streamwise vorticity effects m a y not be universally applicable,

but may instead only apply for certain strut sizes and flow
regimes. t~

Acknowledgments
Sincere appreciation is expressed to Ms. Qi Xie for her contri-
butions to this research, to the General Electric Company for
its support of the first author during this research project, and u
to the National Science Foundation for its support of the second
author under grant No. CTS-9211282.
0 OA ~2 O.3 0~4 Ova o 0 G7 o.a 0.9 .o

References
Baker, C. J., 1979, "The Laminar Horseshoe Vortex," J. Fluid Mech., Vol.
95, Pt. 2, pp. 347-368.
Eibeck, P. A., 1990, " A n Experimental Study of the Flow Downstream of a Fig. 1 The original LMTD correction factor chart for a single-pass cross-
Circular and Tapered Cylinder," ASME Journal of Fluids Engineering, Vol. 112, flow heat exchanger in which both fluids are unmixed (from Bowman et
pp. 393-401. al., 1940)
Eibeck, P. A., and Barland, D. E., 1993, "Turbulent Mixing Behind Two-
Dimensional and Finite Obstacles," Turbulent Mbcing, ASME FED-Vol. 174, pp.
57 -63.
Fisher, E. M., and Eibeck, P. A., 1990, "The Influence of a Horseshoe Vortex heat exchangers with both fluids unmixed (see Fig. 1 and, for
on Local Convective Heat Transfer," ASME JOURNALOF HEAT TRANSFER, Vol. example, Holman (1992), Incropera and DeWitt (1990), Kreith
112, pp. 329--335. and Bohn (1986), and White (1988)), it is highly probable
Mehta, R. D., 1984, "Effect of Wing Nose Shape on the Flow in a Wing/
Body Junction," Aeronautical Journal, Vol. 88, No. 880, pp. 456-460.
that the text authors have unwittingly extended the propagation
Merati, P., McMahon, H. M., and Yoo, K. M., 1988, "Experimental Modeling of small but discernible errors directly traceable to a paper by
of a Turbulent Flow in the Junction and Wake of an Appendage Flat Plate," one of the most recognizable names in the early heat transfer
presented at the A I A A / A S M E / S I A M / A P S 1st National Fluid Dynamics Con- literature, Wilhelm Nusselt. To illustrate by typical example the
gress, Cincinnati, OH, pp. 1255-1264.
Moffat, R. J., 1988, "Describing Uncertainty in Experimental Results," Exp.
extent of the error: For such a crossflow heat exchanger with a
Thermal Fluid Sci., Vol. 1, pp. 3 - 1 7 . capacity ratio of unity (i.e., the dimensionless temperature ratio
Pauley, W. R., 1993, "The Fluid Mechanics and Heat Transfer Downstream R = Athot/Atco~d = 1) in which the change in the cold fluid
of Struts Spanning a Low-Aspect:Ratio Channel," ASME Paper No. 93-HT-41. temperature relative to the difference between the inlet tempera-
Rood, E. P., 1984, "The Separate Spatial Extents of the Trailing Horseshoe
Root Vortex Legs From a Wing and Plate Junction," Paper No. AIAA-84-1526.
tures of the two fluids is P = At~ola/Atin = 0.6, the error in the
Tyszka, D. A., and Wroblewski, D. E., 1994, " A n Experimental Investigation LMTD correction factor, F, read off the chart is 3 percent. At
of Heat Transfer and Fluid Mechanics in the Turbulent Endwall Junction Boundary the extreme combination of low values of R and values for P
Layer Downstream of a Truly Streamlined Body," Fundamentals of Heat Transfer approaching unity, the error is considerably greater: At R = 0.2
in Forced Convection, ASME HTD-Vol. 285, pp. 25-32.
Wroblewski, D. E., and Eibeck, P. A., 1992, "Turbulent Heat Transport in a
and P = 0.975 the error exceeds 10 percent.
Boundary Layer Behind a Junction of a Streamlined Cylinder and a Wall," A SME
JOURNAL OF HEAT TRANSFER, Vol. 114, pp. 8 4 0 - 8 4 9 .
Background
The existence of these errors was discovered in the course
of preparing a partial set of LMTD correction factor charts and
effectiveness-NTU charts for inclusion in a student handout.
Rather than copying the frequently reproduced figures, it was
The L M T D Correction Factor for decided to generate them directly i n a spreadsheet by making
use of the well-established formulae readily available in the
Single-Pass Crossflow Heat literature. This is a quite straightforward task for all but the
Exchangers With Both Fluids case of a crossflow heat exchanger in which both fluids are
unmixed. The two-dimensional temperature field for both fluids
Unmixed in this particular exchanger configuration requires a more com-
plex analysis, which has been attempted by a number of authors.
Notable among these, Nusselt published an analytical solution
A. S. Tucker 1 to the pi-oblem in 1911, the solution being in the form of a
doubly infinite series, which was acknowledged to exhibit very
slow convergence. Almost two decades later (1930), he pro-
duced an alternative solution, again in the form of an infinite
Introduction series but more rapidly convergent than his earlier solution.
The majority of undergraduate texts on heat transfer include This later paper of Nusselt included tabulated numerical values,
a chapter on heat exchangers, almost always containing a sec- derived from his series solution, which were utilized by Smith
tion that explains the concept of the log mean temperature dif- (1934) and have been reproduced in their original three signifi-
ference (LMTD) for simple parallel and counterflow heat ex- cant figure form in at least one reasonably modern heat transfer
changers, and the correction factor, F, which must be applied text (Bayley et al., 1972).
to the LMTD for other heat exchanger types. The landmark paper that originated the most commonly used
Whenever that section of such a text includes a reproduction form of LMTD correction charts was that of Bowman et al.
of the LMTD correction factor chart for single-pass crossflow (1940) and, for the case of the crossflow heat exchanger with
both fluids unmixed, these authors drew on the work of both
Nusselt (1930) and Smith. It is this particular chart of Bowman
t Department of Mechanical Engineering, University of Canterbury, et al. that most heat transfer texts present and acknowledge.
Christchurch, New Zealand. Subsequent to the LMTD analysis approach, the extensive
Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF ME- work of Kays and London ( 1958 ) was published, presenting the
CHANICAL ENGINEERS. Manuscript received by the Heat Transfer Division June
1995; revision received October 1995. Keywords: Heat Exchangers, Numerical often-used effectiveness-NTU (e-NTU) charts, which overcame
Methods. Associate Technical Editor: T. J. Rabas. some shortcomings in the LMTD analysis procedure. To con-

