Heat transfer by free convection from horizontal cylinders in diatomic gases

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Title:
Heat transfer by free convection from horizontal cylinders in diatomic gases
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NACA TM
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73 p. : ill. ; 27 cm.
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Hermann, Rudolf
United States -- National Advisory Committee for Aeronautics
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Washington, D.C
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Aerodynamics   ( lcsh )
Heat -- Transmission   ( lcsh )
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bibliography   ( marcgt )
technical report   ( marcgt )
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Abstract:
Heat transfer by free convection from horizontal cylinders in diatomic gases is investigated theoretically and experimentally. The heat transfer is given by the theoretical equation Nu = 0.37(Gr)1/4 for 10⁴ < Gr < 3 x 10⁸. Experimental determinations of velocity, temperature, and heat transfer are in good agreement with the theory. It is found that the total heat transfer and the boundary-layer development at the upper stagnation point and at the upper edge, respectively, of a horizontal cylinder are equivalent to the respective quantities for a vertical plate which transfers heat on both sides and is 1.31 times the cylinder diameter. A review and discussion of previous investigations of free-convection heat transfer from horizontal cylinders is included.
Bibliography:
Includes bibliographic reference (p. 53-55).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by R. Hermann.
General Note:
"Report date November 1954."
General Note:
"Translation of "Wärmeübergang bei freier strömung am wagrechten zylinder in zweiatomigen gasen." VDI Forschungsheft, No. 379, 1936."

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University of Florida
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Full Text
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1566


HEAT TRANSFER BY FREE CONVECTION FROM HORIZONTAL

CYLINDERS IN DIATOMIC GASES

By R. Hermann


I. DETERMINATION OF HEAT-TRANSFER LAW FOR HORIZONTAL CYLINDER FROM

TESTS WITH PARTICULAR ACCOUNT TAKEN OF THE

TEMPERATURE CHARACTERISTIC Tel

1. Introductory Remarks

The case of the horizontal cylinder is of particular importance in
the study of heat transfer by free convection for the following reasons:
in the first place, next to the rectangular plate it represents the
simplest two-dimensional case; and second, a very wide range of measure-
ments is possible, from the finest electrically heated glow lamp wires
to pipes heated by liquids or gases flowing through them.

To investigate free-convection flow from the point of view of
similarity considerations, it is convenient to consider the case of
small temperature differences between the heated body and the surround-
ings; in this case all the properties of the medium, even the density
(ref. 1, p. 429), in the entire temperature field may be assumed
constant. The case of large temperature differences, for which the
variation in properties over the temperature range can no longer be


"Warmeubergang bei freier Stromung am wagrechten Zylinder in
zweiatomigen Gasen." VDI Forschungsheft, No. 379, 1936, pp. 1-24.
1Accepted as dissertation by the Technical High School at Aachen
with the approval of Prof. Dr.-Ing. C. Wieselsberger and Prof. Dr.
W. Muller. The tests were carried out in the division for applied
mechanics and thermodynamics of the Physical Institute at the University
of Leipzig. The author takes this occasion to thank Prof. Dr. L.
Schiller for the interest which he took in this work and for his valua-
ble advice.







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neglected, is treated separately. As first pointed out by W. Nusselt
(ref. 2), it follows from the differential equations of free convection,
together with the uniquely determining boundary values (ref. 1, p. 428)
for the case of small temperature differences, that the nondimensional
heat-transfer parameter Nu (Nusselt number2) is determined by the lift
coefficient Gr (Grashof number) and the characteristic Pr (Prandtl
number) for the molecular constitution of the gas3:

Nu = F(Gr, Pr) (1)

where

Nu = ad/X, Gr = d3 g p /v2 Pr = v/a (2)

a is the heat-transfer coefficient (cal/cm2)(sec)(oC), d is the cylin-
der diameter (cm), g is acceleration of gravity (cm/sec2), X is the
heat conductivity (cal/(cm)(sec)(oC)), v is the kinematic viscosity
(cm2/sec), a is the temperature conductivity (cm2/sec), 0 is the coef-
ficient of expansion (0C-1), and 8 is the temperature difference (oC)
between body (ty) and medium (t.).

The case of large temperature difference between the body and the
surroundings, for which the gas properties vary over the field, was
similarly first considered by W. Nusselt (ref. 2). The following ex-
pression is obtained for the heat transfer under the restricting con-
dition that the temperature dependence on v and a may be repre-
sented by exponential laws in the absolute temperature, the exponents
of which for gases of the same substance must be equal4;

Nu = F(Gr, Pr, Te) (3)

The form here chosen for the temperature characteristic5


2In the notation of the characteristic numbers, the proposals of
the Heat Conference at Koln in 1931 are followed. See Forschg. Ing.-Wes.,
Bd. 2, p. 380, 1931.

3See eq. (5).

4This is approximately the case, for example, for air, oxygen, and
hydrogen; see reference 1, p. 433.

5The abbreviation Te is chosen to conform to the other abbrevia-
tions for the characteristic numbers and to remind the reader that it
refers to a temperature relation.







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Te = 8/T, (4)

(T,, T.,(K) = absolute temperatures of the body and the surroundings,
respectively) differs inessentially from the nondimensional ratio
T,/T. chosen by Nusselt, namely by the constant -1. Equality of Gr,
Pr, and Te in two different cases means complete similarity, that is,
similarity of the velocity field, the temperature field, and the field
of all the coefficients characteristic of the substance (ref. 1, p. 434),
and therefore also equality of the Nu number. The possibility still
remains, however, that similarity may exist at the same time that the
previously mentioned condition of the power law is not satisfied.

In the present report the characteristics represented in equa-
tion (3) will be computed for the heat transfer from heated wires and
pipes in air, hydrogen, and oxygen; and from these values the most
probable form of the heat-transfer law, obtainable at present, will be
determined for the case of the horizontal cylinder in diatomic gases
(Pr = 0.74).5a The dependence of Nu on the temperature characteris-
tic Te, which follows from the similarity theory and is illustrated by
an example (ref. 1, fig. 1) as a third independent variable, will be ver-
ified for several tests in the range of Gr from 10-4 to 10. For the
smallest values of Gr, the effect of Te considerably outweighs that
of Gr. In the region of large Gr from 104 to 107, on the contrary,
Te is practically without effect. The large scatter of the test points
in the nondimensional representation of the test results of Nusselt
(ref. 2) and Davis (ref. 3) (32 percent in the case of Nusselt) is due
to the fact that the parameter Te is not taken into account in accord-
ance with equation (1). The dependence of Nu on Gr, Pr, and Te
will be theoretically clarified hereinafter. The effect of Te for
small Gr may also be correctly estimated quantitatively.

From the tests conducted by various investigators for different
finite Te values, the limiting law of small temperature difference is
determined in each case by extrapolation to Te = 0. This is of
primary theoretical and practical significance since it is identical
with the theoretically simpler case (1) and is thus free from the pre-
viously mentioned restrictive assumption with regard to the temperature
dependence of the constants defining the properties of the substance.


baFor the values of X and v required for computing the char-
acteristics, values are assumed for the temperature t. (in agreement
with E. Schmidt). The similarity consideration leaves the choice of
the reference temperature free.






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2. Radiation

From the measured over-all heat losses directly determined in the
tests, the amount due to radiation must be subtracted. For this pur-
pose, a well known equation (ref. 4, eq. 38, p. 232) was employed; the
best mean values at present known were substituted as radiation con-
stants for the metal cylinder surfaces, as follows: for platinum, nickel,
silver, and tantalum, the values given by H. Schmidt (ref. 5); and for
copper and iron, the values given by Gr6ber (ref. 6, p. 196). Small
deviations as compared with the values used by the investigators them-
selves are without significance in the range of small values of Gr
because the radiation component in the case of thin wires amounts to
only a few percent of the heat of convection. For the tests of Wamsler
and of Koch on thick pipes, however, for which the radiation losses are
up to 60 percent, an accurate knowledge of the radiation coefficients
is required. These coefficients were obtained by Wamsler (ref. 13)
through his own radiation tests. After correction6 of computations of
reference 13, the values obtained herein for the radiation heat are
0.2 to 2 percent larger.


3. Property-Determining Constants of Gases

In order to compute Nu, Gr, and Pr, values of the following
properties are required: density p, dynamic viscosity u, heat con-
ductivity X, and specific heat c for air, hydrogen, and oxygen.

(a) Constants for air. The density was computed throughout the
temperature and pressure range according tc the ideal gas law7.

The true specific heat for constant pressure was obtained from an
equation given by Holborn and Jakob8 for the mean specific heat.

The temperature dependence of the dynamic viscosity was determined
by graphical adjustment of the values given by Erk9. These values


6Wamsler (ref. 13) computes incorrectly with the previously men-
tioned radiation equation without the factor of the area ratio in the
denominator.

7Landolt-B6rnstein: 5th ed., vol. I, p. 43.

8Landolt-Bornstein: 5th ed., vol. II, p. 1274.

9Landolt-B6rnstein: 5th ed., vol. I, pp. 177-181, Eg. I, pp. 143-
144, Eg. IIa, pp. 138-141.







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differ only slightly (at most, 4 percent) from the graphical mean
values given by Erk from 0 to 7000 C10.

The heat conductivity at high temperatures was obtained by a gas
kinetic extrapolation, since the test values collected by Jakobl1 lie
only between 0 and 2120 C. The Nusselt formula (ref. 1, p. 491;
ref. 6, p. 192), which deviates from the available test values by at
most 2.8 percent, is employed herein.

From the gas kinetic relation12

Pr = i c/X = (n + 2)/(n + 4.5) (5)

where n is the number of degrees of freedom of the molecular motion
according to which the Pr number of a gas is independent of the tem-
perature (and moreover is the same for all gases with the same number
of atoms); there is obtained for air a small decrease of Pr = 0.739
for 0 C to 0.721 at 2000 C and 0.679 at 10000 C (value obtained from
formula for n = 5 is Pr = 0.737), which, however, is of no signifi-
cance because it lies within the uncertainty limits of the p and X
values. The same holds for the differences in the Pr numbers among
air, hydrogen, and oxygen.

(b) Constants for hydrogen and oxygen. For the kinematic vis-
cosity and heat conductivity the values given by Davis (ref. 3) are
employed, which also for air are in good agreement (deviation, at most
2.7 percent) with the chosen values. A knowledge of the specific heat
is unnecessary if the Pr value for 00 C is satisfactory; the values
are 0.717 for hydrogen and 0.731 for oxygen13.

(c) Gas constants for high pressures. From the tests of Petavel,
which extend up to 160 atmospheres, only those up to 40 atmospheres
were computed in order that reliable gas constants would be available.
In this pressure range and up to 10000 C, the computation was conducted


10Wien-Harms: Handb. d. Exp. Phys., IV, 4, p. 531.

llLandolt-Bornstein: 5th ed., vol. II, p. 1304.

12
1A. Busemann in Wien-Harms: Handb. d. Exp. Phys., IV, 1, p. 359.
This relation may be obtained from an equation given by Eucken (Physik
Z., vol. 14 (1913), p. 324.

13A. Busemann in Wien-Harms: Handb. d. Exp. Phys., IV, 1, p. 362.






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with the normal values since the dependence on the pressure of X and
p and the deviations of p from the ideal gas law are either not known
at all or are only partly known, and the known deviations amount to only
a few percent14.


4. Individual Investigations15

(a) General. The nondimensional representation of the results is
effected according to equation (3) by logarithmic graphs with Nu as
a function of Gr and with Te as the only parameter, since Pr
(according to sec. 3a) may be considered as constant. Even if the
slight variation of Pr there indicated is considered as real, there
would, according to a test result of Davis (ref. 3) on the effect of
the Pr number, be obtained only displacements of the curves of the
order of magnitude of 1/100 of those which can actually be observed.

