Distribution of Eddy Viscosity and

Mixing Length in Smooth Tubes

R R. ROTHFUS, D. H. ARCHER, and K. G. SlKCHl
Carnegie Institute of Technology, Pittsburgh, Pennsylvania

Profiles of eddy viscosity and Prandtl mixing length in fluids flowing steadily and iso-
thermally in smooth tubes have been calculated from the velocity data of several investiga-
tors for Reynolds numbers between 1.2 X 10s and 3.2 X 10. In the transition range
unusually high values of eddy Viscosity and midng length are obtained in some p ~ r t i ~ ~
of the stream. In the fully turbulent range the effect of Reynolds number is small and the
mixing length tends toward zero at the center of the tube. The parameters for turbulent
flow between parallel plates have been correlated through the concept of an equivalent
tube. The results are of importance in designing equipment for heat and mass transfer
and mixing.

When dealing with processing equip- number of velocity distributions available mean temporal fluid velocity at the
ment, one often finds i t necessary or in the literature, there have been but few radial distance y from the tube wall and
highly desirable to be able to predict the attempts to translate these into profiles q0the corresponding value of the mean
rate of heat, mass, or momentum transfer of eddy viscosities or mixing lengthe. In local shearing stress. Following the pro-
at a 'particular point in a moving fluid. particular, the effect of Reynolds number cedure used by Murphree (6),the shearing
I n general, this requires some knowledge on the radial distribution of these stress for such a case of flow may be
of the relationship between the designated parameters has not been demonstrated written in terms of the eddy viscosity e as
flux and the corresponding potential adequately.
gradient at the spot in question. If the It is the purpose of this paper to present
flow is truly viscous, a direct solution for explicit correlations of eddy viscosities
the desired rate of transfer is sometimes and Prandtl mixing lengths for the steady
possible. When the flow is turbulent to isothermal flow of constant-density fluids The shearing stress might a h be ex-
any extent, however, empirical informa- in smooth tubes. The cffect of Reynolds pressed in terms of the Pranda mixing
tion is ultimately necessary, even in number on the radial distribution of the length (9). As shown by Schlichting (12),
,ductshaving simple cross-scctional shapes. two quantities has been calculated for the relationship between the eddy vis-
Progress in predicting mixing phe- both the transition and fully turbulent cosity and the mixing length 1 is
nomena as well as local rates of heat and ranges of flow. The supporting data have
m a transfer has not been so rapid as been smoothed and made internally e = p12&)
might be expected. One of the principal consistent in order to increase the ultimate
reasons for this seems to be the lack o f . utility of the calculated results.
sufficient Eonsistent information about If the fluid motion at the point under
eddy viscosities in elementary conduits BASIS OF CORRELATION
consideration is entirely viscous, the
such as smooth tubes of circular cross eddy viscosity and mixing length are zero.
section. In spite of the relatively large The steady, isothermal flow of a Regardless of the prevailing type of
constant-density fluid through a long, flow, the local shearing stress varies
straight tube of circular cross section will linearly with the distance from the wall of
Ksmal G. Sikcbi is a t Amraoti (M.P.), Bombay
State, India. be considered, with u representing the the tube' (IS). Thus, if ng0 represents