488 / Vol. 118, MAY 1996 Transactions of the ASME

Copyright © 1996 by ASME
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 1.0 chart where there are small but significant discrepancies that
. .... lit were found to be insensitive to both retaining even more terms
in the series (typically six were more than sufficient in the areas
LL
0.9 of noticeable discrepancy) and the error tolerance applied on P
in the iteration procedure.
o~
u_ 0.8
Discussion
Possible explanations for the discrepancies are: Nusselt's
0.7~ 1930 series solution is in error; Nusselt's numerical values
I--- : I hl'q hi derived from his series are in error; Bowman et al. made an
error in implementing Nusselt's numerical values; Mason's
0.6-
'" ill I ,l'i rx\ll solution is in error (which would, if true, result in consequential
errors in Kays and London's e-NTU chart); or the numerical
0.5
0 0.1 0.2 0.3 0.4 0.5
I I\l 0'.6 0.7 0.8
I\1 !!
0~.9 1.0
iterative procedure being utilized was erroneous.
The third of these possible explanations can be quickly elimi-
P=(t2"tl)/(T 1 " t 1) nated by comparing Nusselt's numerical results with the curve
- - This study ............... B o w m a n et al. of Bowman et al. for the case of R being unity (implying a
capacity ratio of unity): The agreement is very good, verifying
Fig. 2 A comparison between the correction factor curves of the present
that they had correctly interpreted his tabulated data. When
study and those of Fig. 1
Nusselt's series solution (rather than his table of numerical
results) was subsequently encoded and summed within a
spreadsheet by a procedure very similar to that described above,
struct the e-NTU chart for the crossflow heat exchanger with the resultant curves were indistinguishable from those obtained
both fluids unmixed also requires use of an analytical expression via the Mason solution and, as independent verification of the
for the mean outlet temperatures of the fluids, and Kays and numerical procedure, matched extremely well with selected
London used an even more rapidly convergent infinite series points from an e-Ntu analysis.
solution, this one being due to Mason (1955). This solution For example, for the case R = 1.0 and P = 0.6, the iteration
was also the one that had been utilized by Stevens et al. (1957) procedure outlined above (using either the Mason or the Nusselt
but, like Kays and London, their charts of heat exchanger perfor- series) results in a solution F = 0.8113 at which the number
mance were in terms of effectiveness rather than L M T D correc- of transfer units Ntu = 1.8486 and the effectiveness e = 0.6000
tion. (as it should if P = 0.6 at a capacity ratio of unity, as is implied
Roetzel and Nicole (1975) recognized the potential use- by R = 1.0).
fulness of explicit representations of L M T D correction factors This effectiveness value also corresponds very well with the
in developing computerized packages for heat exchanger de- prediction of 0.6003 obtained fl'om the empirical relationship
sign. In obtaining suitable values for the coefficients to be in-
serted in their equation for the case of crossflow heat ex- ~ 1 -exp{ [exp(-NTU°7~r)- 1].NTU°22/r}
changers, they made use of the doubly infinite series solution
of Nusselt (1930). As will be seen, it is significant that it (where r represents the capacity ratio C,,i~n/C ...... = R = 1 in
was Nusselt's solution series that they used, rather than the this particular case) which usually is accepted as an approximate
representation of the e-NTU relationship for an exchanger of
subsequent tabulated numerical values that Nusselt presented
in his paper. this type (see, for example, Holman, 1992).
Furthermore, when R = 1 and P = 0.6 are substituted into
the general approximate explicit equation for mean temperature
Methodology difference developed by Roetzel and N i c o l e - - b a s e d on Nus-
Because Mason's paper was immediately accessible, his solu- selt's solution but not on his tabulated v a l u e s - - t h e resulting
tion was the one initially chosen for the present exercise of value of F = 0.8117 is in almost exact agreement with the