In table I are summarized the test objects and test conditions of
the individual investigators and the range of Te used. In order to
judge to what extent the condition of infinite extension of the fluid
was satisfied, the ratio of the height H of the surrounding space to
the cylinder diameter d is shown. In figures 1 to 7, approximately
equal Te values are indicated by the same symbols. The decrease in
Nu with increasing Te is particularly evident in the individual
point groups (because of the same wire), whereas the values for different
wires naturally show larger scattering as compared with one another.
The curves Te = constant were obtained by a graphical adjustment pro-
cess and, similarly, the extrapolation points and curves for Te = 0
and the dependence of Nu on Te (relative decrease in Nu for
ATe = 1). The indicated minimum and maximum values hold for the indi-
vidual wires (point groups); the indicated mean values, for the individ-
ual investigators.

(b) Results of Ayrton and Kilgour (ref. 1), figure 1. The authors
of reference 1 give in the form of diagrams for each wire the total heat-
transfer coefficient as a function of the wire temperature; two wires
(0.0206 and 0.0282 cm diam.) because of the very large scattering and a
third wire (0.0102 cm diam.) because of its strong deviation in position
and inclination were excluded from the evaluation of the tests. The


14Landolt-B6rnstein: 5th ed., Eg. I, p. 64 ff, Eg. IIa, p. 144.

15Acknowledgement is made to Dr. K. Winkler, Leipzig, for his
assistance with the computations.








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points computed from the low temperatures of 400 to 800 C were simi-
larly not used on account of the large scattering.

(c) Results of Langmuir (ref. 8), figure 2. Langmuir gives in a
table the adjusted values of the total energy losses as a function of
the absolute wire temperature increasing in stages of 2000 C. Only
the tests up to 13000 K were computed herein in order to have reliable
values available for v and I. The convergence of the curves
Te = 1.33 and Te = 3.35 shows that the dependence of Nu on Te
decreases with increasing Gr, the decrease independently determined
for the individual wires being:

log Gr 0.1-4 0.8-4 0.6-3 0.5-2 0.4-1
Decrease, percent 7.2 7.5 5.4 3.2 2.2

This experimental confirmation of the results following from theore-
tical considerations (sec. 6b) acquires significance in that the tests
of Langmuir are characterized by a large change of the Te number and
smallest scattering. The extrapolation from Te = 1 to Te = O, on
account of the absence of test points for this value, was effected with
the mean value of 13.2 percent taken from reference 7.

Langmuir (ref. 9) also discusses his own tests at various air
pressures (10 to 760 mm Hg) and variable room temperature (t. = -1900 C
to 6000 C) which on account of the strong change in the expansion coef-
ficient (P = 1/T,) would be of great interest and the only tests of
their kind. Unfortunately, these tests have not as yet been published,
and the test data are not available in a form in which they may be
evaluatedl.

(d) Results of Bijlevelt (ref. 10), figure 3. Since the tests of
Bijievelt were primarily for the purpose of investigating forced con-
vection, the experimental setup was designed for the sensitivity re-
quirements of this flow. This explains the greater scatter of the test
points of reference 10 as compared with those of other investigators.
For each individual wire (with the exception of the Ta and Ni wire),
a grouping is nevertheless observed with respect to Te in the usual
sense.

(e) Results of Kennelly, Wright, and Bijlevelt (ref. 11), fig-
ure 4. The investigation of reference 11 is noteworthy in that through
the pressure (p,v) changes, even for constant wire diameter and constant
surrounding temperature, a change is effected in the Gr number so that


16For these data the author is indebted to a personal communication
from Dr. I. Langmuir.







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curves of Te = constant are here obtained from tests on a single
wire. The nondimensional representation of these tests (fig. 4) has
been given previously (ref. 1). The authors give for each wire a dia-
gram with the heating current as a function of the pressure for three
or four different temperature differences, as parameter of which the
values for five different pressures were computed in terms of the
characteristics and were connected by the curves Te = constant. The
extrapolation to Te = 0 shows extremely good agreement for both wires.
The values of the wire of medium diameter (d = 0.02616 cm) were not
plotted because of the irregularities obtained.

(f) Results of Petavel (ref. 12), figure 5. From the over-all
heat-transfer coefficients given for constant pressure as a function
of the temperature difference, values were computed for 2000, 4000,
6000, 8000, and 10000 C temperature differences up to pressures of
40 atmospheres. For the sake of clarity, however, only the curves for
the extreme Te values are plotted in figure 5. Since the heat con-
ductivity and density of hydrogen differ from the values of air by the
factors 6 and 14, respectively, these tests would have been particu-
larly suitable for checking the similarity law. Unfortunately, however,
these tests are evidently unreliable because of the far too small
jacket pipe (l/d = 18.6 as the smallest value of all investigators).
The similarity law for the three gases is confirmed only for Te = 0.69
and Gr > 102. In the case of air and oxygen, there is a splitting
with respect to the dependence on Te, but in the reverse sense from
that observed for all wires of the other investigators. In the case
of hydrogen, a dependence on Te in the correct sense is observable
only at the smallest values of Gr. Finally, the Nu values also for
small Gr lie too high as compared with the values of all other in-
vestigators and the rise in the Nu with increasing Gr has too great
a lag. For these reasons, the results of Petavel for determining the
heat-transfer law were not used.

(g) Results of Wamsler (ref. 13), figure 6. Wamsler's curve,
which was obtained on the basis of correctly computed radiation loss
(sec. 2), shows a dependence on Te such that the mean values for
Te = 0.60 lie about 5 percent higher than those for Te = 0.20. This
small splitting of the effect of Te which occurs in the opposite
sense to that in the case of small Gr, however, in all probability
does not contradict the facts but may be explained by the consideration
that the radiation coefficients in the entire temperature range from
500 to 2700 C were assumed constant, whereas in general for metals there
has been established an increase of the radiation coefficient with the
temperature. If in the previously mentioned temperature range a rise
in the radiation coefficients of about 5 percent is assumed, as follows
for cast iron from Wamsler's determination of the radiation coefficient
and for -wrought iron from the determination of Nusselt (ref. 14), the
observed splitting of the effect of Te is neutralized and the curve
shown, valid for all Te values, is obtained. The test values of the








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copper pipe and of a wrought iron pipe (of 5.9-cm diam.) were not em-
ployed in figure 6 because of the obvious irregularities indicated by
the scattering.

(h) Results of Koch (ref. 15), figure 7. The nondimensional
characteristics were redetermined herein with the values X and v
for wall temperature, since the characteristics given by Koch were com-
puted for a mean temperature of the substances. The test results show
an unusually small scattering which is to be ascribed a careful account-
ing for various factors (e.g., determination of the temperature of the
air in the room and at the walls, and also the end effect of the pipes).
For the two intermediate pipes, after graphical adjustment, a decrease
of Nu for Te = 0.07 to 0.52 by about 2.5 percent is observed, where-
as in the case of the smallest pipe no ordering is observed and in the
case of the largest pipe there is a partial decrease and a partial
scattering. In agreement with the results of Wamsler, the dependence
on Te is thus practically zero also for large Gr (104 to 107).


5. Summary of Test Results

The results of the different investigators as regards their agree-
ment may be compared in two groups. In the region of small Gr
(10-4 to 10), the tests of Ayrton and Kilgour, Langmuir, Bijlevelt, and
Kennelly and coworkers are in agreement. From these investigations are
determined, on the one hand, the curves Te = 0 for the limiting law
of small temperature difference (fig. 8) and, on the other hand, the
curves Te = 0.65 corresponding to an intermediate test range (fig. 9).
For Te = 0 the maximum scatter of the results of the four-investigators
is 10.5 percent, the minimum 5 percent, the corresponding values for
Te = 0.65 being 22 and 8.5 percent, respectively. The middle curves
of figures 8 and 9 obtained for Te = 0 and Te = 0.65 give for this
change in Te an average change in Nu of 14.7 percent (fig. 10).
A relative decrease in the Nu number by 22 percent with increase of
Te from 0 to i is thus obtained as the experimental mean value of the
effect of Te for small Gr.

In the region of large Gr (l04 to 107), the test values of
Wamsler and Koch from energy measurements and the values obtained from
the temperature field measurements of Jodlbauer (ref. 16) by integrat-
ing over the cylinder perimeter are shown in figure 10. As follows
from the discussion of the results of Wamsler and Koch, the values of
Koch must be considered as more reliable. There is nevertheless an
uncertainty in the values due to the uncertainty in the radiation com-
ponents which lie between 40 and 60 percent. On the other hand, the
temperature field measurements of Jodlbauer are free from this source
of error. Since, moreover, the results of reference 16 are in good







NACA TM 1366


agreement (mean deviation, 4 percent) with the theoretical solution
given in Part II they were assigned a greater weight in the evaluation.
Jodlbauer himself, however, considers his energy measurements, which
are about 10 percent higher and lie near the values of Koch, to be
correct and his field measurements in error because of the introduction
of the thermocouple. The test points of Wamsler are, on the average,
18 percent higher than the theoretical values, and those of Koch,
11 percent higher. The interpolation curve is not everywhere satis-
factory as regards its slope and curvature and could be determined only
with a partial deviation from the test values.

The differences still existing, after extensive adjustment, among
the individual investigators and the difficulty of interpolation show
that the heat-transfer curve is still not conclusively determined with
the desirable accuracy of several percent, but that, on the contrary,
further tests are required. The given interpolation curve nevertheless
represents the most probable curve of the heat-transfer law, obtainable
at the present time, for diatomic gases with account taken of the de-
pendence on Te in the range of Gr from 10-4 to 107. The numerical
values of this curve are given in table II.


6. Qualitative Theoretical Interpretation of Heat-Transfer Law

(a) Streamlines. It is necessary to explain theoretically the
dependence of the Nu number on Gr and Te, determined experimentally
in sections 4 and 5 (fig. 10), and also the dependence on Pr as ob-
tained by A. H. Davis (ref. 3) for liquids. A solution of the differ-
ential equations under the assumption of a thin (as compared with the
cylinder diam.) heated laminar layer is given in Part II for diatomic
gases (Pr = 0.74) and small temperature differences (i.e., approximately
constant properties, Te 0) and is valid for mean values of Gr of
about 104 to 3x108. On the basis of these theoretical results and with
the aid of suitable assumptions, the dependence of Nu on small and
large Gr and also on Pr and Te will be discussed theoretically
without any necessary implications of finality in the conclusions.

The streamline plot (fig. 11) obtained from the solution and the
isotherms of figure 12 show that cold air streams from the bottom and
sides of the cylinder with increasing velocity in laminar flow upwards
along its surface, is deflected away from the surface in the region of
the upper stagnation point, and forms a rising current of warm air.
With increasing Gr of the cylinder, the thickness of the streaming
layer relative to the cylinder diameter decreases. Since the velocities
increase at a higher rate, however, the Reynolds number Re of the
boundary layer increases so that the boundary layer finally becomes
turbulent. According to the schlieren photographs of figure 23, this
occurs in the neighborhood of the upper stagnation point at Gr = 3.5x108.







