This procedure lends itself to computer solution, as follows. (1)Input data, W ) , convergence criterion; n = 0. ( 2 ) Reduce all data to ( 7 1 / G ) using eq 4A; n = n 1. (3) Perform straight-line least-squares fit of reduced stress as a function of shear rate; obtain C, Q I , ~ ,and the standard deviation, A%. (4) Store A%(N). If A%(N) lowest so far; store C,
l0.C
+
171,l.
. r
(5) If n > 2 go to (7). (6) Change a, @(N+l) = t. Go to ( 2 ) . (7) If convergence has been achieved, go to (9). (8) Fit a quadratic (A% = a0 ala to lowest A%(i). Use and two nearest neighbors. Solve for best a m i n = ~ 1 / ( 2 ~ 2Set ) . @(N+l) = @'min. GOto (2). (9) Output results. The method is inexpensive (for a reasonable convergence criterion) and reliable; it was used to obtain the results of this paper. Two cautions should be observed, however. First, the method is intended only for power-law fluids. Newtonian fluids are a special case, of course. Second, as evidenced by repeatability studies in this paper, normal data will yield ab f -10%. Hence, a very tight criterion of fit is usually wasteful; a t best, the value of b obtained is a good estimate. However, this seems to compare well with the variability of more difficult experimental methods as based on the limited error analysis for data in Schweyer and Lodge (1974).
+
e
1.0
u
+
0.1 1
0.2
0.4
6
0.6
0 8
1.0
= (T1/G)/i'
Figure 9. Solution S( = 1 - e - { ; {(S).
done by Penwell, Porter, and Middleman [J.Polym. Sci., A2, 9 (1971)l. IV. If it is not possible to draw the ( T / G )power-law line, giving C and Q ~ J ,then a more laborious (yet conceptually simple) method becomes necessary. (1)Guess value of a, giving b . ( 2 ) Correct all data using eq 4A. (4)Repeat steps (1) and (21, adjusting @ until the corrected data have minimum deviation from a best fit power law line.
+
Literature Cited Schweyer, H. E., Busot, J. C., Highw. Res. Rec., No. 361 (1971). Schweyer, H. E., Highw. Res. Rec., No. 404 (1972). Schweyer, H. E., Lodge, R. W., lnd. Eng. Chem., Prod. Res. Dev., 13, 202 (1974). Kafka, F. Y., Master of Science Thesis, University of Florida, 1974. Schweyer, H. E., Moore, J. P., Ling, J. K., ASTM Spec. Tech. Pub., 532 (1973).
Receiued for reuieu! March 31, 1975 Accepted November 24,1975
~~
COMMUNICATIONS
Flow Transition in Finned Tubes
Transition from laminar to turbulent flow in tubes with longtitudinal internal fins has been examined theoretically using Hanks' criterion. The critical Reynolds number is a strong function of the ratio of fin height to equivalent diameter, X/De. The agreement with the experimental data is good. A regression equation for the critical Reynolds number as a function of X/De is given.
Recently, tubes with longtitudinal internal fins (forgedfin tubes) have attracted much attention in the design of compact heat exchangers. Such tubes, when used as heat exchangers, give a much higher ratio of heat transfer to pressure drop as compared with smooth tubes in laminar flow (Watkinson et al., 1974). The experimental work of Watkinson et al. indicated 144
Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
that the transition from laminar to turbulent flow, as shown from their pressure drop measurements, occurs a t much lower Reynolds numbers than the commonly accepted value of 2100 for smooth tubes. The purpose of this study is to examine the transition Reynolds number for such flows by first solving the momentum flow equation and then applying Hanks' fluid flow transition criterion.
0
EXPERIMENTAL DATA (YITKIHSOH E l
LL I
THE09ETICAL ( T i l s STU3Y1
-
\
\
REGRESSION E 3 U A T I O N 6
/ 0 700
-
0 l
I
l
I
1
. I
I
Figure 1. T u b e geometry.