generating the L M T D correction chart from first principles. solution value of 0.8113 obtained here.
Thus an iterative solution procedure was set up within a spread- On the other hand, for the same values of R = 1.0 and P =
sheet to produce the desired chart, in the confident expectation 0.6, Nusselt's numerical value for F (reflected in Fig. 1) is
that it would be indistinguishable from that published by Bow- 0.835 which, for the same Ntu value of 1.8486, corresponds to
man et ah an incompatible value of 0.617 for the effectiveness.
Very briefly, the iterative scheme to generate the curve for a The explanation for the discrepancies is therefore clear: The
particular value of R proceeded as follows: For a particular error lies in the numerical values presented by Nusselt and
value of P at which F was to be evaluated, a trial value for a embraced by Bowman et al., not in the series solution he de-
(defined as the number of transfer units for the hot fluid) was rived. Close examination of Nusselt's paper reveals three princi-
assumed, and Mason's series summed to include sufficient terms pal points at which the errors arose, and these are attributable
that there was no change in the fifth significant figure. From to the severely limited calculator resources available in 1930.
that summed series, P was calculated and compared with the First, it appears from the numbers presented that Nusselt
value for P at which the value for F was being sought. By retained only five terms in evaluating his series and, while this
simply iterating on a in increments that were suitably reduced is of adequate accuracy over most of the domain of interest,
as the final solution was approached, the solution was completed the higher order terms do become significant under particular
when the calculated value for P matched the desired value combinations of his initial input parameters. Second, for three
within an acceptable tolerance (0.00001, with P being in the of his five-term series summation calculations (out of a total of
range from 0 to 1 ). With each point on each curve being calcu- 20) there were not-insignificant outright numerical errors.
lated independently of its neighboring points, there was no pos- Finally, and probably most importantly, the subsequent calcu-
sibility of accumulating en'ors as P was incremented. lation sequence adopted by Nusselt had the potential to intro-
The end result of doing this/'or the same values of R as are duce significant interpolation errors. From the original table of
plotted in Fig. 1 is shown in Fig. 2 (in which the curves from only 20 evaluated data points (many of which were inaccurate
Fig. 1 have been superimposed as accurately as physically read- for either or both of the first two reasons above), Nusselt devel-
ing values off a chart allows). Clearly there are areas of the oped a three-dimensional surface over a square coordinate base,

Journal of Heat Transfer MAY 1996, Vol. 118 / 489

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 1.0 Incropera, F. P., and DeWitt, D. P., 1990, Fundamentals of Heat and Mass
Transfer, 3rd ed., Wiley, New York.
Kays, W. M., and London, A. L., 1958, Compact Heat Exchangers, McGraw-
Hill, New York.
LL
0.9 Kreith, F., and Bohn, M. S., 1986, Principles of Heat Transfer, 4th ed., Harper
and Row, New York.
Mason, J. L., 1955, "Heat Transfer in Crossflow," Proc. 2nd U.S. National
Congress of Applied Mechanics, ASME, New York, pp. 801-803.
LL 8.8
Nusselt, W., 1911, "Der W~irmetlbergang im Kreutzstrom," Zeitschrift des
Vereines deutscher lngenieur, Vol. 55, pp. 2021-2024.
Nusselt, W., 1930, "Eine neue Formel fttr den Warmedurchgang im Kreuzs-
0.7 trom," Technische Mechanik und Thermodynamik, Vol. 1, pp. 417-422.
a Roetzel, W., and Nicole, F. J. L., 1975, "Mean Temperature Difference for
I-- Heat Exchanger Design--A General Approximate Explicit Equation," ASME
_1
0.6-
rl.x 2 I / JOURNAL OF HEAT TRANSFER, Vol. 97, pp. 5-8.
R=t ' \ Smith, D. M., 1934, "Mean Temperature-Difference in Cross Flow," Engi-
neering, Vol. 138, pp. 479-481,606-607.