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With further increasing Gr, the point of turbulence transition travels
upstream (fig. 26) and reaches the equatorial region at Gr = 3x109.
With decreasing Gr of the cylinder, the Re of the boundary layer
likewise decreases, and at the same time the thickness of the layer in
relation to the cylinder diameter increases so that the given solution
of the differential equations finally (below Gr = 104) becomes in-
valid. At very small Gr (very small Re), the isothermal picture of
the free flow in the neighborhood of the body becomes increasingly
similar to the concentric isothermal picture of the pure heat con-
ductionl7 so that almost static heat-conduction relations may.here be
supposed. An estimate of the Te effect made under this assumption
at small Gr likewise gives good agreement with the measurements (see
following section b). The following fundamental difference is never-
theless to be observed. For initially given cylinder temperature and
heat transfer, the solution of the heat-conduction equation has as the
potential function at infinity a singular point (negatively infinite
temperature). That is, even at a very large distance from the cylinder
the room temperature continues to decrease, whereas actually in the
case of free flow the room temperature at large distance from the body
very soon becomes constant.

(b) Heat transfer. Dependence on Gr. For the dependence of Nu
on Gr, the solution of the differential equations under the familiar
assumptions of the Prandtl boundary-layer theory gives

Nu Grl/4 (6)

as is also approximately shown by the tests of Wamsler and Koch (fig. 10).
At very large Gr with at least partial turbulent flow, the heat trans-
fer must rise more strongly on account of the increased mixing. With
increasing turbulent mixing the dependence of the heat-transfer coef-
ficient on the position must also decrease. The assumption that a is
independent of the position (or on the cylinder diam.), as is obtained
from the tests of Griffiths and Davis (ref. 17) for a vertical plate,
gives on the basis of dimensional considerations

Nu Grl/3 (7)

Above Gr = 3.5x108, the proportionality of the Nu number to the
fourth root of Gr corresponding to the upstream travel of the point
of transition to turbulence must gradually become a proportionality to
the cube root. Experiments of W. King (ref. 18) on vertical cylinders,


17See photographs by R. B. Kennard; Bur. Stand. J. Res., vol. 8
(1932), p. 787. Report on this by M. Jakob: Forschg. Ing.-Wes., vol. 4
(1933), p. 45.






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plates, and blocks at Gr up to 1012 confirm the cube-root law as
already pointed out by M. Jakob and W. Like (ref. 19). For very small
Gr, for which the amount of heat transferred by convection is always
below that transferred by conduction, the Nu number in accordance
with its definition must gradually become constant, as may be seen from
the discussion of data below Gr = 10-3 (fig. 10).

Dependence on Pr. The dependence of Nu on Pr for
Gr = constant and Te = constant may be understood by assuming a
change in X with all the remaining magnitudes kept constant. The
increase in Pr, through a decrease in X, gives in the region of
predominant heat conduction (very small Gr) a proportional decrease
in the quantity of heat transferred, that is, Nu = constant. In the
region of large Gr, a decrease in )X (on account of the smaller con-
duction component as compared with the convection component) gives a
relatively smaller decrease in a, that is, an increase in Nu. In
agreement, investigations by Davis refss. 3 and 20) on liquids of very
different Pr (Pr = 0.74 to 7940) for small Gr give the limiting
value of Nu which is independent of Gr and also of Pr. For medium
and large values of Gr, an increase of Nu with Pr was obtained.

Dependence on Te. The dependence of Nu on Te for
Gr = constant and Pr = constant for diatomic gases is determined by
the temperature dependence on X, v, a, and also by the temperature
chosen in computing the characteristics Gr and Nu (Pr is independ-
ent of the temperature) for the fluid properties X and v (0 = 1/T.
is constant in the entire field, ref. 1); the wall temperature tw or
T, was chosen herein.

An increase of Te = (Tw/T,)-l is considered to be due to a de-
crease in T. with Tw constant so that the values of X and v
employed for the computation of Nu and Gr remain unchanged. The
associated increase of p and 0 is compensated by a decrease in the
gravity field g so that gp8, that is, the lift acceleration and Gr
remain constant. Hence colder outer layers with smaller X and v
but larger p now take part in the heat transfer. In the region of
very small Gr with predominating heat conduction X, the magnitude
a, and hence Nu, must therefore decrease with increasing Te. This
is also shown by the data discussed previously between Gr = 10-4 and
10 (see fig. 10). For the region of very large Gr with smaller vis-
cosity and heat-conduction effect as compared with the turbulent momen-
tum and heat mixing, it also follows18 from the momentum equation,


18Under the reasonable assumption that with the change of Te the
profiles of the velocity and temperature to a first approximation under-
go affine variations. The 'standard velocity W' may thus be taken for
example as the maximum value of the tangential velocity at the equa-
torial region of the cylinder.






NACA TM 1366


because of the constancy of the lift acceleration, that the standard
velocity W is constant. The participation of colder layers with in-
creased value of p in the heat transfer here means an increase in the
heat-transfer coefficient a, which is proportional to cpW, and there-
fore in Nu.

The decrease of Nu with increasing Te for very small Gr
contrasts, therefore, with an increase for very large Gr. The effect
of Te for small Gr values must therefore, in agreement with the
test results of Langmuir, first decrease with increasing Gr values
and finally, somewhere in the range of medium Gr, must practically
vanish, as is evidently the case according to the experiments between
Gr = 104 and 107 (fig. 10).

For the range of very small Gr, the order of magnitude of the
dependence of Nu on Te under the approximating assumption of purely
static heat conduction may be estimated when a mean conductivity is
used,


m 1 T T X(T)dT (8)


the following expression is obtained for the Nu numbers corresponding
to two different Te values Tel and Te2 for equal wall temperature
T = Tw2

Nu(Tel) X(Tel)
Nu(Te2) = m(Te2) (9)

The value Tel = 0 is chosen, that is, vanishing temperature difference
or T1w = T,; and Te2 = 1, that is, Tw2/T.2 = 2. For the temperature
dependence of X, the previously (ref. 1) given exponential form is
used9

X/O0 = (T/To)n (10)

with n = 0.73820


190 is the value of X for the arbitrarily chosen temperature
TO.

2Determined from the previously given exponent for a of 1.738
neglecting the temperature dependence on c.







NACA TM 1366


From
4

Xm(Te2) = T 1 fW2 X dT
m T 2 T T

2

with TO = T = T2 in equation (10) and by integration:


X(Twl) Tw2 T-, T. n +
km(Te2) = n+ T,' (I)


Since Xm(Tel) = X(Tw ), the following expression results after sub-
stitution of the numerical values:

m(Te2) 2 J- 1 7
1= T = 0.81 (1)
n(Tel) = 1.738 G
The decrease of Nu for Gr = constant with increase from Te = 0
to Te = 1 is thus theoretically obtained as 19 percent. The pre-
viously discussed tests of the four investigators between Gr = 10-4
and 10 gave values between 13 percent and 26 percent with a mean
value of 22 percent (see fig. 5). This good agreement justifies the
assumption of approximately static conducting conditions at small Gr.


II. THEORETICAL SOLUTION OF THE BOUNDARY EQUATIONS FOR THE

HORIZONTAL CYLINDER (CASE OF STEADY MOTION)

1. Abstract

The large number of experimental investigations of heat transfer
in free convection correspond to only a single physically satisfactory
theoretical solution of the differential equations, namely the solution
for the vertical plate (two-dimensional steady-flow case) for moderate
temperature differences, which was given by Schmidt and Beckmann
(ref. 21) with the help of Pohlhausen. The solution is based on the
approximation of the Prandtl boundary-layer theory laminarr flow in a
layer which is thin as compared with the distance from the lower edge;
velocity normal to the wall small as compared with that in the direction
of the principal flow along the wall). The results of the theory as
regards the velocity and temperature fields and therefore the heat
transfer agree very well with the corresponding measurements.








NACA TM 1566


For the case of the horizontal cylinder, schlieren photographs in
air (see ref. 22 and figs. 23 and 25) show that, for Grashof character-
istics (Gr = d3gpe/v2) of about 104 and above, the heating of the air
extends to only a relatively thin layer around the cylinder, a fact
which permits the mathematical simplification of the problem. On the
assumption that the heat-transfer and the flow processes are restricted
to a thin film, as compared with the cylinder diameter, with laminar
flow (boundary-layer assumption) the differential equations (ref. 23)
for moderate temperature differences can be solved approximately for
the velocity and temperature fields; and therefore the heat transfer
can be computed which, in the range of Gr 104 to 35108, within which
the initial assumptions are satisfied, is in good agreement with
experiment.


2. Setting up of Differential Equations

To start, the hydrodynamic differential equations are used in a
form previously given (ref. 1)21 which shows a "lift term", charac-
terizing the free convection, that arises from the combining of the
gravity term and the hydrostatic component of the pressure drop, so
that only the gradient of the dynamic pressure p* remains as the
pressure force. This dynamic pressure is equal to zero where there
is no motion and no temperature difference with respect to the medium
at a large distance. is was indicated in detail in reference 1 (p.
429), the density and other characteristics of the substance in the
entire field may be considered as constant (case of small temperature
difference) and the equations will continue to describe the free-
convection problem. The equations in vector form are then22

w grad w = 1/p grad p* v rot rot w gp8.

div w = 0 (12)

w grad 0 = aA e

with the boundary conditions: w = 0, 0 =9 on the cylinder surface; and
w = 0, 8 = 0, p* = 0 at infinity. (w, velocity; 0, temperature differ-
ence with respect to temperature at infinity; p*, dynamic pressure;
p, density; v, cinematic viscosity; P, expansion coefficient; a, temper-
ature conductivity; g, gravity acceleration vector; g, its absolute
value; r, cylinder radius.)


21In equation (3c) in reference 1 a minus sign mistakenly appears
on the right side.

22In place of Aw the invariant form is written grad div w -
rot rot w.








16 NACA TM 1366


For treating the two-dimensional flow about the cylinder, these
equations are conveniently written in arc length coordinates with the
arc length s on the cylinder perimeter (taken in clockwise direction
with the lower stagnation point as origin) and the normal distance n
from the cylinder surface, and the velocity components u and v
parallel and normal to the wall, respectively. There is then obtained,
without neglecting any terms23:

r ou ou u*v 1 op* r s
u + V 6- + r +---- = + gp sin +
r + n rs /n +n sr n r
1.1 6.1/6 1.6


f2u 1 ou
Sn2 r + n n
62 1/62 1/6


r 6v ov u2
----- u -- + v _- -
r + n +s 1n r + n
1. 6. 1 1


1 6v r2 62v
+ n (r + n)2 s2
r + n an (r" + n)2 ;82


r2
(r + n)2


u 2r
(r + n)2 + (r + n)2


6v
3s


2 -


1 bp* s
J gpo cos +
r



v 2r
(r + n)2 (r + n)2


uls
3s


2 1


r u v+ v
-+n + r+
r + n -s n r + n
1 1 6


r o0 a0 f 9
--- u + v = a -
r + n 7s 6n 2
1.1 S.1/6 E2 1 E62


1 0
+ +
r + n Tn
1/e


The approximations made according to the boundary-layer theory are now
introduced. The cylinder radius r and hence also s (with the ex-
ception of the lower stagnation point) and u are assumed to be of the
order of magnitude 1; n is assumed to be of the order of magnitude of


23The hydrodynamic equations agree, except for the dynamic pres-
sure and lift terms, with the equations (10), (11), and (12) for
r = constant of W. Tollmien: in Wien-Harms, Handb. d. Exp. Phys., IV,
1, p. 248.