Vol. 4, No. 1 A.1.Ch.E. Journal Page 27

the shearing stress a t the wall and T~ can be represented approximately by an and coworkers (1, 7, 8) for flow between
the tube radius, equation of the form parallel plates. The Reynolds-number
range included is 600 to 3,240,000 for
U' = A + B In Y' (10) tubes and 6,960 to 53,400 for parallel
plates. The notable smooth-tube data,
where A and B are constants. Under such of Deissler ( 2 ) can be used to establish
Since the skin friction ngocan be obtained conditions the effect of the viscous term velocity profiles but have not been
readily from pressure-drop correlations, in Equation (8) is negligible and Equa- presented in a form adaptable to the
the cited equations permit calculation of tions (8) and (9) reduce to the simple calculation of velocity gradients.
the eddy viscosity and mixing length relationships Nikuradse has presented profiles of the
from experimental velocity distributions. total effective viscosity as well as of a
I n the present work it has been found Prandtl type of mixing length calculated
convenient to correlate the results in by means of the equation
terms of two dimensionless groups, E
and L, defined as follows: and 1 = u , d l - y/r,/(du/dg)
His tables contain several numerical
L = (3)41 - 3 (12) errors, however, and his velocity gradients
B To yo
near the center of the tube appear to lack
and To the extent that Equation (10) is satisfactory precision. The graphs of
correct, therefore, E and L should prove mixing length against position in the
to be independent of Reynolds number; stream indicate a consistent increase of
(5) the maximum value of E should be l/ro with increasing y / r o a t constant
obtained a t y / r o = 0.50 and that of L at Reynolds number in the turbulent range.
y/ro = 0.67. Since the logarithmic Sage and coworkers (8) have calculated
The symbol u* denotes the friction eddy-viscosity profiles from their excellent
velocity distribution cannot be valid at
velocity 4 3 The groups E and L the center of the stream or close to the velocity data for flow between broad,
have the same form as those used by wall of the tube, even at high Reynolds parallel plates. The velocity distributions
Nikuradse (6) and others, except for the numbers, these conclusions can be taken for tubes and parallel plates have been
introduction of the velocity ratio (uJV),. only as rough approximations. They do, found to coincide under the conditions
The latter ratio was used by Rothfus however, afford a starting point for any imposed by Rothfus and Monrad as
and Monrad (11) in modifying the usual attempt to describe the behavior of the noted in a previous paragraph. By virtue
u+,y+ correlation of velocity distribution dimensionless groups E and L in the of Equations (8) and (9), therefore, the
in order to remove the effect of Reynolds lower ranges of Reynolds number. corresponding profiles of E and L must
number in the fully turbulent range. At high Reynolds numbers the thick- also be coincident on this basis. The data
They found that the relationship between ness of the buffer layer and laminar film for parallel plates can thus be used in
the new parameters (if any such film exists) is a very small establishing the eddy-viscosity and mix-
part of the distance from the center of ing-length profiles for tubes as well.
the stream to the wall. For practical The literature contains no explicit
purposes the main-stream behavior can information about the characteristics of
be taken to apply over the entire cross the eddy viscosity and mixing length in
and section of the fluid. At lower Reynolds the range of laminar-turbulent transition.
numbers the buffer layer and laminar The velocity data of Senecal and Rothfus,
film are thicker. Therefore, the eddy however, provide the raw materia1
viscosity and mixing length can be necessary for the calculation of these
expected to be zero or very small for a profiles at tube Reynolds numbers up
could be adequately represented by a measurable distance from the wall. I n to 4,000.
single curve at Reynolds numbers greater v?ew of this, it is probable that the
than 3,000 in smooth tubes. It was also maximum point in the profile of E or
shown that the same correlation could L shifts toward the center of the stream COMPUTATIONS A N D RESULTS
be used for flow between parallel plates as the Reynolds number is decreased In both the transition and turbulent
if (V/um), were taken to be that in a through the lower turbulent and tran- ranges of flow, the eddy viscosity and mixing
tube having the same radius as the half sition ranges of flow. length parameters were calculated by means
clearance between the plates and operat- of Equations ( 8 ) and (9) respectively. The
ing a t the same friction velocity u*,with technique used in establishing the proper
a fluid of the same kinematic viscosity SOURCES OF VELOCITY DATA value of the gradient dU+/dY+ was de-
p / p as that between the plates. pendent on the precision of the experimental
I n terms of the U+ and Y+parameters, It is apparent that accurate values of information available in the Reynolds
EquatioG (1) and (2) take the forms the velocity gradient must be available number range under consideration.
before the eddy viscosity and Prandtl
mixing length can be computed satis- The Transition Range
factorily from experimental velocity data. In the transition range, at Reynolds
Suitable gradients can be obtained only numbers from 1,200 to 4,000, the velocity
and when the velocity profile is established by data of Senecal and Rothfus were first
numerous points of sufficiently high plotted as curves of U+ against Reynolds
precision. This requirement makes it number at constant values of y/ro ( =
(9) necessary to reject the results of several ' Y+/Y,+). Cross plots of U+ against y/ro at
constant values of the Reynolds number
investigations which could profitably be were then constructed and the curves on
The symbol Y,+ denotes the maximum included if velocity distribution alone both diagrams were adjusted t o achieve
(i.e., center-line) value of the modified were the subject of primary concern. smoothness, consistency, and the best
friction distance parameter Y+. The velocity data of Nikuradse (6) and agreement with the data. Figure 1 shows
At very high Reynolds numbers the of Senecal and Rothfus (14) can be the final form of the velocity correlation.
velocity distribution in the main stream used for smooth tubes and those of Sage The gradient (dU+/dY+) was obtained

