were determined from Eq. (533) and the results of Table 114 by the described procedure, and are presented in Table 115 and Fig. 99. B. Regular Polygonal Ducts with Central Circular Cores Gaydon and Nuttall [491] proposed a method for estimating the volu metric flow rate for fully developed laminar flow through a cylindrical multiply connected duct. The method yields the upper and lower bounds of the flow rate derived from the Schwartz inequality of the variational method. They obtained such bounds for a square and a hexagonal duct with central circular cores. These upper bound / Re factors are in excellent agreement with the more accurate / R e factors of Table 116. Cheng and Jamil [109] analyzed fully developed laminar flow through regular polygonal ducts having a central circular core (see Fig. 100). They employed a point-matching method for the analysis. Ten points on one-half side of the regular polygon were used. For the heat transfer problem, they considered the (Hi) thermal boundary condition, as defined in Table 5 and also on p. 328 for doubly connected ducts, and tabulated / Re, Nu H1 , and the flow rate for 0 < α/ξί < 0.5, 3 < n < 20. Here a is the radius of the circular core and ξ1 is the radius of an inscribed circle in an n-sided regular polygon. Cheng and Jamil also presented graphically shear stress distribution, fluid temperature gradients at the wall, and fluid velocity and temperature profiles for typical duct geometries. Their / Re factors for the limited range of α/ξί are in excellent agreement with those of Table 116. Nu H1 of Cheng and Jamil are presented in Table 117 and Fig. 101. Ratkowsky and Epstein [112] independently investigated the flow characteristics of the same problem by a discrete least squares method. Their / R e factors [492] are presented in Table 116 and Fig. 100. Two limiting cases of this geometry (see Fig. 100), α/ξι = 0, and 1, are of interest. When α/ξχ equals zero, the corresponding geometry is the n-sided regular polygonal duct, which has been discussed in Chapter X. The / Re factors for this case are in excellent agreement with those of Table 75. Ratkowsky and Epstein have analyzed the other limiting case {α/ξγ = 1) in detail. The / R e factors for n = 3 through 18 are presented in Table 116. They showed
TABLE 116 REGULAR POLYGONAL DUCTS WITH CENTRAL CIRCULAR CORES : / R e
FOR FULLY DEVELOPED LAMINAR FLOW (FROM RATKOWSKY [492]) fRe
a
h
n=3
n=4
1*000 0.99 1 0.987 0.98 1 0.979
7.80
7.10
0.969 0.950 0.900 0.87 5 0.800
9.18
9.86
15.32
16.21 19.08
0.750 0.700 0.600 0.511 0.500
16.58
20.25
18.98 19.69
0.400 0.318 0.300 0.200 0.100 0.050 0.025 0
— — — -
— 1 2— .93
— — — -
— —
n=6 6.62
— — 1 0— .89 _ 15.35
19.68
—
n=8 6.48
— — _ 18.79
11.04
22.04
—
22.48
23.41
21.82
22.92 23.14 23.23
23.54 23.59 23.54
22.02
23.12
23.39
19.90
21.89
22.88
23.18
19.73 19.22
20.92
22.53 22.01 21.13
22.88 22.42 21 . 5 9
16.89 13.33
14.23
20.31 19.59 15.05
20.79 20.07 15.41
— —
— _
— —
— — _ 18.40
—
—
—
—
n=18 6.48 17.82
— _ 23.53
21.59
23.89
— _ — 23— .70 23.36 — 2 2— .16 1 5— .86 23.92
24.0 l· 22.0l· 20.0 L 18.0 h fRe k ie.ol·14.0 k 12.0 k 10.0 h 8.0 L 60'
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
' ' 1.0
FIG. 100. Regular polygonal ducts with central circular cores: / R e for fully developed laminar flow (from Table 116).
348
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 117
REGULAR POLYGONAL DUCTS WITH CENTRAL CIRCULAR CORES: Nu H1
FOR
FULLY DEVELOPED LAMINAR FLOW (FROM CHENG AND JAMIL [109])
Nu H 1
a/ξχ n=3
n=7
n=8
n=9
n=10
n=20
4.002 6.303 6.670 6 . 722 6.778
4.102 6.431 6.794 6.845 6.898
4.153 6.525 6.885 6.935 6.986
4.196 6.588 6.946 6.99 5 7.045
4.227 6.635 6.992 7.040 7.090
4.329 6.778 7.142 7 . 189 7.236
6.644 6. 705 6.755 6 . 760 6.593
6. 834 6.887 6.924 6.906 6.697
6.951 6.998 7.027 6.995 6.75 7
7.037 7 .080 7.102 7.060 6.801
7.094 7.136 7 . 154 7.104 6.831
7 . 138 7 . 177 7 . 192 7 . 138 6.854
7.28 0 7.314 7.321 7.251 6.9 33
5.488
5.498
5.480
5.465
5.453
5.444
5 . 4 19
n=4
n=5
3 . 1 11 4 . 9 38 5.296 5.354 5.417
3.608 5.723 6 . 104 6 . 163 6.228
3.859 6.094 6.467 6. 523 6.582
1/3
5.486 5.560 5.626 5.655 5.51 1
6.29 7 6 . 3 70 6.438 6 . 4 74 6.351
0.5
4.466
5.31 7
0.00 0.05 0. 1 1/9
0 . 125 1/7 1/6 0.2
0.25
n=6
T—i—i—ι—i—T—i—i—i—i—i—ι—i—i—i—i—i—i—i—i—i—i—i—r
0/f FIG. 101. Regular polygonal ducts with central circular cores: Nu H1 for fully developed laminar flow (from Table 117).