62V
VaIn2
62 1/6


(13)


r2
(r + n)2


1Ye

3s2
1







NACA TM 1366


the boundary-layer thickness s <<1, so that n + r may be re-
placed by r. The estimate of the order of magnitude of the individual
terms according to the method given by Prandtl and his coworkers then
gives the results written underneath the individual terms. In order
that the friction and inertia forces may be of equal effect, it is
necessary that v (and correspondingly a) be of the order of magnitude
of C2. In order that the lift forces for the motion in the direction
of the main stream be of significance as compared with the inertia and
friction forces, that is, in order that 'free convection' exist at all,
it is necessary that gpAe 1. From the two momentum equations it
then follows that the pressure terms are at most. 1. Since p* is
zero at the outer edge of the boundary layer, it follows from

1 )p*


by integration from the outer edge of the boundary layer to the cylinder
surface over the length e that I*/p in the boundary layer can at
most be of the order of magnitude 6, that is, so that 1/p p 6/os = 6.

Since in what follows only the terms of the order 1 are retained,
the following result, important for simplifying the mathematical treat-
ment, is obtained: The tangential drop of the dynamic pressure is to
be neglected as compared with the tangential lift, inertia, and friction
forces. The term p* alone is contained in the equation of the normal
momentum. Hence, u, v, and 0 are to be computed from the equation of
the tangential momentum, the continuity equation, and the heat-transfer
equation (eq. (14)). Thereafter p* may, if desired, be computed from
equation (15) of the normal momentum. Hence, the boundary-layer equa-
tions for the horizontal cylinder for free convection are
2
)u Ou --u .+ S
u s + v = t + ge sin
s n2 r

6u 6v
+ = 0 (14)


u + v w- = a ,-
as on 2
oan


u 1 3p* (15)
r = -5T- g- cos r







18 NACA TM 1366


A corresponding estimate of equations (13), first rendered non-
dimensional, gives the two corresponding conditions:

1 2 Gr 1
-- and --- 1, i.e. r (b)
Re Re2 4

For a maximum e, in the sense of the approximations, of 0.1 (approxi-
mation to about 10 percent), this gives Re = 100 and Gr = 104 as
lower limits of the computations. The approximation is therefore better
the larger Gr, that is, the smaller the boundary-layer thickness e.
The upper limit is then attained if the boundary layer becomes turbu-
lent, which (according to part IV) is the case for Gr = 3.5xl08 at
the upper stagnation point, for Gr = 3x109 at the equator. The range
of applicability of the laminar-boundary-layer computations therefore
lies between Gr = 104 and Gr = 109.


3. Reduction to Ordinary Differential Equations

The usual introduction of the stream function

U V (16)
on 3s

eliminates the continuity equation. By introducing nondimensional
magnitudes with Gr' = r3gg3/v2 by means of the transformation24:

x= n Gr1l/4 = Gr'-1/4 (17)
r r V g

there remain of the five constants (v, gp, a; 0, r) of differential
equations (14) only two in the nondimensional combination of the Prandtl
characteristic Pr = v/a.
y 82y 32 3
S- + sin x
dydx dy d y2 y3
(18)
BYi J Yt br 1 o22
by bx bx by Pr dy2

Equation (17) is not an arbitrary transformation which carries
equation (14) over into equation (18), but a definitely determined
transformation with the aid of which it is possible to eliminate Gr


24Gr' referred to r is more convenient for the computation than
Gr referred to d. In the final result, the more usual Gr is again
introduced.







NACA TM 1366


so that it no longer occurs in equation (18). The solution of equa-
tion (18) therefore likewise no longer contains Gr. It is therefore
unnecessary to know the solution of equation (18) if it is of interest
to know only how the solution (e.g., velocity, temperature, boundary-
layer thickness, heat-transfer coefficient) of the initial equation (14)
depends on the Gr number. This dependence of Gr is already com-
pletely represented by the transformation (17), whereas equation (18)
contains the further dependence on Pr and on the space coordinates.
In other words, the solution of equation (18), obtained for a definite
Pr value24a gives, by means of equation (17), at the same time. the
velocity and temperature fields for all Gr values. On the other hand,
the transformation (17) without the solution of equation (18) already
gives important general information on the flow condition and the heat
transfer for free convection with respect to the dependence on Gr.
From equation (17) there follows, if the following expression is set up,

U (19)

that

n yGrl-1/4 = U Gr'1/2 vr = V Gr1l/2 r = Gr1,/4
(20)

That is, the relative boundary-layer thicknesses decrease with Gr-1/4
the nondimensional velocities increase with Grl/2; and the non-
dimensional temperature drop increases with Grl/4, from which there
follows directly the well-known 1/4-power law of heat transfer in free
convection:

Nu m Grl/4 (21)

These theoretical results agree with those for the rectangular
plate and are not restricted to the latter and the horizontal cylinder
alone, but depend on more general assumptions (ref. 19). They are
valid, in general, for all two-dimensional cases of free flow with a-
boundary-layer character (ref. 23), that is, where the thickness of the
heated laminar layer is small compared with the distance from the inci-
dence edge and the radius of curvature of the wall, and the latter does
not change discontinuously in the flow direction. The existence of
sharp edges about which the flow occurs is therefore excluded. In this
more general case, there are on the right side of equation (13) addi-
tional terms which are determined by the radius of curvature and which


24alt is here assumed as self evident from physical considerations
that one and one only such solution exists. On the mathematical side
of these existence and uniqueness proofs, see the discussion in refer-
ence 1 (p. 428).







NACA TM 1366


vary with the arc length (or/6s) but which drop out in the estimate
made under the initial assumptions. Furthermore, in place of the sine
and cosine of the lift term there appear functions which determine the
direction of the surface element with respect to the horizontal and
vertical but which similarly do not change anything in the transfor-
mations (17) and the conclusions from equations (20) and (21) based on
these transformations.

In agreement with these theoretical conclusions equation (21) has
already been confirmed for vertical plates (ref. 21), horizontal cyl-
inders25, and cylindrical layers (ref. 24), and even for the heat
transfer from the two sides of a square horizontal thick plate with
edges (ref. 25) and for the heat transfer with evaporation at a verti-
cal cylinder in water and carbon tetrachloride (ref. 26) for which the
previously mentioned assumptions are not all satisfied.

A reduction of the partial differential system (18) to systems of
ordinary differential equations is attempted by setting

q = y g(x) (22a)

Y(x,y) = p(q)f(x) (22b)

:(x,y) = t(q) (22c)

for example,

U = = f(x)g(x)p'(q) (22d)

based on the notion that all the profiles of the stream function |(y),
tangential velocity U(y), and temperature T(y) for the different
azimuths x of the cylinder are produce by affine distortions by
means of the azimuth functions f(x) and g(x) from a single "base
profile" of the stream function p(q) or the velocity p'(q) and one
base profile of the temperature t(q) (q = normal coordinate of the
base profile). Substitution of equation (22) in equation (18) gives
(primes of p and t denote differentiation with respect to q; primes
of f and g denote differentiation with respect to x):

p,2(f2gg, + ff'g2) pp"ff'g2 = p"'fg3 + t sin x
/Prgt" + f't'(23)
1/Pr gt" + f'pt' = 0


25See Part I, section 5, figure 10.







NACA TM 1366


A separation into functions of the two independent variables x
and q, that is, a successful application of the assumed expression
(eq. (22)) for solving the equations, is possible only if the two func-
tions f(x) and g(x) can be uniquely determined from the four ordinary
differential equations

f'(x) = ag(x) (24a)

f(x)f'(x)g2(x) = c sin x (24b)

f2(x)g(x)g'(x) = b sin x (24c)

f(x)g3(x) = d sin x (24d)

(where a,b,c,d are initially undetermined constants) in spite of the
redundancy of the equations26. The possibility of such unique determi-
nation, which is the necessary assumption for the applicability of
equation (22), will be discussed in more detail after equation (29) is
considered. If such determination is possible, there remain for p(q)
and t(q) two ordinary differential equations:

(b + c)p'2(q) cp(q)p"(q) = dp"'(q) + t(q)
(25)
t"(q) + a Pr p(q)t'(q) = 0 j

In section 4, equation (24) will be solved and in section 5 equation (25)
will be discussed.


4. Determination of the Azimuth Functions F(x) and G(x)

(a) Setting up of the two differential equations for' F(x). The
boundary values of f(x) and g(x) which are required for solving equa-
tion (24) are determined from physical considerations. From
=- dt(q) x
-=y dq x)

it follows that

g(x = 0) = go o 0 (26a)


26The author is indebted to Dr. A. J[aumann, Leipzig, for several
mathematical suggestions.







NACA TM 1366


Otherwise, the temperature of the cylinder surface would also be the
temperature in the field along the entire normal at the lower stagna-
tion point, which cannot be the case. From equation (24a) there is
then obtained

f'(x = 0) = agg (26b)

Since the tangential velocity at the lower stagnation point must, for
reasons of symmetry, be different, there fellows from equation (22d):

f(x = 0) = 0 (26c)

The system of equation (24) with the boundary conditions (26a), (26b),
and (26c) thus contains five free available constants a,b,c,d, and go,
which are determined in subsequent computation.

By introducing the normed functions27:

F(x) = f(x) G(x) g(x) (27)
ago g0

equation (24) may be transformed to:

F' = G (28a)

F2 G G' = a sin x (28b)
a2g04

F F' G2 = sin x (28c)
a2g04

F G3 = sin x (28d)
ago4

with the boundary conditions

F(x = 0) = 0 F'(x = 0) = 1 G(x = 0) = 1 (29)

A necessary assumption for the applicability of the expressions of
equation (22) for solving the equations is that the two functions F
and G can be uniquely determined from the four equations (28a) to
(28d). This would be the case if it could be proven that the four


27By normed is meant fully determined also with respect to a
factor.







NACA TM 1366


equations are not independent of one another, but that two equations
follow from the other two. This proof is partly possible only inso-
far as one equation (for suitable choice of the constants) is a con-
sequence of the remaining equations. Thus, elimination of G with
the aid of equation (28a) gives

F2 F' F" = b sin x (30a)
a2g04

F F'3 = C sin x (30b)
a2g04
a9
/
F F'3 d sin x (30c)
ago4

Equation (30c) is thus a consequence of equation (30b) in the case that
the following first condition for the constants is satisfied:

d = c/a (31)

Setting for briefness

c/a2g04 = a (32a)

b/a2g04 = ( (32b)

there then remain for the determination of F the two equations

F F' = a sin x (331)

F2 F' F" = 0 sin x (3311)

with freely available values of a and P. As a necessary assumption
for the applicability of equation (22), there remains the further con-
dition that the two equations I and II possess two equal (or at least
approximately equal) sDlutions.

This proof will be obtained by solving equations I and II by a
power-series development above and below the equator x = 4/2. For
this purpose, with p = x n/2, I and II are transformed into

F(q)F'3(p) = a cos p (341)

F2(p)F'(()F"(g) = P cos 9 (3411)

where the primes now denote differentiation with respect to 9.







NACA TM 1366


(b) Solution of differential equation (I) for F. The value of
a, by the boundary values of equation (29), is first determined as


S= 1


(32c)


as is seen from the first term of the power-series development of
equation (331) about the point x = 0. From the expression for solving
equation (341):


FI()) = a0(1 + Cl ( + c2 92 + .)


there follows


FI ( = 0) = ao


F '(p = 0) = a0 c1


(36a)


and by substituting in equation (341)


a04 c13 = a = 1


(36b)


Substitution of expression (35) in differential equation (341)
gives the unknown coefficients, which are taken up to the fifth, in-
clusive, and which can be successively determined in terms of Cl:


1
c2 = c1


1 5
c3 = 18 cic8


1 5 1
c5 = 6 c 13 +
5 3_601 324 1] 2


= 1 2 5 .4
S(37)
3
. 1L


The boundary condition FI(P = -r/2) = 0 according to equation (29)
gives for the value of cl the solution of an algebraic equation of
the fifth degree which is solved by trial


CI = 0.581


(38)


Of the five possible real roots it is necessary to choose
which no further zero of FI is obtained between cp = 0
The coefficients then assume, according to equation (37),


c = + 0.581


c2 = 0.05626


that one for
and 9 = -n/2.
the values


c3 = 0.01412


c4 = 0.00165


(39)


C5 = 0.00066


According to equations (36a), and (36b), there is obtained


FI'( = 0) = a0 c1 = 0.874


(35)


FI(9 = 0) = ao = 1.504


(40)







NACA TM 1366


and the numerical solution of FI(cp) is thereby completely given (see
table III).