Page 28 A.1.Ch.E. Journal March, 1958

from the derivative dU+/d(y/ro), since by
definition

dU' 1 dU'
dY' - Y,' d(y/r,) (14)
Values of Uf were read from Figure 1 at
even increments of 0.1 y / r ~and difference
tables were constructed from which values
of d U+/d(ylro) were calculated numerically.
Any one of three interpolation formulas,
two by Newton and one by Stirling, was
used in these calculations. The choice of a
formula depended on its applicability in
the particular range of y/ro being investi-
gated. I n regions of the stream where more
than one formula could be used equally
well, the average value of the gradient
obtained by the various means was com-
puted. The quantity E was then calculated
by the use of Equations (8) and (14) for
each point in the cross section at which the
gradient had been determined. These 1000 1500 2000 2500 3000 3500 4000
values were smoothed with respect to both Reynold) Number. N
,
radial position and Reynolds number in a
manner similar to that described for the Fig. 1. Cokelation of ~ o c dvelocities for transition flow in smooth tubes; dotted lines are
velocity distribution data. The consistent, calculated from viscous-flow equation (15).
smoothed results for Reynolds numbers
between 1,600 and 4,000 are shown in
Figure 2. Corresponding values of L,
calculated by means of Equation (9), are
shown in Figure 4.

The Turbulent Range

r In the fully turbulent range, a t Reynolds
numbers between 4,000 and 3,240,000,
values of the eddy viscosity parameter E
were calculated largely from the velocity
data of Nikuradse. The parallel-plate
velocity data of Sage and his coworkers
were taken to be equivalent t o smooth-tube
data on the basis proposed by Rothfus
and Monrad. Upon appropriate transforma-
tion, values of E obtained from Sage's data
were used to supplement those calculated
from Nikuradse's results.
Nikuradse presented values of the velocity
gradient du/dy but did not clearly indicate
his method of obtaining the derivative.
His tabulated results scattered sufficiently,
however, to suggest that his method was
not precise or that the velocity data were
not smoothed before the derivatives were
calculated. For more reliable values of the
velocity gradient from Nikuradse's data,
the following graphs were drawn at each
Reynolds number investigated by him:
in the range
0.3 < y/r0 < 1.0, In u against (y/ro)
in the range
0.02 < y/ro < 0.3, u against ( y / r 0 )

Straight l i e s representing the data as

closely as possible were drawn on both
diagrams. Curves of residual velocity show-
ing the deviation of the data from the two
straight lines as a function of radial position
were than drawn and smoothed. The slopes
of these curves were measured by means of a OD 01 a2 a a4 clj a6 a7 QR 0.9 10
prismatic tangent meter. The value of the Distance Ratio y/r.
derivative du/d( y/ro) a t a particular y/ro
was then obtained by adding the slope of Fig. 2. Effect of Reynolds number on the radial distribution of the eddy-viscosity parameter
the residual curve to the slope of the E in the transition flow range.
appropriate straight line. The gradient
du/dy followed directly. The values of
duldy thus obtained were checked by means
of difference tables as previously described