For a Newtonian fluid flowing in steady, fully developed, laminar flow in a horizontal duct the momentum equation, in cylindrical coordinates, is 1 a2u’ + -1- au’ -+ - = -a%’ a@ r’ ar’ arr2
1dp p dz‘
where f * is a body force per unit mass, e.g., gravity. For a Newtonian fluid flowing in fully developed steady, laminar flow in a horizontal duct, eq 3 reduces to
Putting R2 d p u =ut/(;z)
and
Introducing dimensionless quantities, eq 4 becomes r = r’/R
eq 1 becomes where De‘ = 4A/C, De = De’/R, and Re = pDe’ D’lp. Using Hanks’ approach, the critical Reynolds number a t which transition to turbulent flow occurs is then given by Due to symmetry, only the flow region OABCEO of Figure 1 needs to be analyzed. Consequently, eq 2 is subject to the boundary conditions au _ -0
(along OA, symmetry)
u =0
(along ABCE, no slip)
a0
au _ a0
-- 0
du
ar
0
(5)
where K is a unique constant evaluated by Hanks as 404. After the dimensionless velocity u was obtained a t each of the grid points of the flow field, the term
(along OE, symmetry)
and -=
I [ (p)’+(;$)z]1’21
(Re)crit= KDeDImax u
(at r = 0, symmetry)
In order to facilitate the solution of the momentum equation, the fin was taken to form part of the cylindrical coordinate system. Equation 2 was put in a finite difference form using second order central difference equations (Roache, 1972) and solved numerically. A total of 21 and 11 points in the radial and angular directions were used, respectively. The convergence criterion taken was when the velocity difference a t each grid point between two successive iterations was less than 10-j. The finite difference scheme and the tube geometry are shown in Figure 1. The numerical solution provided the axial dimensionless velocity a t each of the grid points and the average tube velocity, D, was obtained using Simpson’s integration formula. Hanks (1963) showed for the case of a rectilinear flow that (3)
was evaluated a t each of the mesh points using Lagrange 3-points differentiation formulas and its maximum value then used to evaluate the critical Reynolds number as given by eq 5 . The flow field in a finned tube for various fin geometries was obtained and the corresponding (Re),.,it was evaluated. Figure 2 shows the dependence of the critical Reynolds number on the fin geometry as represented by X / D e , where X is the fin height. The experimentally determined values of the critical Reynolds number as obtained by Watkinson e t al. are also shown in Figure 2 for comparison with present theoretical results. I t can be seen that the critical Reynolds number decreases as XIDe increases. The general agreement between the theoretical values obtained in this study and the corresponding experimental data of Watkinson et al. for straight fins is fairly good. Here, Hanks’ criterion is shown to give an excellent indication on the onset of turbulence in straight internally finned tubes. Such a criterion is, of course, very useful in any rational design pertaining to fluid flow in ducts. Table I gives the values of i and j at which 2 was found to be a maximum. Such a maximum value occurs near the fins and the variation of 2 in this region was a very weak Ind. Eng. Chem., Fundam., Vol. 15,No. 2, 1976
145
Table I. Values of (Re),,it and fRe for the Various Fin Geometries
fi, d e g 9 9 9 9 9
6 6 6
6 4.5 4.5 4.5
No. of
fins 8 8 8 16 16 12 12 12 24 16 16 16
X
De
A
C
XIDe
0.10 0.15 0.20 0.15 0.2
1.558 1.397 1.263 1.0435 0.883 1.413 1.224 1.077 0.852 1.292 1.089 0.938
3.002 2.967 2.915 2.794 2.689 3.022 2.967 2.915 2.794 3.022 2.966 2.915
7.757 8.495 9.232 10.71 12.18 8.557 9.695 10.83 13.10 9.359 10.90 12.43
0.0642 0.1074 0.1583 0.1437 0.2265 0.0708 0.1225 0.1858 0.1761 0.0774 0.1377 0.2132
0.1
0.15 0.2 0.15 0.1 0.15 0.2
function of position. The values of fRe together with the fin geometry are also shown in Table I. The regression equation representing the critical Reynolds number as a function of XIDe as obtained in this study is given by ( R e k t = 2060 - 8530 ( X I D e )
+ 13380 ( X / D e ) 2
(6)
for 0.065 IXIDe I0.22. It is interesting to note that the regression equation for fRe, namely
De’ (z)
1.17
fRe = 16.07
obtained in another study (Nandakumar and Masliyah, 1975), where “triangular” fins were employed rather than the present “trapezoital” fins, predicts fairly well the fRe values given in Table I. In the above mentioned study, a finite element method technique was employed in the solution of the momentum equation.
Nomenclature A = flow area, dimensional C = wetted perimeter, dimensional De = dimensionless equivalent diameter, ( = De‘IR) De’ = equivalent diameter, dimensional f = fanning friction factor f * = body force per unit mass
146
Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
fRe
11.3 10.4 9.87 6.76 5.88 9.7 8.56 7.97 4.69 8.38 7.16 6.5
(i,j) 13J1 12,11
13,11 11J1
10,11 13,11 12,11 12,11 11,11
12,11 11,11 11,11
i , j = coordinates of a grid point; i in radial direction; j in angular direction
K = constant ( = 404) p = pressure, R = tube radius, dimensional r = radial distance. dimensionless (= r’lR) r’ = dimensional radial distance Re = Reynolds number, pDe‘u‘/H (Re)crit = critical Reynolds number u = dimensionless axial velocity, u = v‘/(R2/pdpldz’) u’ = axial velocity, dimensional 0’ = average axial velocity, dimensional 0 = dimensionless mean velocity, d = u’/(R2/pdpldz’) X = dimensionless fin length (= xV/R) X’ = fin length, dimensional z’ = axial distance, dimensional
Greek L e t t e r s @ = fin angle, deg V = gradient 0 = angle p =
Acknowledgment The authors are indebted to the National Research Council of Canada and to the University of Saskatchewan for the continual financial support. T h e authors are also indebted to Dr. A. P. Watkinson for his encouragement in initiating the study of finned tubes and to Noranda Research Centre for providing us with their experimental findings.
(Re1crit 1606 1407 1245 1085 905 1462 1245 1070 872 1331 1094 920
fluid viscosity
p = fluiddensity
L i t e r a t u r e Cited Hanks, R. W., A.I.Ch.f. J.,9, 45 (1963). Nandakumar,K., Masliyah, J. H., Chem. fng. J., 10, 113 (1975). Roach, P. J., “Computational Fluid Dynamics,” Hermosa Publishers, 1972. Watkinson. A. P.. Miletti, D. L., Kubanek. G. R., Internal Report No. 303 (1974), Noranda Research Centre, Pointe Claire, Quebec, Canada, 1974.
Department of Chemistry and Chemical Engineering Uniuersity of Saskatchewan Saskatoon, Saskatchewan, Canada S7N OW0
J. H. Masliyah* K. Nandakumar
Receioed for reuiew February 28, 1975 Accepted December 22, 1975