0.5 I
\ 1\141 Stevens, R. A., Fernandez, J., and Woolf, J. R., 1957, "Mean-Temperature
Difference in One, Two, and Three-Pass Crossflow Heat Exchangers," Trans.
0 o.1 0.2 o.a 0.4 d.s 0.8 0.7 8.8 0.9 1.o ASME, Vol. 79, pp. 287-297.
P = (ta-t0/(-r 1 -t 1) White, F. M., 1988, Heat and Mass Transfel; Addison-Wesley, Reading, MA.

Fig. 3 The correct form of the LMTD correction factor chart for a single-
pass crossflow heat exchanger in which both fluids are unmixed. (This
figure repeats the corrected curves generated in this study and included
in Fig. 2, with the erroneous curves of Bowman et al. removed so as to
provide an uncluttered chart for other users.)
Optimum Design of Radiating
Rectangular Plate Fin Array
and this surface had pronounced curvature at two of its edges. Extending From a Plane Wall
By interpolating over this surface, he generated two subsequent
tables each containing 100 points, the vast majority of which
did not coincide with any of the comparatively sparse data
C. K. Krishnaprakas i
points from which the surface had been constructed.
Consequently, Nusselt's final tables of values presented to
three significant figures imply a calculation accuracy that is, in
fact, quite unjustified. Indeed, were Nusselt to submit his paper Nomenclature
to A S M E JOURNAL OF HEAT TRANSFER under its current policy
of requiting that papers containing numerical solutions have an Ap = profile area of fin array, m 2
assessment of numerical errors, it is probable that it would not b fin spacing, m
be considered for review! dFaul-,l~z = elemental view factor from d/.Zl to d#2
h = height of fin array, m
J = radiosity, W / m 2
Conclusion K = thermal conductivity of fin material, W / m K
It must be acknowledged that in current heat exchanger de- 1 = length of fin, m
sign practice, rarely would use be made of L M T D correction N = number of grid points along fin length
factor charts such as the one discussed here. Normally it would Nc = conduction-radiation number = Kt/12~rT~
be expected that L M T D correction factors, if used, would be Np = dimensionless profile area = Ap/w 2
represented by the appropriate equations incorporated into com- Nw = dimensionless width = crTb3w/K
puter-based design packages. Alternatively, the need for an t = semithickness of fin, m
L M T D correction factor can be bypassed through the use of the T = temperature of fin, K
e-NTU method of analysis. Tb = temperature of base, K
Nevertheless, and although generally they are not major in w = width of fin = b + 2t, m
their magnitude, the errors revealed unexpectedly as a result of x = coordinate along the length of the fin
this work do illustrate that even a respected name in the heat fl = dimensionless radiosity = J/aT~
transfer literature was not totally infallible, and that errors can c = emittance
easily propagate through the literature for decades; it is hoped 0 = dimensionless temperture = T/Tb
that now there is no need for those errors to propagate further, # = dimensionless coordinate length = x / l
particularly in undergraduate heat transfer texts where the cr = Stefan-Boltzmann constant = 5.67E-8
L M T D correction factor charts are commonly presented. For W/m2K 4
this purpose, Fig. 3 has been prepared. ~b = dimensionless rate of heat loss from fin and base
Finally, out of fairness to Nusselt and out of respect for his
standing, it must be said that current computers enable us easily
to undertake tasks that would not have been even contemplated Introduction
a few decades ago. Had Nusselt attempted his task with access Radiating fins are used in spacecraft and space vehicles for
to the facilities now available to even the most humble of re- rejecting on-board waste heat to deep space. Since weight is at
searchers, his published results almost certainly would have a premium, it is important to have minimum fin mass in these
been accurate and unchallengeable. applications. Several investigators have analyzed the problem
of minimization of fin mass in the past; however, there exist a
References
Bayley, F. J., Owen, J. M., and Turner, A. B., 1972, Heat Transfer, Nelson, Thermal Systems Group, ISRO Satellite Centre, Bangalore-560 017, India.
London. Contributed by the Heat Transfer Division of THE AMERICANSOCIETY OF ME-
Bowman, R. A., Mueller, A. C., and Nagle, W. M., 1940, "Mean Temperature CHANICALENGINEERS. Manuscript received by the Heat Transfer Division August
Difference in Design," Trans. ASME, Vol. 62, pp. 283-294. 1994; revision received August 1995. Keywords: Numerical Methods, Radiation
Hohnan, J. P., 1992, Heat Transfer, 7th ed., McGraw-Hill, New York. Interactions. Associate Technical Editor: M. F. Modest.

490 / Vol. 118, MAY 1996 Transactions of the ASME

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