D. ELLIPTICAL DUCTS WITH CENTRAL CIRCULAR CORES
349
that when n -► oo, / R e -> 56/9 = 6.222 for this case. However, the number of sides n of a regular polygon cannot be smaller than 3 ; the corresponding case is discussed in the following section. It is interesting to note that the presence of a small circular core (α/ξ1 ~ 0), centrally located in a regular polygonal duct, dramatically increases / Re and Nu H1 . This behavior is similar to that observed for concentric annular ducts. C Isosceles Triangular Ducts with Inscribed Circular Cores As mentioned in the preceding section, Ratkowsky and Epstein [112] also analyzed laminar flow through a section bounded by a regular polygon and its inscribed circle (α/ξί = 1), and reported / R e factors for n > 3. They also considered the isosceles triangular duct with the side angle φ < 60° (see figure with Table 118). They reported the study of Bowen [493] for this geometry. Bowen employed a finite difference method to obtain the / R e factors presented in Table 118. TABLE 118 ISOSCELES TRIANGULAR DUCTS WITH INSCRIBED CIRCULAR CORES : / Re FOR FULLY DEVELOPED LAMINAR FLOW (FROM BOWEN [493])
Φ 0 2 5 10 15
fRe 12.0 11.4 11.3 10.65 10. 16
Φ
fRe
20 30 40 50 60
9.79 9. 14 8.69 8.26 7.80
^È>\
D. Elliptical Ducts with Central Circular Cores Sastry [88,89] and Shivakumar [489] analyzed fully developed laminar flow through an elliptical duct with a central circular core by the conformai mapping method. Sastry employed the Schwarz-Neumann alternating method, an approximate method of conformai mapping, and determined the velocity profile with the first four approximations. Shivakumar also employed the conformai mapping method, although a different one, to analyze the same problem. He presented an equation for the volumetric flow rate through this cross section and evaluated numerical results for some geometries. Based on his tabulated results, / R e factors were determined and are presented in Table 119 [494]. Table II of Shivakumar [489] has some errors; the values for the area of the cross section should be multiplied by π, and the flow rate values in the last three columns should be divided by π [494].
350
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 119
ELLIPTICAL DUCTS WITH CENTRAL CIRCULAR CORES : / Re FOR FULLY DEVELOPED LAMINAR FLOW (FROM SHIVAKUMAR [489,494])
a*
r*
0.9 0.9 0.9 0.9
0.5 0.6 0.7
0 .95
23.519 23.435 23.159 16.816
0.7 0.7 0.5
0.5 0.7 0.5
21.694 19.402 19.32 1
fRe "2ao a
r*=
E. Circular Ducts with Central Regular Polygonal Cores Cheng and Jamil [109,111] employed a point-matching method to analyze fully developed laminar flow through an annulus having a circle as the outside boundary and a concentric regular polygon as the inside boundary. Both surfaces of the annulus were heated to obtain the (HI) boundary con dition defined in Table 5 and also on p. 328. They presented / R e and Nu H1 graphically. The / R e factors were obtained for the limited range 0 < ξί/α < 0.5 and are in excellent agreement with those of Table 120. The tabulation of / R e factors of Cheng and Jamil is available in [13,449]. Their Nu H1 are presented in Table 121 and Fig. 103. Hagen and Ratkowsky [113] studied the flow characteristics of the same problem by a discrete least squares method. Their / Re factors are reported in Table 120 and Fig. 102. Note that while Cheng and Jamil used ξι/α as an independent variable, Hagen and Ratkowsky employed ξ2/α. Here, ξ1 and ξ2 are the radii of inscribed and circumscribed circles, respectively. Cheng and Jamil [111] encountered some difficulties with their pointmatching method as the number of sides of the regular polygonal core decreased from 20 to 3. They found that the shear stress distributions and normal temperature gradients along the inner regular polygonal boundary exhibited a wavy character, with a very small region having a negative shear stress distribution. In spite of the difficulty with local values, Cheng and Jamil stated that the integrated overall quantities, such as / R e and Nu H1 were sufficiently accurate for practical purposes. Hagen and Ratkowsky [113] did not encounter the negative shear stress distribution and its wavy character; however, the least squares matching procedure became more difficult as the number of sides of the polygon became smaller. It may be noted that the presence of a small regular polygonal core at the center of the circular duct significantly increases the flow resistance ( / Re) and heat transfer (Nu H1 ).