It may be remarked27a that for equation (331) a solution was also
obtained by a power-series expansion at the point x = 0 In powers of
sin (x/2) which, because of the fact that the curvature of F at that
point is zero, converges with particular rapidity:

F*(x) = 2 sin + 1 sin2 + sin4 + (4
2 10 2 100 2

This solution assumes at the equator the values

F*(x = r/2) = 1.500 F*'(x = 4/2) = 0.874 (e)

which are in good agreement with the values of equation (40) of the
series expansion at the equator. In this way, the sufficient conver-
gence of FI(() with the five computed terms is assured, and a good
check is obtained against computation errors. The solution F* will
not be used in what follows; this is because of the singular behavior
of equation (3311) at the point x = 0, and the identity proof must
therefore be carried out by the series expansion at the equator.

(c) Solution of differential equation II for F. For equa-
tion (3411), corresponding to equation (35), the following expression
is assumed:

FI(,) = b0(1 + dP + d2 + .) (42)

Substitution in the differential equation gives the following values
for the constants: bO and di are arbitrary

d2= 1
2 2b04dI

d d = 2 (d2d + d22) (43)
13 3 1

where, on account of the length of the expressions, d4 and d5 are
not given as functions of di and d2. The function FII thus depends


27aAcknowledgment is made to Prof. W. Muller, Aachen, for the
suggestion that the solution of equation (331) may be reduced to two
quadratures by separation of the variables. One of the quadratures is
not in closed form, however, but can be carried out only by a series
expansion so that this method would not be essentially different from
the one employed.








NACA TM 1366


on the as yet freely available values of bo, dl, and B. The solu-
tions F and FII are equal if the corresponding coefficients of the
series are equal. Therefore, the as yet undetermined coefficients are
equated as follows:


bo = a0


di = cl


(44)


The condition d2 = c2 gives

-L.= -I 2
2bo4d = 6

or with equations (36b) and (44)

n = 1/3 a = 1/3

All the coefficients may now be expressed in terms of dl:


(f)


(45)


d3= 5 dl3


d5 7 1 + L 5
540 di 216


d = d2 d 4
4 =+ -21 ~T2_ 1


(46)


Comparison of the coefficients of the two series for
gives


b0 ao = 0


di cI = 0


d2 c2 0


F and FII


1I
d3 c3 1 '1
Id ~ T 3=7Bc1


d41 2- 4
d4 c4 = -1T


S-1 1 3
d C5 = 60 1 + 405


This result states that the solutions of the two different differen-
tial equations (341) and (34II) agree at the equator in the value of the
function, in the tangent, and in the curvature, but not in the higher
derivatives in which they differ fundamentally to a small degree. FI
and FII are approximately, but not rigorously, equal. The system (24)
is, strictly speaking, overdetermined and has only approximately unique
solution. This solution of expression (22) is not rigorously, but
only approximately, valid. Because of the agreement of F- and FI
at the equator up to and including the curvature, however, these differ-
ences become more appreciable in the neighborhood of the lower and
upper stagnation points where the initially assumed boundary-layer
assumptions are not satisfied. The method that has here been employed
thus gives an approximate solution of the boundary-layer equations with-
in the accuracy limits determined by the physical approximations of the
boundary-layer assumptions.


S 12
d2 = 6 1d


(47)






NACA TM 1366 27


The numerical computation of equations (44) and (46) with a0
and cl according to equations (40) and (39) gives:

bo = 1.504 d1 = 0.581 d2 = 0.05626 d3 = + 0.01816

d4 = -0.00321 d5 = + 0.00144 (48)

Table III shows the values of the functions FI(q) and FII(p) computed
according to equations (39), (40), and (48), with sufficient agreement
between x = 30, 1500 and even at 1650 (graphically interpolated for
F I). The negative value of FII for x = 0 has no significance
since, on account of the singular behavior of equation (3411) for
F = 0, the series (42) beyond this singular point no longer represents
the solution of the differential equation. For the further equations,
FI is employed because only this function possesses the required
property of vanishing at the lower stagnation point.

(d) Determination of C. The term G is determined according
to equation (28d) where, according to equation (49), d = ag04.
Table III shows the computed values based on F .


5. Determination of Basic Profiles p(q) and t(q)

The constants b, c, d which occur in equation (25) can now, with
the aid of equations (31), (32a), to (32c), and (45) be expressed in
terms of a and go:

b = 1/5 a2 g04 c = a2 go4 d = a g04 (49)

Equation (25) then becomes

2/3 a2 0g4 p'2 a2 g04 p p = a gO4 p"' + t
(50)
t" + a Pr p t' = 0

with the boundary conditions:

q = 0: p = 0 p' = 0 t =1

q = : p' = 0 t = 0 (g)

Equation (50) for a = 1 and go = 1 may now be numerically integrated.
The following values were chosen for the as yet free remaining constants
a and gg







NACA TM 1366


a = 3 g = 3-1/4 (51)

equation (50) becomes

p"' + 3 p p" 2 p2+t = (52)
(52)
t" + 3 Pr p t' = 0

This system with the same boundary conditions and for Pr = 0.733
(diatomic gases) has already previously been numerically integrated by
E. Pohlhausen for the vertical plate so that, because of the introduc-
tion of the normed functions F and G and the consequent availability
of the constants, a new numerical integration is unnecessary. The
stream function p(q) and temperature function t(q) correspond to the
functions t(() and 0e() of Pohlhausen. Their values are given in
reference 21 in tabular form (ref. 4, p. 187). Graphical representa-
tions of the velocity profile P'(q) and of the temperature profile
t(q) are given in figures 13 to 16 (continuous curves).


6. Complete Solution for Velocity and Temperature Fields

With the values of equations (51) for a and go, the azimuth
functions f(x) and g(x) according to equation (27) are now uniquely
determined. For

a g = 33/4 = 2.280 g = 3-1/4 = 0.760 (53)

there is obtained

f(x) = 2.280 F(x) g(x) = 0.760 G(x) (54)

Table IV shows the values of f(x), g(x), and f(x)-g(x) computed from
FI and G of table III. Figure 17 shows these values graphically.

With the values f(x) and g(x) from table IV, the values p(q)
and t(q) from the table of Pohlhausen and the transformations (16),
(17), and (22) there is obtained the complete solution of the boundary-
layer equations (14) for the velocity (u and v) and temperature
fields (8) obtained without any empirical value. Collecting results
yields (see also eq. (i ) and (20) for simultaneous passage from Gr'
to Gr (Gr = 8Gr'):







NACA TM 1366 29


u Gr_/2
u(s,n) = F 2 f(x)g(x) p'(q)



v(sn) = r l4ff) ) + ()g'(x) P) (55a-
v'sn =' 77 f '(x)p(q) + l ^q 5(q))


(sn) Gr1/4
O(s,n) = @t(q) (srn) = -- g(x)t'(q)


s n Gr1/4'
with x = and q = --7 g(x)
r r

For the determination of the normal velocity v it is required to
know f'(x) and g'(x). The term f'(x) is, according to equation (24),
determined by means of g(x); g' (x) was determined by graphical differ-
entiation of g(x)28. Here the qualitative description of the normal
velocity obtained from the streamline picture is sufficient. It can be
shown that the boundary conditions of u, v, e for q = 0 are cor-
rectly assumed. If the assumption n to the solution of equation (14) is applied for very large q, there
are obtained for u and 0 the correct value zero; but for v, in
contradiction to the boundary conditions (12), a finite value is ob-
tained which is related to the fact that the cylinder curvature was
neglected.

The boundary-layer thickness, tangential velocity, temperature,
and heat-transfer coefficients according to equation (55) behave as
follows. With regard to the dependence of these magnitudes on Gr,
there hold first of all the considerations of section 3 in connection
with equations (20) and (21). The dependence on the cylinder azimuth
is such that all profiles of the tangential velocity and temperature
are obtained by affine distortions from the basic profiles p'(q) and
t(q).

The value l/g(x) (fig. 17) represents the 'extension' of the tem-
perature and velocity profiles normal to the surface with increasing
azimuth, that is, the development of any characteristic distance from
the wall, for example, of the place of maximum velocity (q = 0.95) or
of the boundary-layer thickness to be defined later (q = 2.18). In
accordance with this, the boundary-layer thickness at the lower stag-
nation point possesses a finite approximately constant value over a


28The values are not given here.






NACA TM 1366


large azimuth range but increases at first gradually then more and more
rapidly and attains at the upper stagnation point a theoretically in-
finitely large value (upward current of warm air). The function g(x),
since it is proportional to the temperature drop, likewise represents
the variation of the local heat-transfer coefficient with the cylinder
azimuth. At the lower stagnation point, g(x) is constant and with
constant'slcpe connects with the values of the other side of the cyl-
inder. At the upper stagnation point (x = i) this is no longer the
case; g(x) has a sharp peak which is associated with the fact that the
boundary-layer assumptions at this point strongly deviate from the
actual conditions (normal velocity large as compared with the tangen-
tial velocity). Under actual conditions this peak is balanced out by
a very sharp minimum value as is shown by the value of Nu along the
cylinder perimeter computed from the schlieren pictures of E. Schmidt
(ref. 22) No quantitative comparison of this experimental curve
of g(x) with the theoretical curve given herein can be given because
the conditions of the two-dimensional problem are so little satisfied
for the schlieren pictures (pipe length = 2 times cylinder diam.) that
the computed heat-transfer coefficient is 40 percent greater than that
of the tests of Koch (ref. 22).

The development of the tangential velocity with the azimuth, that
is, the increase of any characterizing velocity, for example the max-
imum velocity, is described by f(x)- g(x) (see fig. 17). The velocity
increases from the value zero at the lower stagnation point (up to 600
approximately linearly) to a maximum at about 1280 and, on approaching
the upper stagnation point (convergence of the flow on either side)
rapidly decreases to zero. This physically correct behavior of the
solution at the upper stagnation point is not introduced as a boundary
condition but is obtained as a necessary consequence of the theory.

An over-all picture of the velocity and temperature fields is
given by the streamlines and isotherms (figs. 11 and 12). They are
computed from the equations

(x,n/r) = f(x)Grl1/4 p [1 Gr,1/4 g(x)] = const

O(x,n/r) n Gr'1/4 g(x) = cnst (55e,f)


for various values of the constants30. The scale of the representation


2See figure 13 for maximum Gr = 16xl06.

30The streamline picture for azimuth intervals of 150 and in addi-
tion for 50, 100, 1700, 1750; the isotherm picture for azimuth intervals
of 300 and in addition for 165, 1700, 1750.








NACA TM 1366


holds for Gr = 104; for Gr = 106 the distances from the cylinder
surface should be reduced by a factor of 3.16 and for Gr = 108, by 10.
At the lower and equatorial neighborhood of the cylinder up to an azi-
muth of about 1050, the flow near the wall is directed toward the sur-
face, while above it is directed away from the surface upwards. In
this respect the theoretical solution for the cylinder differs funda-
mentally from that for the vertical plate, which gives only a flow
toward the plate. The isotherms at the lower part form almost con-
centric circles. At the upper stagnation point (upward warm current)
the isotherms theoretically go toward infinity, whereas actually (corre-
sponding to the gradual dissolution and spreading out of the warm up-
streaming air) they close at a relatively large distance over the
cylinder. The isotherm 0.53 gives at the same time the place of the
maximum tangential velocity.