Vol. 4, No. 1 A.1.Ch.E. Journal Page 29

for the transition region, and excellent
agreement was notedl
dimensionless groups (p + ‘a)/rou,p and
Z/ro respectively. The latter group was
maximum eddy viscosity and mixing
length shift toward the center of the
In order to show the precision of the obtained from Equation (13) and there- stream in the transition range. Particu-
results and the nature of Reynolds-number fore involves the mixing length defined larly in the case of the mixing length,
dependence, values of E in both the tran- by that equation. At high Reynolds this is accompanied by an increase in
sition and turbulent ranges of flow are
plotted in Figure 3 as functions of Reynoldsnumbers, where the contribution of the magnitude of the parameter L. Thus
number at constant values of y/ro. The viscous shear is negligible, Nikuradse’s the central region of the tube in tran-
solid lines represent the recommended groups are almost independent of Reyn- sition flow becomes a zone of large-scale
correlation, smoothed with respect to both olds number except near the center of mixing. Dye studies (10) indicate that
Reynolds number and radial position. I n the tube, where they increase with the flow in this region may be sinuous
the placement of the recommended lines increased Reynolds number to a greater at one instant and part of a large dis-
due consideration was given t o the value ofextent than do E and L. This simply turbance eddy at another. I n either case,
dU+/dY+ derivable from the generalized reflects the small rotation of the lines in however, relatively long segments of
U+, Y + correlation through the use of
difference tables. Figure 3 can be used Fighre 3 resulting from inclusion of the the dye f i h e n t move as units without
directly as a working graph at Reynolds velocity ratio (V/u,), in the ordinate E . appreciable rapid small-scale diffusion;
numbers above 4,000. The nature of the The most striking departure from but it should be recognized that these
correlation, however, makes it advisable Nikuradse’s conclusions is evident in low-frequency transient effects influence
to use Figure 2 in the transition range. Forthe radial distribution of the mixing- the film thickness, eddy viscosity, and
purposes of comparison, Figure 2 also length parameter near the center of the mixing length in the transition range.
contains curves showing the radial distribu-stream. His published results indicate The values indicated in the cited graphs
tion of the eddy viscosity group at Reynoldsthat values of l/ro calculated from are time averages and are therefore
numbers of 6 X 103, 1 X lo4 and 3 X lo6. Equation (13) increase with distance dependent on the frequency with which
Figure 4 shows values of the mixing-
length group L, calculated by means of from the wall all the way to the center strong disturbance eddies are cast off
Equation (9), for several Reynolds numbers of the tube. Careful reevaluation of his and move downstream.
in the transition range. The radial distri- data, however, has led to the conclusion Figure 1 can be used as a working
bution at a Reynolds number of 3 X 108 is that the mixing-length parameter goes correlation of local velocities in the
also included. I n every case the lines cor-through a maximum point and tends transition range of flow. This type of
respond to the recommended values of E strongly toward zero where the center of correlation aids interpolation, as it
shown in Figures 2 and 3. To show a the stream is approached, as shown in eliminates the crossing over of neighbor-
comparison with Nikuradse’s data, points Figure 4. The fact that the velocity ing lines. The graph illustrates the com-
representing the average of his results for gradients calculated in the present work plexity of correlation in the transition
Reynolds numbers above 100,000 are
included in Figure 4. are more consistent than Nikuradse’s region and emphasizes the fact that in
lends support to this conclusion. I n the
The values of the velocity ratio (V/u,,,), transition flow the functional relation