TABLE 120 CIRCULAR DUCT WITH CENTRAL REGULAR POLYGONAL CORES: / R e
FOR FULLY DEVELOPED LAMINAR FLOW (FROM RATKOWSKY [492])
h.
fRe
a
|
15 Q I
n=3
n=4
n=6
n=8
n=l8
l.ooo 0.975 0.950 0.925 0.900
15.75
— 18— .42
15.60
— 21— .71
15.57
1 7 . 15
15.67 16.89 17.96 18.90 19.71
15.52 22.93 23.66
0.85 0.80 0.75 0.70 0.65
19.50 20.36 21 . 0 5 21.60 21.92
20.94 21.80 22.34 22.72 22.93
2 2 . 71 23.21 23.44 23.54 23.58
2 3.76 23.78 2 3.74
0.60 0.55 0.50 0.45 0.40
22.20 22.34 22.36 22.38 22.35
2 3 . 10 23.09 2 3 . 10 23.04 22.98
23.57 2 3.52 23.49 23.41 23.34
23.71 23.70 23.60 23.54 23.46
0.35 0.30 0.25 0.20 0 . 15
22.37 22.09 22. 1 1 21 . 9 8 21.90
22.91 22.83 22.68 22.55 22.31
23.25 2 3 . 12 22.98 22.80 22.55
2 3.37 23.26 23.1 1 22.92 2 2.65
0.100 0.075 0.050 0.000
21 . 64
21 . 9 5 2 1. 74
2 2 . 17
22.24
22.32
21.54 16.00
21 . 5 4 16.00
21.56 16.00
T
1
i
i
0.0
—
21 . 0 8 16.00
1
1
i
i
0.1
1
i—i
0.2
1
1
1
21 . 2 1
—
1 6— .00
1
— 23— .47 _ 23.68
1 9 . 74
1
1
1
— _ 23.94 2 3— .90 -
23.93
23.84
23.4 1 2 3 -. 0 5 -
2 3.75 2 3.62
—
1
1
1
—
1
1
1
1
1
Γ
i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—I
0.3
0.4
0.5
f2/a
06
0.7
0.8
0.9
1.0
FIG. 102. Circular duct with central regular polygonal cores:/Re for fully developed laminar flow (from Table 120).
352
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 121
CIRCULAR DUCT WITH CENTRAL REGULAR POLYGONAL CORES : Nu H1
FOR FULLY DEVELOPED LAMINAR FLOW (FROM JAMIL [449])
Nu
ξχ/a
Hl
n=4
n=5
n=6
n=7
n=8
n=9
0.0 1/9 1/8 1/7 1/6
4.364 7.337 7.398 7.467 7 . 5 38
4.364 7 . 321 7 . 3 85 7.457 7.537
4.364 7.318 7.385 7.459 7.543
4.364 7 . 3 20 7.388 7.464 7.550
4.364 7.390 7.467 7.555
4.364 7.325 7.395 7.057 7.558«
1/5 1/4 1/3
7.616 7.69 3 7.738
7.626 7. 723 7.814 7.718
7.638 7 . 744 7.858 7.910
7.648 7.759 7.884 7.989
7.654 7.769 7.898 8.027
7.660 7.776 7.909 8.049
1 1/2
~i
1
1—r—i
1
i
-T
1
1
r—
,n = oo
8.0
7.0
6.0
n- sided regular polygon
5.0
4.364 4.0
_i
0.0
0.1
i
i
_i
i_
0.2
0.3
i
i
i i 0.4
i
i_ 0.5
Ci/a FIG. 103. Circular duct with central regular polygonal cores: Nu H1 for fully developed laminar flow (from Table 121).
F. Miscellaneous Doubly Connected Ducts Sastry [87] employed the Schwarz-Neumann alternating method, an approximate method of conformai mapping, to analyze the fully developed laminar velocity and (Hi) temperature problems for a circular tube with
F. MISCELLANEOUS DOUBLY CONNECTED DUCTS
353
central elliptical cores. He employed boundary conditions for the velocity problem as u = 0 on the circle and u = u0 Φ 0 on the ellipse. Hence his results for the velocity and temperature problems are not relevant for stationary ducts. Sastry [88] investigated fully developed laminar flow for a circular tube having a central square core with rounded corners, and considered the no-slip boundary condition at both inner and outer boundaries. He analyzed the fundamental boundary conditions of the fifth kind (see Table 4) for the temperature problem. He employed the Schwarz-Neumann alternating method and presented the approximate solutions for the velocity and temperature distributions, mean velocity and cross-sectional average fluid temperature. He did not present explicit formulas for / R e and Nujf.
Fully developed laminar flow through confocal ellipses, Fig. 97, (also referred to as concentric ellipses) has been analyzed by Piercy et al. [74], Sastry [80,89], Shivakumar [489], and Topakoglu and Arnas [490]. The conformai mapping method was used by the first three investigators to arrive at the flow characteristics. Closed-form solutions to the momentum and energy equations were obtained by Topakoglu and Arnas by the use of elliptical coordinates. While Piercy et al. employed ζ' = c cosh ζ as the mapping function, Sastry [89] employed z' = c(£ + l/ζ). The equations for the velocity profile confocal : a^ - b02 = a,2 - bj2 r
*
=
_bj_ b
a
* _ b0
o '
FIG. 97. A confocal elliptical duct. 341
a
o
342
XIV.