7. Heat-Transfer Law

From the expression for the quantity of heat dQ passing from
an element of area df,

dQ = ]) df (56)
dI = 0 )-

the value of the temperature gradient at the wall obtained from
equation (55c)

a (n = 0) = Gr'l/4 g(x)t'0 (57)

(t'0 = t'(q = 0) = -0.508), and the defining equation for locallyy
variable heat-transfer coefficient a(x)

a(x) = d (58)

there is obtained
a()r =g(x)(-to') Cr'1/4 (59)


From this there is obtained for the dependence of the nondimensional
local heat-transfer coefficient on the cylinder azimuth x and the
Gr number (Grd = 8Gr')30a:


30aSince several formulas of this section will be used later in
Part III, and in order to avoid misunderstandings, indices (e.g., d,h,H)
for the characteristics Gr and Nu are written which indicate the
lengths with which these magnitudes are formed.







NACA TM 1366


a -x = Nud(x) = 0.604 g(x) Grdl/4 (60)


see figure 18.

To compute the mean heat-transfer coefficient averaged over the
cylinder perimeter,


am = (x) cx (61)


there is required the mean value

1 JA
g = g(x) dx (62)


which, by planimetering g(x) (fig. 17), is found to be

g = 0.616 (63)

It may also be computed by the following relation from equation (24a)
with a = 3 according to equation (51)

3 it g = f(it) (64)

which gives g = 0.620. However, the value 0.616 is preferable because
this value takes into account the total curve g(x) and does not depend
on the end point of the series expansion f(x). From equation (60) the
following is then obtained for the mean heat-transfer coefficient as a
function of Gr for free flow at a horizontal cylinder

am- = Nud = 0.372 Grdl/4 (65)

A comparison of this theoretical heat-transfer coefficient for diatomic
gases (Pr = 0.74) with test results is given in Part I, section 5; see
also figure 10.


8. Comparison of Theoretical Velocity and Temperature

Fields with Available Measurements

The comparison of the computed velocity and temperature fields is
carried out with the measurements of Jodlbauer (ref. 16) which are the







NACA TM 1366


only ones thus far available3. Of the fields measured for six differ-
ent Gr values, the following ones were computed:

2r = 9 cm, tw = 99.2, t = 18.1, Gr = 3.76x106 (figs. 13, 14)
2r = 5 cm, tw = 104.6, t = 18.10, Gr = 6.54x105 (figs. 15, 16)

The best agreement with theory is to be expected for the maximum Gr.
On the basis of the theory it would be desirable to have measurements
for the largest possible laminar Gr, that is (see Part IV) values of
about 5.5x10 which could be realized in air at 200 C at a surface
temperature of 100 C with cylinders of 42 centimeters diameter.

The comparison with the theory was carried out in such manner that
all velocity profiles u(s,n) and temperature profiles e(s,n) corre-
sponding to a value of Gr were, with the aid of equation (55), re-
computed for the theoretical basic profiles p'(q) and t(q) in the
representation of which they must all coincide.

In the case of the velocity profiles (figs. 13 and 14), there is
a regular deviation in that the measured velocities are greater than
the computed ones, the deviation from the theory in the maximum velocity
amounting on the average to 22 percent for the smaller and 17 percent
for the larger value of Gr. A part of the deviations occurring for
large distances from the wall beyond the maximum value should be
ascribed to the uncertainties in the difficult measurements of such
small air velocities, as can be seen from the following discrepancies.
The measured velocities at the azimuth 600 for Gr = 5.76x106 lie
considerably above the theoretical curve, whereas for Gr = 6.54x105
they lie on the theoretical curve. On the other hand, the velocities
measured at x = 30:' for Gr = 6.54x105 show strong deviations upward,
whereas for Gr = 3.76x10l they are in agreement with the-theory. The
measured distance from the wall of the maximum value of the velocity is
everywhere in very good agreement with the theory.

The measured temperature profiles show very good agreement with
the theory even for the azimuth of 1650. A small regular deviation is
noted for all azimuths and Gr numbers for medium distances from the
wall. The small deviations at small distance from the wall can be
ascribed partly to the fact that the condition for the isothermal sur-
face was not quite satisfied in the tests. The measured temperature


31
1Acknowledgment is made to Prof. E. Schmidt and Dr. K. Jodelbauer
for their aid in providing the data.







NACA TM 1366


drops at the wall for Gr = 6.54xi05 are on the average 3 percent
smaller than the values given by the theory, while for Gr = 3.76X06
they are 1 percent higher32.

It is likewise of interest to compare the solution based on the
boundary-layer theory for the vertical plate with the corresponding
test results. Reference 21 (figs. 20 to 23) also shows the velocity
fields to have stronger deviations than the temperature fields. For
the small plate with smaller deviations, the measured velocities also
lie somewhat higher than the theoretical values; but for the larger
plate with larger deviations the contrary is true. For the temperature
fields, particularly for the large plate, no regular deviation can be
established between theory and experiment. The deviations occurring
for the plate are clearly smaller in comparison with those for the
cylinder. This is to be ascribed to the fact that in the case of the
plate only the boundary-layer terms of the order 2 << 1 (rectilinear
flow) were neglected, whereas for the case of the cylinder there were,
in addition, neglected terms of the order el << 1 that were related
to the curvature.


III. HYDRODYNAMIC AND THERMAL COCMPARISCOI BETWEEN VERTICAL

PLATE AND HORIZONTAL CYLINDER FOR FREE CONVECTION

AND FOR PLATE IN PARALLEL FLOW

1. Abstract

Now that the theoretical boundary-layer solutions for the vertical
plate and the horizontal cylinder for free convection have been obtained,
it is of advantage to compare these two standard bodies of two-
dimensional free flow with regard to the shape of the stream and the
heat transfer. Similarly, a comparison of the boundary-layer develop-
ment between the vertical plate for free convection and for the plate
in parallel flow (Blasius solution of the bourdary;-layer equations) is
of interest. The following features are characteristic for the three
types of flow.



32The fact that the values of the heat-transfer coefficient ob-
tained from the field measurements of J.dlbsuer through integration of
the temperature gradients at the wall over the yliinder perimeter lie,
on the average, 4 percent higher than the thecretical values (see
fig. 10) is due to the deviation between the theoretical and the actual
conditions in the neighborhood of the upper stagnation point.







NACA TM 1366


The boundary-layer thickness of the plate in parallel flow in-
creases with the square root of the distance from the incidence edge;
that of the vertical plate in free convection increases, however, as
the fourth root of the height above the lower edge (cf. eqs. (69) and
(71)). Both start with the thickness zero, which for the plate in
parallel flow results in infinitely large velocity gradients, and for
the free plate results in an infinitely large temperature drop and an
infinitely large local heat-transfer coefficient at the lower edge.
In the case of the cylinder, on the contrary, the boundary layer starts
at the lower stagnation point of the horizontal cylinder with a finite
thickness and thereafter with finite velocity and temperature gradients
and a finite local heat-transfer coefficient (fig. 17). It increases
according to a complex law (l/g(x), see eqs. (70) and (82)) and at the
upper stagnation point reaches a theoretically infinite thickness, with
vanishing velocity and temperature drops normal to the surface (upward
stream of warm air) and with a vanishing heat-transfer coefficient.

In spite of these fundamental differences it is possible to set
up a number of relations between the three cases. These show on the
one hand a close hydrodynamic and thermal kinship which is due to the
boundary-layer character of the differential equations underlying the
theoretical solutions. On the other hand, they give practical view-
points with regard to the application of rectangular plates or hori-
zontal cylinders for the heat transfer. For this purpose it is first
of all necessary to represent the Reynolds numbers of the characteristic
length, of the boundary-layer thickness, and of the nondimensional local
and average heat-transfer coefficients as functions of the Grashof
numbers of the plate and cylinder, which in turn requires the introduc-
tion of a boundary-layer thickness (flow discharge thickness) for the
velocity profile of the free convection. A knowledge of the dependence
of Re of the boundary-layer thickness on Gr of the plate, as well
as of the azimuth and Gr of the cylinder, is in addition required
for the later evaluation of the tests as regards the occurrence of
turbulence (Part IV) for free convection at the plate and the cylinder.


2. Boundary-Layer Thickness for Free Convection

It is first necessary to determine the magnitude to be associated
with the boundary-layer thickness. For the velocity profile u(y) of
the plate in parallel flow (U = maximum velocity), Prandtl and his
coworkers, as is known, introduced the displacement thickness 8*,
defined as
r-

5 U = (U = u) dy (66)


which is a measure of the decrease in the flow discharge as a result
of the friction.







NACA TM 1366


For the velocity profile of the free convection u(y), a flow
discharge thickness 6 is introduced which is hydrodynamically equiv-
alent to the displacement thickness, being like the latter character-
istic of the development of the friction layer, which in this case
takes up the entire flow:

SU = u dy (67)


or expressed in words: Through the flow discharge thickness 6 would
flow the same fluid mass with the maximum velocity as actually flows
with the total stream.

In the nondimensional velocity profile p'(q) (q = ncrndimensional
distance from the wall, p = stream function) which occurs in the theo-
retical solution, the discharge flow thickness is denoted by q., the
maximum velocity by p',. From equation (67) there is then obtained:


qmax p (q)dq = p(-) p(0)

and with the numerical values:

p' = 0.275
max

p(m) = 0.6033

p(0) = 0

the value of qc is obtained as

qg = 2.18 (68)

For orientation purposes34, it is assumed that the velocity maximum
lies at q = 0.95 and the point of inflection of the profile at
q = 1.85.

With the values of q_ for the plate and cylinder from equa-
tions (72) and (82), respectively, there is obtained for the vertical
plate, where h is the distance from the lower edge:


33Extrapolation from last computed value p(6.0) = 0.5928.


34See continuous curves in figures 13 and 15.







NACA TM 1366


5 21/2 r/4 = 2.18
h 2


and for the horizontal cylinder of diameter d at azimuth x


Sg(x)21/4 Crl/4 = 2.18
.g =


(70)


3. Nondimensional Representation of Theory for Vertical Plate

(a) Reynolds numbers. From the theoretical solution for the
vertical plate given by Schmidt and Pohlhausen (ref. 21) the following
expression is obtained according to their equations (23), (24), and
(31)35, since c4 = Grh/4h3, for the velocity u and the distance
from the wall y:


u = p' (q) Grhl/2


S= qif, Grh-1/4


For the flow discharge thickness y = 8 and the maximum velocity
u = U, there is then obtained


U 2v Grhl/2


U -8 pm 1/4
V 9v2 Pmax q5 Grh


h = q4,/" Grh-1/4



U-h = 2 Pax Grh1/2
V


and using the abbreviations Rh = U-h/V and R= = U-6/v
tuting the numerical values for pma and -q


(72)



(73)


and substi-


Rg = 1.695 Grh1/4

Rh = 0.550 Grhl/2


(74)

(75)


Elimination of Grh from the last two equations gives the following
expression as the relation between Re of the flow thickness and the
height of the plate for a vertical plate in free flow:


35The distance from the lower edge is denoted by h instead of x,
the nondimensional distance from the wall 4 by q, the nondimensional
stream function C by p.