which were used in computing E and L are last analysis, however, it remains for between Uf and Y +depends on the value
shown in Table 1. Obtained directly from more precise measurements of local of Reynolds number-unlike the relation
experimental data, these represent the velocities to furnish the basis for a firm in turbulent flow, where the relation
results of several reliable investigations judgement. It should be noted that the between U+ and Y + is virtually inde-
(3,4,6,14, 16). Prandtl and von Karman logarithmic pendent of Reynolds number.
velocity distributions predict zero mixing The dashed lines in Figure 1 indicate
length at the center of the tube. Since the velocity distribution calculated by
DISCUSSION OF RESULTS
both theories also indicate a finite the equation
Figure 3 shows that the radial distri- velocity gradient at the center, they
bution of the eddy viscosity group E cannot be considered valid in the latter
is not entirely independent of Reynolds vicinity; however, they do predict quali-
number in the higher turbulent range. tatively the relation between l/ro and
At Reynolds numbers greater than y/ro which has been calculated from
100,000, however, the variation is small experimental data obtained in the central
for y/ro values from 0.3 to 0.7. It is portions of smooth tubes.
apparent that the effect of the (V/u,,,), At Reynolds numbers less than 10,000, This expression can be obtained from
ratio in the eddy viscosity group is to there is a decided effect of Reynolds Equation (8) by assuming that E = 0
minimize the effect of Reynolds number number on the magnitude and radial and integrating at a given Reynolds
in the central portion of the stream,-where distribution of the parameters E and L. number from the tube wall, where Ucand
y/ro is greater than 0.7. At the same time The nature of the variation is illustrated Yf are zero, to any arbitrary point in
values of E close to the wall are main- in Figures 2, 3, and 4. The values of E the fluid stream where Yf = (y/ro)(Y,+)
tained reasonably independent of Reyn- over the cross section of the tube increase and U+ = U+. I n the calculations the
olds number, the variation at y/ro = 0.1 rapidly as Reynolds number is increased actual values of f and (V/u,), for smooth
being in the neighborhood of 10% over above 1,600. At a Reynolds number of tubes at the given Reynolds number were
the range 106 < NRs < 3 X 106. between 3,000 and 10,000, depending on used. A U+ calculated from Equation
Since viscous effects are negligible at the value of y/ro, E reaches a maximum (15) should be the correct one in the
high Reynolds numbers, the small varia- and then decreases with a further increase regions adjacent to the wall, where the
tion of E with Reynolds number must be of N R s . This behavior of E in the tran- assumption that E = 0 is reasonably
the result of a Reynolds-number effect on sition and lower turbulenbflow regions accurate. The profiles calculated by
the Uf, Y + velocity correlation. This cannot with certainty be interpreted in Equation (15) aid in interpreting the
effect is imperceptible in the velocity terms of existing theories of turbulence. experimentally determined velocity dis-
distribution itself but is magnified when The effect of laminar-film thickness in tribution in the transition range. Since
local gradients are taken. Even though the transition zone is clearly indicated in the actual profiles depart so gradually
the U+, Yf correlation is not unique, no Figures 2 and 4.The radial distributions from the viscous distributions in the lower
part of the main stream shows much of the eddy viscosity and mixing-length transition zone, i t is difficult to establish
effect of Reynolds number on E and L parameters are set out a greater and the laminar-film thickness very precisely.
in the upper turbulent range of flow. greater distance from the wall as the film But it does appear, however, that the
Nikuradse correlated eddy viscosities thickens with decreased Reynolds number. modified friction velocity parameter U,+
and mixing lengths in terms of the It is notable also that the points of taken at the edge of the laminar film i s