OTHER DOUBLY CONNECTED DUCTS
and volumetric flow rate of these investigators are so highly complex that numerical values were not computed. Shivakumar [489] employed z' = c(C + λ/ζ) as a mapping function, and obtained numerical results for seven different confocal elliptical geometries. The fRe factors, calculated from Shivakumar's volumetric flow rate — δπβ/^, are in excellent agreement with those of Table 113. Topakoglu and Arnas [490] obtained closed-form solutions to the mo mentum and energy equations written in an elliptical coordinate system. They presented equations for the velocity profile, coordinates for the points of maximum velocity, and the mass flow rate through the cross section. Note that in the present terminology the definition of Reynolds number employed by Topakoglu and Arnas is —ροχ(α0 + ο0)3/(32μ). The following expression for fRe is derived from the equations of Topakoglu and Arnas:
(522)
'--d£i*7
where
41ηω
(1-ω2)2(ΐ-^-| \ or
, 22)) (1l + 3—? '—s = 7- ( 1l -- a ω T ) 2 V (a0 + b0) 4 \ ω2
(524) YYl
(a0 + b0)
2\
2 {1 + m z )Ê! + ( 1 + —2 )ω£, [
m=
(523)
{ï^)
(525)
(526)
aj + bi
ω =a +K 0 a*r* + [l - a * 2 ( l - r * ) 2 ] 1 / 2 1 +a*
(527)
El and Εω are the complete integrals of the second kind, which are evaluated for the arguments (1 — b02/a02) and (1 — b2/a2\ respectively, where
H'-£)/K) The / Re factors, calculated by the present authors, are presented in Table 113 and Fig. 98.
343
A. CONFOCAL ELLIPTICAL DUCTS TABLE 113 CONFOCAL ELLIPTICAL DUCTS: / R e
FOR FULLY DEVELOPED LAMINAR FLOW
[FROM EQ. (522)] fRe r*
a*=0.2
0.40
0.60
0.80
0.90
0.95
0.02 0.05 0.10 0.20
19.419 19.433 19.452 19.478
19.468 19.534 19.622 19.759
20.291 20.454 20.662 20.965
21.766 22.071 22.388 22.750
22.436 22.825 23.151 23.454
22*620 23*049 23*366 23*643
0.30 0.40 0.50 0.60
19.495 19.507 19.516 19.525
19.871 19.973 2 0 . 0 72 20.171
21.201 21.404 21.585 21.749
22.974 23*135 23.257 23*353
23.610 23.708 23.773 23.819
23*777 23.855 23*903 23*934
0.70 0.80 0.90 0.95 0.98
19.534 19.544 19.555 19.561 19.565
20.268 20.365 20.460 20 · 50 6 20.534
21.896 22.029 22.148 22.203 22.234
23*429 23*490 23.539 23.560 23.572
23*85 1 23*874 23*890 23*896 23*900
23.953 23.966 23.973 23.975 23.976
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
r* FIG.
Confocal elliptical ducts: / R e for fully developed laminar flow (from Table 113).
2. HEAT TRANSFER
Sastry [80] analyzed the (Hi) temperature problem for a region bounded by confocal ellipses by employing the method of complex variables. He used the boundary condition for the velocity problem as u = 0 on the outer ellipse, u = u0 Φ 0 on the inner ellipse. Hence, his results for the velocity and temperature problems are not relevant to stationary ducts. Topakoglu and Arnas [490] analyzed the fundamental problem of the fifth kind f for confocal ellipses. The solution to the energy equation was f One wall at axially constant q\ the other wall insulated; the temperatures along the periph eries uniform at a section but different (see Table 4).
344
XIV. OTHER DOUBLY CONNECTED DUCTS
obtained in closed form by the use of elliptical coordinates. Based on their equations, the numerical results for the fundamental solutions of the fifth kind were determined by the present authors and are presented in Table 114. The other quantities of interest are [βίΒ5/-^] = ηβ&-0ίο5)]
(529)
Nul?» = „,.,' „,(5 gÎ 5)
(530)
1 Nui 5 » = 0(5) _ 0(5)
(531)
Nu05) = NuS05) = 0
(532)
0< - 0 ·>
where P* is the ratio of inner to outer ellipse perimeters. TABLE 114 CONFOCAL ELLIPTICAL DUCTS : FUNDAMENTAL SOLUTIONS OF THE FIFTH KIND FOR FULLY DEVELOPED LAMINAR FLOW (CALCULATED BY THE PRESENT AUTHORS BASED ON THE WORK OF TOPAKOGLU AND ARNAS [490])f θ(5).θ(5)
θ (5) -θ (5)
io
H»i?> 11
0.02 0.05 0 . 10 0.20 0 .30 0.40
.0084733 .0084179 .0083411 .0082406 .00820 38 .008 22 33
5.0556 5.0571 5.0566 5.0 4 8 1 5.0327 5.0126
4.7885 4 . 7 9 14 4.7957 4.8031 4.8097 4.8160
.93277 .93313 .93421 .93782 .94292 .94915
.018465 .018190 .017833 .017434 . 0 1 7 3 70 .017561
5.1361 5. 1449 5. 1466 5.1210 5.0760 5.0240