(69)


(71)







NACA TM 1366


RB = 2.29 Rh1/2 (76)

(b) Nusselt characteristics. For the local heat-transfer coef-
ficient a(h) at the height h above the lower edge, the following
equation holds (ref. 21)*:

a(h) = X(- to) h) 1/ (77)

(tO = -0.508 is the value of the derivative of the nondimensional
temperature function t(q) for q = O) or, in nondimensional form, the
numerical values may be substituted:

a(h).h = 0.359 Crhl/4 (78)

The variation of the heat-transfer coefficient along the variable
height h of a plate of total height H is most conveniently obtained
by introducing a local heat-transfer coefficient which has been made
nondimensional by dividing through by H
a.(H = Nu(h) = 0.359 Grll/4 )-1/4 (79)


The heat-transfer coefficient (fig. 18) at the lower edge of the plate
with zero boundary-layer thickness is theoretically infinitely large
and then continuously drops with the reciprocal of the fourth root of
the height.

The mean heat-transfer coefficient corresponding to a plate of
height H
a 1 = ft a(h)dh (j)


is obtained from equation (77) as

.4 L (-to0) 1 (80)
a,=: 3 2 t)

or, nondimensionalized and with numerical values substituted (fig. 18),
as
=H =0.479 r1/4
X = NuH = 0.479 GrH1/4 (81)


*Extrapolation.







NACA TM 1366


4. Nondimensional Representation of Theory for Horizontal Cylinder

(a) Reynolds numbers. The theoretical solution given previously
for the tangential velocity u (eq. (55a)) and the distance n from
the wall (eq. (55d)) for the azimuth x of a horizontal cylinder,
with the values n = 8 and the maximum velocity u = U, may be trans-
formed to

U = f(x)g(x)pa Grl/2 6 = q Grr-1/4 (82)


For the upper stagnation point x = t, because g(t) = 0 and
f(n) = 5.84, the boundary-layer thickness becomes infinite and the
tangential velocity zero. The quantity flowing through U6 and there-
fore the Re of the boundary layer U8/v, however, remain finite. To
introduce Re for the characteristic length results in no simplication
for the cylinder because, in addition to the characteristic length, the
cylinder radius occurs as an additional characteristic length of the
system. From equation (82),

U&/v = f(x)Pax q Grrl/4 (83)

Substituting the numerical values and using Grd result in the follow-
ing expression for Re of the boundary layer at azimuth x:

Rg(x) = 0.357 f(x) Ord1/4 (84)

and for Re of the boundary layer at the upper stagnation point
(with f(t) = 5.84):

R5 = 2.08 Grdl/4 (85)

where R8 is written for R8(r). Although the theoretical solution
for x = i is no longer valid, equation (85) because of the small
change of f(x) gives the correct relations in the region of the upper
stagnation point (up to about 1650), so that for the sake of simplic-
ity the formula will be used in the following discussion.

(b) Nusselt characteristics. Figure 18 shows the variation of
the nondimensional local heat-transfer coefficient according to equa-
tion (60) over the developed semicircumference of the cylinder. In
the region of the lower stagnation point, it has a finite, practically
constant value; and at the upper stagnation point, it drops rapidly to
zero in accordance with the upstreaming warm air at that point. In the
same figure the mean heat-transfer coefficient Nud for a cylinder
is plotted according to equation (65).







NACA TM 1366


5. Hydrodynamic Comparison

(a) Between vertical plate and horizontal cylinder in free con-
vection. Although the boundary-layer growth for a vertical plate and
a horizontal cylinder, as already remarked (Part III, sec. 1) and as
seen from equations (72) and (82), differ fundamentally in character,
it is nevertheless of interest to compare the flow condition (always
characterized in what follows by Rg) at the upper edge of the plate
and in the region of the upper stagnation point of the cylinder. The
following three questions will here be considered:

1. The behavior of the flow at the upper edge of the plate and at
the upper stagnation point of the cylinder for the case that the height
of the plate is equal to the cylinder diameter.

2. The behavior of the flow at the upper edge of the plate and at
the upper stagnation point of the cylinder for the case that the height
of the plate is equal to the developed semicircumference of the cyl-
inder, that is, the characteristic lengths are equal.

3. The relation of the height of a plate whose flow condition at
the upper edge is equal to that of the upper stagnation point of a
cylinder to the diameter of the cylinder.

Before each of these questions is considered, the ratio of the
boundary-layer Re numbers at the upper stagnation point of the cyl-
inder RB(Z) and at the upper edge of the plate RB(P) are written,
with the aid of equations (74) and (85):

R5(Z)/R5(P) = 1.226 Grd1/4/Grhl/4 (86)

where R5(Z) and Grd refer to the cylinder, and R%(P) and Grh
refer to the plate. The coefficient 1.226 has the following signifi-
cance (cf. eqs. (85) and (83) for the cylinder, eqs. (74) and (73) for
the plate):

1.22 2.08 2-3/4pmax-'q8f() f() (8
1.226 = 16--5= ax = (87)
1.695 23/2p'. q 421/4
2 Pmax 4.21/4

1. For equal Gr of plate and cylinder, Grd = Grh, equation (86)
gives


R5(Z) = 1.226.R5(P)


(88)








NACA TM 1366


that is, for equal Cr of plate and cylinder, for example, for a plate
of the height of the cylinder diameter h = d under otherwise equal
conditions (with regard to temperatures and materials), Re at the
upper stagnation point of the cylinder is 22 percent greater than that
of the upper edge of the plate (fig. 19). This result states nothing
about the hydrodynamic relation between the two, since the comparison
refers to different characteristic lengths. It is useful only for the
rapid comparison of plate and cylinder for the usual values of Gr.
It is then known that for equal Gr the cylinder possesses the more
developed flow, so that the flow at the cylinder may be turbulent while
that of the plate is still laminar.

2. For equal characteristic length h = A/2d there is obtained
from equation (86)

Rg(Z)/Rg(P) = 1.226 (2/t)3/4 = 0.874 (89)

The boundary-layer Re at the upper stagnation point of a cylinder is
thus 13 percent smaller than that of a plate of the height of the
developed semicircunf'erence of the cylinder (fig. 19). This retar-
dation of the boundary-layer development as compared with the vertical
plate is due to the fact that the boundary layer at the lower stagnation
point of the cylinder already has a finite thickness, whereas in the
case of the plate the thickness must increase from zero.

3. For equal boundary-layer Re values for the plate and cylinder,
equation (86) gives

Grd/Grh = 1.226-4 = 0.441 (90)

At the upper stagnation point of the cylinder, there is therefore the
same flow condition as at the upper edge of the plate if Gr of the
cylinder is 0.14 times that of the plate. This permits a rapid con-
version of the flow data for the plate into those for the cylinder and
vice versa. If, for example, it is known from heat-transfer experiments
on a plate that the departure from the laminar Gr1/4 law occurs at
(Grh),r = Sxl18, then it can be concluded without experiment from equa-
tion (90) that this must be the case for the horizontal cylinder for
(Grd)kr = 3.5x0I-. If the plate and cylinder are under otherwise equal
conditions, it is possible from equation (90) to derive a simple rela-
tion for the height of a plate ha which is hydrodynamically equivalent
to a cylinder:


d3:ha3 = 0.441 hg:d = 1.311


(91)








NACA TM 1366


In the neighborhood of the upper stagnation point of a horizontal cyl-
inder, the same flow condition prevails (Rg) as under otherwise equal
conditions at a vertical plate of height 1.31 times that of the cylin-
der diameter (fig. 19). As was to be concluded from the answers to
questions 1 and 2, this height must lie between the cylinder diameter
and the developed semicircumference.

The answer to the three questions on the flow condition at a
cylinder and at plates of various heights is shoum in figure 19 (Rg
of the cylinder set equal to i).

(b) Hydirodynamic comparison between a vertical plate in free con-
vection and a plate in a parallel flow. The relation (76) between Re
of the boundary-layer thickness and that of the height of the plate
(characteristic length) Rh for the vertical plate in free convection
is strongly analogous to the corresponding relation (92) between Re
of the boundary-layer thickness RB*(referred to the displacement
thickness 8*) and the characteristic length Rx for a plate in a
stream parallel to its plane (Blasius solution of the boundary-layer
equations):

R = 2.29 Rhl/2 (76)II R* = 1.73 Rxl/2 (92)


In spite of the already mentioned entirely different rate of growth of
the boundary-layer thickness, in the case of the parallel flow as the
square root of the distance from the incident edge ani in the case of
the free convection as the fourth root of distance from the lower edge,
the development of the Reynolds number of the boundary-layer thickness
as a function of the Reynolds number of the characteristic length in
the case of these two very different types of flow follows the same
exponential law and with approximately equal magnitude.


6. Thermal Comparison between Vertical Plate and Horizontal Cylinder

(a) Mean heat-transfer coefficients. For equal Gr of plate and
cylinder (GrH = Grd), there is obtained from equations (81) and (65) the
following ratio of the nondrimensional heat-transfer coefficients of the
horizontal cylinder Nud and of the vertical plate INuH

NuJdNuH = 0.777 (93)

Under otherwise equal conditions, therefore, the mean heit-transfer
co-fficienL of a cylinder is only 78 percent of that of a plate of the







NACA TM 1366


height of the cylinder diameter (fig. 18). In agreement with this, the
comparison given by M. Jakob and W. Linke (ref. 19) of convection tests
on vertical plates and horizontal cylinders in a single Nu-Gr diagram
(with R and d denoting the length dimension) gives a lower position
of the cylinder tests. The modified determination of the critical num-
ber, which follows from these results, will be discussed later (Part IV,
sec. 5).

For equal mean heat-transfer coefficient am of plate and cylinder
under otherwise equal conditions, there is obtained from equations (81)
and (65)

0.479 H-1/4 = 0.372 d-1/4 H/d = 2.76 (94)

The mean heat-transfer coefficient of a cylinder is therefore, under
otherwise equal conditions, equal to that of a plate of height 2.76
times that of the cylinder diameter. On account of the small decrease
of the local heat-transfer coefficient at large plate heights, this
factor of 2.76 is not very sharply determined. Thus Jodlbauer (ref. 29)
found it possible, without considering the previously given theoretical
solution for the cylinder, to represent the mean heat-transfer coef-
ficients for cylinders from tests by Koch by the formulas for the ver-
tical plate if H is replaced by 2d (instead of theoretically by
2.76d). As a matter of fact, the am values thus determined for the
cylinder lie only 8.3 percent above the values given by the theory
according to equation (65), as can be seen by replacing H by 2d in
equation (81).

(b) Total heat transfer. The total heat transfer Q(Z) of a
cylinder of diameter d along the entire circumference and the total
heat transfer Q(P) along the two sides of a plate of the height H,
under otherwise equal conditions, are in the ratio

Q(z) am(Z)id = Nud(Z)
Q(P am(P)2H 2 NuH(P)

and from equations (81) and (65),

Q(Z)/Q(P) = 1.221 Grdl/4/GrHl/4 (95)

where the significance of the factor 1.221 is as follows (cf. eqs. (81),
(80), (65), and (59)):

S0.372 21/4g(-to') 3_ g )
1.221 = (96)
2 0.479 24 f 421/4
5 2 (-t







NACA TM 1366


A comparison of this factor 1.221 with the factor 1.226 occurring in
the ratio of the boundary-layer Re of cylinder and plate (eq. (86))
according to equation (87) shows that both are identical, since accord-
ing to the theoretical solution, the following relation holds as an
immediate consequence of the validity of the heat-transfer equation:

3i5g = f(:K) (97)

The small difference of 1/2 percent is explained by the fact that (as
mentioned in connection with eq. (64)) g was computed not according
to equation (97) but was obtained by planimetering.