Page 30 A.1.Ch.E. Journal March, 1958

related to the center-line value Y,+ of

%
the modified friction distance parameter
through the expression
Uf+Y,+ = 900 (16) 0 10

within the limits of observation. Dye W 00

studies by Prengle and Rothfus (10)
and pressure-drop data by Senecal and
Rothfus (14) have yielded a value of

i4
about 1,200 for the constant in the last
equation. In both cases the experimental
technique might be expected to produce ? 005
somewhat higher values than those
3 004
obtained from velocitv measurements. I 1
The agreement can therefore be con- o.o& ?
sidered satisfactory in view of the various
uncertainties involved. Since the flow
within the laminar layer is parabolic, 0.0I
the radial distance from the center of
the stream to the edee of the laminar I 0'
1 I I I dd
Id
I I 1 1 1 1 1

10.
I I I ,,,,,,I
d
, I , I,,,,
I8
I , , , , , , , ,]
loT
Reynolds Number. N,.
film rf can be obtained from Equation
(16) and is given by the relationship Fig. 3a.

1- t)'
2 = 14'400
(NRc fl)2(17)

Prengle and Rothfus report a value of

19,600 for the constant on the right-hand
side of the equation.
It is apparent from Figure 1 that the
divergence of the actual velocity profiles
from the viscous-flow extrapolations be-
come less pronounced a t very small values
of y/ro. This suggests that a higher
Reynolds numbers where the laminar
film, if any, is extremely thin, velocity
measurements become poor indicators of
the film thickness. Instead, it can be
predicted that such measurements should
lead to the notion that a much thicker
film exists than is actually the case, since
the effects of weak eddies in the buffer
layer next to the film will not be picked
up by ordinary means. Consequently, it
is not surprising that velocity data seem Reyndds Number
to indicate that the modified friction
Fig. 3 b.
distance parameter Y,+ at the edge of the
apparent film is approximately constant Fig. 3. Effect of Reynolds number on the eddy-viscosity parameter E at fixed position in
the stream.
TABLE1
THE RATIOO F BULKAVERAGETO and independent of Reynolds number a t in Figure 3 do not coincide with Nikul
MAXIMUM J k m m I E s IN SMOOTH TUBESReynolds numbers above, say, 10,000. radse's data near the center of the tube.
N R ~ (J'/u,)p N R ~ (Vl~m),
The eddy viscosity and mixing-length It is believed that the data of Senecal
parameters used in Figures 2, 3, and 4 and Rothfus are more reliable than
2,m 0,500 8,000 0.774 were chosen because they yield relation- Nikuradse's in this region. The recom-
2y100 O.so2 9~000 0 778 ships which are essentially 'independent mended lines have, therefore, been drawn
2i200 0.567 lojooo 0.780 of Reynolds number in the higher tur- to progress smoothly to Senecal's values
2,300 0.604 15,000 0.792
o, 798 bulent range and are relatively easy to a t the upper limit of the transition region.
2,400 o. 636 20,000
2,500 o.667 25,000 o,803 interpolate. To aid in the placing of the The parallel-plate data of Sage and his
2,600 0.685 30,000 0,808 recommended lines on these graphs, the coworkers were found to be in excellent
2,700 0.697 40,000 0 812 generalized diagram of U+ against Y+ agreement with tube data on the equiv-
2,800 0 705 50,000 0.818 was used as a guide in the middle range alent basis suggested by Rothfus and
2,900 0.710 60,000 0 822 of y/ro values. Since the assumption of a Monrad. This result was to be expected
3,000 0.717 80,000 0.826 unique U+, Yf relationship is only a in view of the excellent agreement in
3,250 x lo6 o.830 rough approximation near the wall and velocity data obtained on U+, Y +
37500 0.731 lo6 0.841 a t the center of the stream, the slope of coordinates. The correspondence of eddy
3,750 0 738 4 X lo6 0.852
4,000 o,744 106 o,857 the generalized U+, Y + curve is not viscosities, however, constitutes a more
4,m o,752 106 o,864 sufficiently accurate to be of much help stringent check on the postulated equiv-
5,000 0.758 2 X lo6 0.872 in these zones* alence of tubes and parallel plates.
6,000 0 765 3 X lo6 0.877 In the lower turbulent range it is Isakoff and Drew (3) have measured
7,000 0.770 apparent that the recommended curves velocity profiles for mercury flowing

Vd 4, No. 1 A.1.Ch.E. Journal Page 31

 average flow velocity in a
smooth tube, dimensionless
u* = friction velocity parameter
GiZ, ft./sec.
U+ = velocity parameter u/u,, di-
mensionless
U+ = modified velocity parameter
u+ (V/um),, dimensionless
V = average flow velocity within
a tube, ft./sec.
2 = distance measured along the
tube axis, ft.
Y = distance measured in a radial
direction from the tube wall:
(ro - r ) , ft.
2/+ = distance parameter yu,p/p,
dimensionless
Y+ = modified dimensionless dis-
tance parameter (um/V), y+,
dimensionless
Y,+ = maximum value of Y + in a
tube (rou*p/d (um/V),, di-
mensionless
Greek letters
B = eddy viscosity, lb. mass/(ft.
(sec.)
P = coefficient of viscosity, lb.
mass/(ft .) (sec.)
P = fluid density, Ib. mass/cu. ft.
7 = shear stress at any point in
the tube, Ib. force/sq. ft. 1

To = shear stress at the tube wall,

lb. force/sq. ft.