4.3924 4.4062 4.4271 4.4649 4.5010 4.5370
.79690 .79852 .80321 .81765 .83636 .85764
0.50 0.60 0.70 0.80 0.90 0.95
.008 2 9 2 5 .0084058 .0085579 .0087441 .0089584 .0089174
4.9895 4.9645 4.9 3 8 3 4.9U6 4.8849 4.8679
4.8224 4.8289 4.8358 4.8430 4.8505 4.8507
.95628 .96411 .97251 .98135 .99054 .99524
.0 1 7 9 4 6 .018473 .019101 .019796 .020531 .020920
4.971 1 4.9206 4.8740 4.8319 4.7946 4.7779
4.5736 4.6108 4.6485 4.6864 4.7242 4 .7433
.88046 .90413 .92820 .95236 .97635 .98823
0.02 0.05 0.10 0.20 0.30 0.40
.040344 .039484 .038432 .037328 .037073 .037311
5.4154 5.4462 5.4553 5.4003 5.3156 5.232 3
4.1682 4.2063 4.2620 4.3593 4.4480 4.5322
.62794 .63250 .64482 .67905 .71917 .76138
.078352 .075656 •072488 .068620 .066252 .064555
6.0394 6.1164 6 . 1 0 19 5.9296 5.7568 5.6206
4.3262 4.40 38 4.5023 4.6477 4.7622 4.8599
.42663 .43907 .46833 .53686 .60658 .67360
0.50 0.60 0 .70 0.80 0 .90 0.95
.037818 .038453 .039129 .039790 .040407 .040696
5.159 7 5.0994 5.0506 5.0116 4.9806 4.9677
4.6128 4.6897 4.7627 4.8315 4.8960 4.9267
.80391 •84585 .88668 .92610 .96392 •98218
.063199 .062035 .060992 .060036 .059152 .058731
5.5169 5.4378 5.3 766 5.3285 5.2901 5.2737
4.9459 5.0224 5.09 1 1 5.1530 5.2088 5.2345
.73703 .79675 .85282 .90534 .95439 .97762
0 .02 0.05 0.10 0.20 0 .30 0.40
.099392 .094830 .089767 .083280 .078792 .075289
6.79 10 6.8 2 7 1 6.6459 6.2544 5.9841 5.7999
4.5920 4.6769 4.7654 4.8791 4.9648 5.0380
.29979 .32361 .37128 .46681 .55461 .634 76
.107724 .102188 .096232 .088414 .082898 .078598
7.7718 7.5342 7.0391 6.4203 6.0796 5.8656
4.7438 4.8133 4.8760 4.9588 5.0268 5.0884
.21908 .25646 .31993 .43272 .53017 .61683
0.50 0 .60 0.70 0 .80 0.90 0.95
.072399 .069942 .067816 .065955 .064318 .063569
5.6687 5 . 5 7 14 5.4970 5.438 5 5 . 3 9 18 5.37 17
5.1033 5.1626 5.2168 5.2665 5.3121 5.33 33
.70833 .77619 .83894 .89701 .95064 .97585
.0 7 5 0 8 7 .072141 .069629 .06 7 4 6 2 .065582 .064735
5.7187 5.6119 5.5311 5.4680 5.4178 5.3964
5.1454 5.1986 5.2483 5.2946 5.3375 5.35 78
.69524 .76690 .83273 .89331 .94900 .97508
r*
mo
Nu<5> OO
P./P l'
O
mo
io
a*=0.20
Nu(5> oo
V*o
a*=0.40
a*=0. 60
a*=0. 80
a*=0.90
f
Hui?> 11
a*=0.95
K2 of Eq. (6.2) of [490] should have(l - ω4) instead of (1 - ω 2 ).
345
A. CONFOCAL ELLIPTICAL DUCTS
As mentioned on p. 292 for concentric annular ducts, additional solutions of practical interest can be obtained by the superposition of the fundamental solutions. The solution for the case of heat flux specified constant on both walls is obtained in the same manner as that for concentric annular ducts (described on p. 294), except that the superscript designating the kind of fundamental solution is changed from (2) to (5). When constant axial heat fluxes q^'i^qJ/Po) and q"{ = q\IPx) are specified on each wall, the corre sponding Nusselt numbers are obtained from Eqs. (519) and (520) with the superscript 2 replaced by 5. The Nusselt numbers for the cases of q^lql = — 0.5 and —2.0 analyzed by Topakoglu and Arnas can be obtained by the use of Eqs. (519) and (520) and the results of Table 114. Similarly, the Nusselt numbers Nup b) and NuJ,5b) for the equal wall temperature case analyzed by Topakoglu and Arnas can also be determined from Eqs. (519) and (520)
TABLE 115 CONFOCAL ELLIPTICAL DUCTS: Nu H1
FOR FULLY DEVELOPED
LAMINAR FLOW [FROM EQ. (533)] NuH1 r* α*=0.2
0.40
0.60
0.80
0.90
0.95
0.02 0·05 0.10 0.20 0.30 0·40
5.1237 5.1248 5.1252 5.1230 5.1185 5.1130
5.1231 5.1311 5.1395 5.1479 5.1541 5.1626
5.4782 5.5121 5.5534 5.6162 5.6770 5.7441
6.5083 6.6178 6.7384 6.8973 7.0218 7.1325
7.1933 7.3679 7.5273 7.6945 7.7961 7.8696
7.4100 7.6216 7.7940 7.9574 8.0427 8.0955
0·50 0*60 0.70 0.80 0.90 0.95
5.1072 5.1016 5.0965 5.0921 5.0885 5.0788
5.1751 5.1922 5.2137 5.2390 5.2676 5.2836
5.8179 5.8966 5.9779 6.0597 6.1404 6.1801
7.2325 7.3224 7.4023 7.4724 7.5336 7.5606
7.9259 7.9699 8.0046 β.0320 8.0536 8.0621
8.1306 8.1546 8.1711 8.1825 8.1901 8.1928
Ό·0|
■ _ ι —■—I—'—I—'—I—■—I—'—I—'—I—'—I—'—I— Γ
FIG. 99. Confocal elliptical ducts: Nu H1 for fully developed laminar flow (from Table 115).
346
XIV.
OTHER DOUBLY CONNECTED DUCTS
(with superscript 5) with q"/q0" obtained from Eq. (518) (with superscript 5). Nu H1 is then computed from the following equation, where h is defined by Eq. (76): Nu<,5b) + P*NuS5b) NuH1= 1 + p«
_„„, (533)
Nu H1 for confocal elliptical ducts were determined from Eq. (533) and the results of Table 114 by the described procedure, and are presented in Table 115 and Fig. 99. B. Regular Polygonal Ducts with Central Circular Cores Gaydon and Nuttall [491] proposed a method for estimating the volu metric flow rate for fully developed laminar flow through a cylindrical multiply connected duct. The method yields the upper and lower bounds of the flow rate derived from the Schwartz inequality of the variational method. They obtained such bounds for a square and a hexagonal duct with central circular cores. These upper bound / Re factors are in excellent agreement with the more accurate / R e factors of Table 116. Cheng and Jamil [109] analyzed fully developed laminar flow through regular polygonal ducts having a central circular core (see Fig. 100). They employed a point-matching method for the analysis. Ten points on one-half side of the regular polygon were used. For the heat transfer problem, they considered the (Hi) thermal boundary condition, as defined in Table 5 and also on p. 328 for doubly connected ducts, and tabulated / Re, Nu H1 , and the flow rate for 0 < α/ξί < 0.5, 3 < n < 20. Here a is the radius of the circular core and ξ1 is the radius of an inscribed circle in an n-sided regular polygon. Cheng and Jamil also presented graphically shear stress distribution, fluid temperature gradients at the wall, and fluid velocity and temperature profiles for typical duct geometries. Their / Re factors for the limited range of α/ξί are in excellent agreement with those of Table 116. Nu H1 of Cheng and Jamil are presented in Table 117 and Fig. 101. Ratkowsky and Epstein [112] independently investigated the flow characteristics of the same problem by a discrete least squares method. Their / R e factors [492] are presented in Table 116 and Fig. 100. Two limiting cases of this geometry (see Fig. 100), α/ξι = 0, and 1, are of interest. When α/ξχ equals zero, the corresponding geometry is the n-sided regular polygonal duct, which has been discussed in Chapter X. The / Re factors for this case are in excellent agreement with those of Table 75. Ratkowsky and Epstein have analyzed the other limiting case {α/ξγ = 1) in detail. The / R e factors for n = 3 through 18 are presented in Table 116. They showed
TABLE 116 REGULAR POLYGONAL DUCTS WITH CENTRAL CIRCULAR CORES : / R e
FOR FULLY DEVELOPED LAMINAR FLOW (FROM RATKOWSKY [492]) fRe
a
h
n=3
n=4
1*000 0.99 1 0.987 0.98 1 0.979
7.80
7.10
0.969 0.950 0.900 0.87 5 0.800
9.18
9.86
15.32
16.21 19.08
0.750 0.700 0.600 0.511 0.500
16.58
20.25
18.98 19.69
0.400 0.318 0.300 0.200 0.100 0.050 0.025 0
— — — -
— 1 2— .93
— — — -
— —
n=6 6.62
— — 1 0— .89 _ 15.35
19.68
—
n=8 6.48
— — _ 18.79
11.04
22.04
—
22.48
23.41
21.82
22.92 23.14 23.23
23.54 23.59 23.54
22.02
23.12
23.39
19.90
21.89
22.88
23.18
19.73 19.22
20.92
22.53 22.01 21.13
22.88 22.42 21 . 5 9
16.89 13.33
14.23
20.31 19.59 15.05
20.79 20.07 15.41
— —
— _
— —
— — _ 18.40
—
—
—
—
n=18 6.48 17.82
— _ 23.53
21.59
23.89
— _ — 23— .70 23.36 — 2 2— .16 1 5— .86 23.92
24.0 l· 22.0l· 20.0 L 18.0 h fRe k ie.ol·14.0 k 12.0 k 10.0 h 8.0 L 60'
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
' ' 1.0
FIG. 100. Regular polygonal ducts with central circular cores: / R e for fully developed laminar flow (from Table 116).