LITERATURE CITED

1. Corcoran, W. H., F. Page, Jr., W. G.

Schlinger, and B. H. Sage, Ind. Eng.
Chem., 44,410 (1952).
Distance Ratio y/, 2. Diessler, R. G., Trans. Am. Soc. Mech.
Fig. 4.Effect of Reynolds number on the radial distribution of the mixing-length parameter Engrs., 73, 101 (1951).
L in the transition flow range. 3. Isakoff, S. E., and T. B. Drew, “Proc.
General Discussion on Heat Transfer,”
Institution of Mechanical Engineers,
-
London (1951).
through smooth tubes a t ten different NOTATION 4. Klimaszewski, I. C., *V.S.thesis, Car-
nenie Inst. Technol., Pittsburgh (1950).
Reynolds numbers in the range of 125,000 A , B = constants 5. MGrphree, E. V., Ind. Eng. chem., 24,
to 500,000. They present a graph which 726 (1932).
shows a calculated eddy momentum
E = eddy viscosity parameter
e/rouu,p (u,,,/V),, dimension- 6. Nikuradse, J., V.D. I . Forschungsheft,
diffusivity ratio ( e / p ) / ( ~ / p ) ~=~ E/E,,,,,
= 356, 1 (1932).
less
as a function of y/ro. They were unable 7. Page, F., Jr., W. H. Corcoran, W. G.
to detect a variation of this ratio with f = fanning friction factor, di-
mensionless
Schlinger, and B. H. Sage, I d . Eng.
Reynolds number in the range of their Chem., 44,419 (1952).
investigation. Their calculated values of 90 = conversion factor, 32.2 8. Page, F., Jr., W. G. Schlinger, D. K.
E/E,,,.,agree with those presented in (lb. mass)(ft.) Breaux, and B. H. Sage, Ind. Eng.
Chem., 44, 424 (1952).
Figure 3 with an accuracy of &7% (lb. force)(sec.*) 9. Prandtl, L., Z . angew. Math. u. Mech.,
except in the center of the tube, where 1 = Prandtl mixing length, ft. 5, 137 (1925).
the Isakoff and Drew calculations are L = Prandtl mixing-length param- 10. Prengle, R. S., and R. R. Rothfus,
about 15% greater than the results eter (Z/ro)(um/V),, dimension- Ind. Eng. Chem., 47, 379 (1955).
presented here. Since Isakoff and Drew less 11. Rothfus, R. R., and C. C. Monrad,
state that their diffusivity data scatter ibid., 1144.
by as much as +lO%, it can be con- N R s = Reynolds number for flow in 12. Schlichting,Hermann, “BoundaryLayer
a tube 2roVp/p, dimensionless Theory,” p. 387, McGraw-Hill Book
cluded that there is substantial agreement
between their work and the results r = distance to a point measured Company, Inc., New York (1955).
from the tube axis, ft. 13. Ibid., p. 400.
presented here. Furthermore they con- 14. Senecal, V. E., and R. R. Rothfus,
clude that (e/p),= varies as the 0.83 rr ’ = distance from the tube axis to
Chem. Eng. Progr., 49, 533 (1953).
power of the Reynolds number. Figure 3 the edge of the laminar film,
ft. 15. Stanton, T. E., and J. R. Pannell,
predicts that for a given fluid ( ~ / p ) ~ ~ * Trans. Roy. SOC.(London), A214, 199
will vary with about the 0.9 power of ro = tube radius, ft. (1916).
the Reynolds number over the range of Urn = maximum flow velocity with-
Reynolds numbers investigated by Isa- in the tube, ft./sec. Manuscript submitted February, 1957; rewision
received August 16, 1957; paper arcepted September $0,
koff and Drew. (urn/V), = ratio of the maximum to the 1957.

Page 32 A.1.Ch.E. Journal March, 1958