348
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 117
REGULAR POLYGONAL DUCTS WITH CENTRAL CIRCULAR CORES: Nu H1
FOR
FULLY DEVELOPED LAMINAR FLOW (FROM CHENG AND JAMIL [109])
Nu H 1
a/ξχ n=3
n=7
n=8
n=9
n=10
n=20
4.002 6.303 6.670 6 . 722 6.778
4.102 6.431 6.794 6.845 6.898
4.153 6.525 6.885 6.935 6.986
4.196 6.588 6.946 6.99 5 7.045
4.227 6.635 6.992 7.040 7.090
4.329 6.778 7.142 7 . 189 7.236
6.644 6. 705 6.755 6 . 760 6.593
6. 834 6.887 6.924 6.906 6.697
6.951 6.998 7.027 6.995 6.75 7
7.037 7 .080 7.102 7.060 6.801
7.094 7.136 7 . 154 7.104 6.831
7 . 138 7 . 177 7 . 192 7 . 138 6.854
7.28 0 7.314 7.321 7.251 6.9 33
5.488
5.498
5.480
5.465
5.453
5.444
5 . 4 19
n=4
n=5
3 . 1 11 4 . 9 38 5.296 5.354 5.417
3.608 5.723 6 . 104 6 . 163 6.228
3.859 6.094 6.467 6. 523 6.582
1/3
5.486 5.560 5.626 5.655 5.51 1
6.29 7 6 . 3 70 6.438 6 . 4 74 6.351
0.5
4.466
5.31 7
0.00 0.05 0. 1 1/9
0 . 125 1/7 1/6 0.2
0.25
n=6
T—i—i—ι—i—T—i—i—i—i—i—ι—i—i—i—i—i—i—i—i—i—i—i—r
0/f FIG. 101. Regular polygonal ducts with central circular cores: Nu H1 for fully developed laminar flow (from Table 117).
D. ELLIPTICAL DUCTS WITH CENTRAL CIRCULAR CORES
349
that when n -► oo, / R e -> 56/9 = 6.222 for this case. However, the number of sides n of a regular polygon cannot be smaller than 3 ; the corresponding case is discussed in the following section. It is interesting to note that the presence of a small circular core (α/ξ1 ~ 0), centrally located in a regular polygonal duct, dramatically increases / Re and Nu H1 . This behavior is similar to that observed for concentric annular ducts. C Isosceles Triangular Ducts with Inscribed Circular Cores As mentioned in the preceding section, Ratkowsky and Epstein [112] also analyzed laminar flow through a section bounded by a regular polygon and its inscribed circle (α/ξί = 1), and reported / R e factors for n > 3. They also considered the isosceles triangular duct with the side angle φ < 60° (see figure with Table 118). They reported the study of Bowen [493] for this geometry. Bowen employed a finite difference method to obtain the / R e factors presented in Table 118. TABLE 118 ISOSCELES TRIANGULAR DUCTS WITH INSCRIBED CIRCULAR CORES : / Re FOR FULLY DEVELOPED LAMINAR FLOW (FROM BOWEN [493])
Φ 0 2 5 10 15
fRe 12.0 11.4 11.3 10.65 10. 16
Φ
fRe
20 30 40 50 60
9.79 9. 14 8.69 8.26 7.80
^È>\
D. Elliptical Ducts with Central Circular Cores Sastry [88,89] and Shivakumar [489] analyzed fully developed laminar flow through an elliptical duct with a central circular core by the conformai mapping method. Sastry employed the Schwarz-Neumann alternating method, an approximate method of conformai mapping, and determined the velocity profile with the first four approximations. Shivakumar also employed the conformai mapping method, although a different one, to analyze the same problem. He presented an equation for the volumetric flow rate through this cross section and evaluated numerical results for some geometries. Based on his tabulated results, / R e factors were determined and are presented in Table 119 [494]. Table II of Shivakumar [489] has some errors; the values for the area of the cross section should be multiplied by π, and the flow rate values in the last three columns should be divided by π [494].
350
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 119
ELLIPTICAL DUCTS WITH CENTRAL CIRCULAR CORES : / Re FOR FULLY DEVELOPED LAMINAR FLOW (FROM SHIVAKUMAR [489,494])
a*
r*
0.9 0.9 0.9 0.9
0.5 0.6 0.7
0 .95
23.519 23.435 23.159 16.816
0.7 0.7 0.5
0.5 0.7 0.5
21.694 19.402 19.32 1
fRe "2ao a
r*=
E. Circular Ducts with Central Regular Polygonal Cores Cheng and Jamil [109,111] employed a point-matching method to analyze fully developed laminar flow through an annulus having a circle as the outside boundary and a concentric regular polygon as the inside boundary. Both surfaces of the annulus were heated to obtain the (HI) boundary con dition defined in Table 5 and also on p. 328. They presented / R e and Nu H1 graphically. The / R e factors were obtained for the limited range 0 < ξί/α < 0.5 and are in excellent agreement with those of Table 120. The tabulation of / R e factors of Cheng and Jamil is available in [13,449]. Their Nu H1 are presented in Table 121 and Fig. 103. Hagen and Ratkowsky [113] studied the flow characteristics of the same problem by a discrete least squares method. Their / Re factors are reported in Table 120 and Fig. 102. Note that while Cheng and Jamil used ξι/α as an independent variable, Hagen and Ratkowsky employed ξ2/α. Here, ξ1 and ξ2 are the radii of inscribed and circumscribed circles, respectively. Cheng and Jamil [111] encountered some difficulties with their pointmatching method as the number of sides of the regular polygonal core decreased from 20 to 3. They found that the shear stress distributions and normal temperature gradients along the inner regular polygonal boundary exhibited a wavy character, with a very small region having a negative shear stress distribution. In spite of the difficulty with local values, Cheng and Jamil stated that the integrated overall quantities, such as / R e and Nu H1 were sufficiently accurate for practical purposes. Hagen and Ratkowsky [113] did not encounter the negative shear stress distribution and its wavy character; however, the least squares matching procedure became more difficult as the number of sides of the polygon became smaller. It may be noted that the presence of a small regular polygonal core at the center of the circular duct significantly increases the flow resistance ( / Re) and heat transfer (Nu H1 ).
TABLE 120 CIRCULAR DUCT WITH CENTRAL REGULAR POLYGONAL CORES: / R e
FOR FULLY DEVELOPED LAMINAR FLOW (FROM RATKOWSKY [492])
h.
fRe
a
|
15 Q I
n=3
n=4
n=6
n=8
n=l8
l.ooo 0.975 0.950 0.925 0.900
15.75
— 18— .42
15.60
— 21— .71
15.57
1 7 . 15
15.67 16.89 17.96 18.90 19.71
15.52 22.93 23.66
0.85 0.80 0.75 0.70 0.65
19.50 20.36 21 . 0 5 21.60 21.92
20.94 21.80 22.34 22.72 22.93
2 2 . 71 23.21 23.44 23.54 23.58
2 3.76 23.78 2 3.74
0.60 0.55 0.50 0.45 0.40
22.20 22.34 22.36 22.38 22.35
2 3 . 10 23.09 2 3 . 10 23.04 22.98
23.57 2 3.52 23.49 23.41 23.34
23.71 23.70 23.60 23.54 23.46
0.35 0.30 0.25 0.20 0 . 15
22.37 22.09 22. 1 1 21 . 9 8 21.90
22.91 22.83 22.68 22.55 22.31
23.25 2 3 . 12 22.98 22.80 22.55
2 3.37 23.26 23.1 1 22.92 2 2.65
0.100 0.075 0.050 0.000
21 . 64
21 . 9 5 2 1. 74
2 2 . 17
22.24
22.32
21.54 16.00
21 . 5 4 16.00
21.56 16.00
T
1
i
i
0.0
—
21 . 0 8 16.00
1
1
i
i
0.1
1
i—i
0.2
1
1
1
21 . 2 1
—
1 6— .00
1
— 23— .47 _ 23.68
1 9 . 74
1
1
1
— _ 23.94 2 3— .90 -
23.93
23.84
23.4 1 2 3 -. 0 5 -
2 3.75 2 3.62
—
1
1
1
—
1
1
1
1
1
Γ
i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—I
0.3
0.4
0.5
f2/a
06
0.7
0.8
0.9
1.0
FIG. 102. Circular duct with central regular polygonal cores:/Re for fully developed laminar flow (from Table 120).
352
XIV.
OTHER DOUBLY CONNECTED DUCTS TABLE 121
CIRCULAR DUCT WITH CENTRAL REGULAR POLYGONAL CORES : Nu H1
FOR FULLY DEVELOPED LAMINAR FLOW (FROM JAMIL [449])
Nu
ξχ/a
Hl
n=4
n=5
n=6
n=7
n=8
n=9
0.0 1/9 1/8 1/7 1/6
4.364 7.337 7.398 7.467 7 . 5 38
4.364 7 . 321 7 . 3 85 7.457 7.537
4.364 7.318 7.385 7.459 7.543
4.364 7 . 3 20 7.388 7.464 7.550
4.364 7.390 7.467 7.555
4.364 7.325 7.395 7.057 7.558«
1/5 1/4 1/3
7.616 7.69 3 7.738
7.626 7. 723 7.814 7.718
7.638 7 . 744 7.858 7.910
7.648 7.759 7.884 7.989
7.654 7.769 7.898 8.027
7.660 7.776 7.909 8.049
1 1/2
~i
1
1—r—i
1
i
-T
1
1
r—
,n = oo
8.0
7.0
6.0
n- sided regular polygon
5.0
4.364 4.0
_i
0.0
0.1
i
i
_i
i_
0.2
0.3
i
i
i i 0.4
i
i_ 0.5
Ci/a FIG. 103. Circular duct with central regular polygonal cores: Nu H1 for fully developed laminar flow (from Table 121).
F. Miscellaneous Doubly Connected Ducts Sastry [87] employed the Schwarz-Neumann alternating method, an approximate method of conformai mapping, to analyze the fully developed laminar velocity and (Hi) temperature problems for a circular tube with
F. MISCELLANEOUS DOUBLY CONNECTED DUCTS
353
central elliptical cores. He employed boundary conditions for the velocity problem as u = 0 on the circle and u = u0 Φ 0 on the ellipse. Hence his results for the velocity and temperature problems are not relevant for stationary ducts. Sastry [88] investigated fully developed laminar flow for a circular tube having a central square core with rounded corners, and considered the no-slip boundary condition at both inner and outer boundaries. He analyzed the fundamental boundary conditions of the fifth kind (see Table 4) for the temperature problem. He employed the Schwarz-Neumann alternating method and presented the approximate solutions for the velocity and temperature distributions, mean velocity and cross-sectional average fluid temperature. He did not present explicit formulas for / R e and Nujf.