For equal Grd and GrH values in equations (86) and (95), the
following is thus obtained:

Q(Z)/Q(P) = R8(Z)/RB(P) (98)

that is, the total heat transfer of a horizontal cylinder and that of
the two sides of a vertical plate under otherwise equal conditions are
in the same ratio as the boundary-layer Re values at the upper stag-
nation point of the cylinder and at the upper edge of the plate,
respectively.

On the basis of relation (98) it is now possible, without further
computation, to answer the three questions on the total heat transfer
of a cylinder and plates of different height, analogous to the questions
referring to the hydrodynamic comparison (see fig. 19; the heat transfer
of the cylinder has been set equal to 1).

Corresponding to equation (88),

Q(Z) = 1.22 Q(P) (99)

The total heat transfer of a cylinder is thus 22 percent greater than
that for the two sides of a plate of a height equal to the cylinder
diameter under otherwise equal conditions (fig. 18).

Corresponding to equation (89),

Q(Z) = 0.87 Q(P) (100)

The total heat transfer of a cylinder is 13 percent smaller than that
of the two sides of a plate of height equal to the developed semi-
circumference of the cylinder (equal characteristic length) under other-
wise equal conditions.







NACA TM 1366


Corresponding to equation (91) for

Q(Z) = Q(P)

there is obtained the height of a plate Hg thermally equivalent to
a cylinder:

Eg : d = 1.31 (101)

The total heat transfer of a cylinder under otherwise equal conditions
is therefore as great as for the two sides of a vertical plate of
height 1.51 times that of the cylinder diameter. In summary it may
therefore be stated that a horizontal cylinder and a vertical plate,
which transfers heat on both its sides, of a height equal 1.31 times
the cylinder diameter are equivalent both thermally (with reference to
the total heat transfer) and hydrodynamically (with reference to the
boundary-layer development at the upper stagnation point and at the
upper edge, respectively).


IV. THE OCCURRENCE OF TURBULENCE IN FREE FLOW ABOUT A HORIZONTAL

CYLINDER AND ALONG A VERTICAL PLATE

1. Introductory Observations

For the determination i:f the transition from laminar to turbulent
flow in the case of free convection, tests were carried out on a verti-
cal plate and a horizontal cylinder of sufficient size. Two main
reasons underlay the investigation. In the first place, it was desired
to determine the upper limit of the validity of the laminar boundary-
layer theories for plate and cylinder and therefore the upper limit of
applicability of the heat-transfer formulas developed from these theo-
ries for practical application. The second reason was of a more
theoretical nature. The numerous turbulence investigations, both ex-
perimental and theoretical, undertaken in recent decades have been
concerned almost exclusively either with the flow between two parallel
walls with linear velocity distribution (Couette flow) or with the flow
in pipes, channels, about rotating cylinders, along plates, cylinders,
or other resistance bodies which have in common a velocity profile which
rises uniformly from the value zero at the wall up to a maximum value.
Differing essentially from these profiles are evidently velocity pro-
files for which the velocity rises from the value zero at the wall to
a maximum value and then, after passing through a point of inflection,
again drops to the value zero at a large distance from the wall. Such
profiles are typical in the case of heat transfer by free convection.







NACA TM 1366


Some further observations on the setting up of turbulence in free con-
vection will be made later in connection with the discussion of results
(sec. 5).


2. Test Procedure

For the investigation, the schlieren method developed b,- E. Schmidt
was applied, which as an optical method provides the possibility of an
instantaneous view of the entire field, and is therefore of advantage
for determining the critical number.

The schlieren method has been described in detail by E. Schmidt
(ref. 22) so that only the essential points will be given here. It is
based on the deflection suffered by a light ray in passing through a
field with density stratification (i.e., variation of the index of
refraction) normal to its direction of propagation toward the colder
layer. In the test, a parallel light beam is allowed to pass tangen-
tially along the surface of the heated body. The rays in the neighbor-
hood of the wall, because of the maximum temperature drop at that place,
are most strongly deflected from their initial direction; those farther
away from the wall are less deflected, while those rays which are out-
side the temperature field undergo no deflection. On a screen which
is set up at a sufficient distance behind the heated body, the rays
near the wall appear the farthest from the wall. The following picture
is obtained: the heated body throws a completely dark shadow which, as
compared with that of the cold body, is increased by the thickness of
the temperature boundary layer. This shadow is bounded by a first
"interior caustic curve" adjacent to which is formed a medium-bright
region (rays which have passed through the temperature field), which in
turn is bounded toward the outside by a second "external caustic curve"
originating from the tangential rays in the neighborhood of the heated
surface. Still farther toward the outside is the uniformly bright field
of the undisturbed illumination.

For an accurate evaluation, the method is suitable only for bodies
having one of its length dimensions (two-dimensional problem), so that
the effect of the temperature fields at the ends on the path of the
light ray is small compared with the deflection undergone in passing
through the distance along the length. On the other hand, the length
should be chosen only large enough so that the light ray at the end of
the body still is approximately at the same distance from the wall as
at the initial point of the body.







NACA TM 1366


3. Test Setup and Procedure36

(a) Plate. The plate was 100 by 100 centimeters and 1.0 centi-
meter thick. It consisted of two zinc sheet plates 1 millimeter thick
which by interposing a frame of 8-millimeter U-brass were soldered onto
each other. In the interspace there was placed a heating coil, insu-
lated by asbestos boards, which consisted of 36 meters of chrome-nickel
wire of 1-millimeter diameter with a total resistance of 51 ohms. In
air of room temperature, the plate could be kept at 1000 C surface tem-
perature with 5.6 amperes. The plate was suspended on two bicycle
wheel spokes which were attached to the upper strip of the frame.
Through the latter the current was also conducted to the interior. For
measuring the surface temperature a silver-constantan thermocouple was
used which was flatly soldered on about 5 centimeters from the upper
rim of the plate and the wires of which extended a few more centimeters
quite close to the plate. This simple temperature measurement could be
applied because extreme accuracy was not required. Figure 20 shows
the plate suspended with the current-supplying wires, and the thermo-
couple and two pendulums each with two spheres which served to determine
the vertical and made possible the mutual comparison of the different
photographs.

(b) Cylinder. The cylinder consisted of two layers of 0.7-
millimeter-thick zinc sheet which was rolled over two bicycle wheel
rims and soldered to their upper sheathing strip. Between the sheets
there was placed the heating coil insulated by asbestos board, the coil
being of 54 meters of chrome-nickel wire of 1-millimeter diameter with
a total resistance cf 77 ohms, which brought the cylinder in air of
room temperature to 1000 C surface temperature with a current of 4.1
amperes. The cylinder was suspended on two bicycle wheel spokes. The
current leads and thermocouple were similarly located on the upper
sheathing strip. The mean length of the cylinder was 100.2 centimeters,
and its mean outer diameter 58.45 centimeters (fluctuating between
58.35 in the horizontal and 58.75 over the soldered seam). The frontal
surfaces were closed off with asbestos board. Figure 21 shows a view
with the suspension, current leads, and the two pendulums.

(c) Setup. Since it is very difficult to produce a parallel
bundle of rays of the diameter of the body here employed by means of
lenses, such a beam was replaced by the light from a strongly screened
(as a rule of 3-mm aperture) arc lamp 32 meters from the heated body.
The distance between the body and the screen was 8 meters. The appa-
ratus was set up on the largest floor, of 42-meter length, of the


36Acknowledgment is made to Herr Dr. R. Weise and Dr. H. Kurzweg,
Leipzig. For the test setup, acknowledgment is made to Master Mechanic
C. Hentsch, Leipzig.








NACA TM 1366


Institute. Because of the window in one of the .alls and the connec-
tion to the staircase halls, it was not possible to exclude entirely
currents of the room air so that only at high temperature differences
(above 500 C) could useful schlieren photographs be obtained. The
existing dissymmetry of the critical points on the two sides of the
bodies, both for the plate and cylinder (see table V) was due to a
unilateral convection flow from the window wall. The horizontal
adjustment of the test bodies was effected by means of a water balance;
the adjustment parallel to the light cone, by measuring the shadow of
the cold body.

(d) Photography37. The schlieren pictures were obtained by two
methods. In one method the shadow pictures were taken on the screen
by means of a lens and camera (13x18 cm, Zeiss-Tessar 1:2.7). In spite
of highly sensitive plates (Agfa-Superpan 16/1C DIm), exposure times
of 5 to 25 seconds were required on account of the low brightness, be-
cause of the strong screening and large distance from the arc lamp.
For determining a time mean value of the fluctuating critical points,
these time pictures were, however, particularly well suited. The
photograph was taken either somewhat from the side (plate) or, in order
to avoid distortions, exactly central (partly in the case of the cyl-
inder); the camera was set up in the shadow of the cylinder so that the
legs of the tripod threw an additional shadow (fig. 23). The second
method consisted of a direct illumination of one or several photoplates
simultaneously (Agfa-Isochron 13/10 DIN, 10x15 cm), which were brought
to the position of the screen. The greater brightness obtained in this
way made possible instantaneous photographs (through light flashes from
the arc lamp), which in connection with their greater scale pro.'ided
a view in instantaneous detail of the transition f'r:m laminar to turbu-
lent flow.


4. Evaluation of the Schlieren Pictures

Figures 22 and 23 show time exposures of the plate and cylinder
taken by means of lens and camera. In the center the magnified shadow
of the body by the thickness of the temperature boundary layer, the
"inner caustic curve" surrounding it, and further toward the outside
the "outer caustic curve" arising from the rays near the wall. The
dissymmetry of the lower edge of the plate is due to the disturbances
of the room air that were mentioned previously. The upper edge of the
plate is immediately below the frame of the picture in the "necking"
of the central shadow. In the upper stagnation point of the cylinder
are seen the shadows of the suspension wires, current leads, and thermo-
couple wires; below are seen the three tripod legs for the central


37For valuable advice in connection with the photography, ac-
knowledgment is made to Herr Dr. A. Naumann, Leipzig.








NACA TM 1366


photograph. Figure 24 shows for simultaneously obtained instantaneous
photographs of the left38 side of the plate (in the case of direct
illumination about 25 cm from the lower edge starting with small inter-
spaces of about 1 cm), a region of about 50-centimeter height: at 24(a)
and the lower part of 24(b), laminar; in the upper part of, 25(b), a
transition; 24(c) and 24(d), turbulent. Figure 25 shows three simul-
taneously taken instantaneous pictures of the cylinder with direct
illumination: 25(a) at the lower stagnation point, laminar; 25(b) at
the left azimuth of 120, turbulent; 25(c) at the upper stagnation
point, warm air stream.

As shown by figures 22 to 25, the outer caustic curve of the rays
near the wall is particularly suitable for determining the critical
location. If this line is broken up, then turbulence certainly exists;
whereas the flow regions farther from the wall are more easily disrupted
through external disturbances. In the laminar (lower) region, the
caustic curve is sharp and is completely stationary; in the turbulent
(upper) region it is torn into several pieces and in disordered motion.
In the time photographs, this means complete washing out. The critical
position itself fluctuates upwards and downwards, at one time very
rapidly and at another slowly. For evaluating the time photographs
there was taken as the "time mean value of the critical position" the
end of the region in which the existence of an outer caustic curve
through differences in brightness could still be determined outwards
and inwards arrowss). Above the critical position the heated layer
rises as a whole outwards as broad, uniformly bright strips. Comparison
of the photographic observation thus determined with the mean critical
position estimated from naked eye observation over a period of time
showed good agreement and therefore the justification of this procedure.

The critical positions obtained by this procedure are'collected
together with the remaining test values in table V. Each critical point
is the mean value of four determinations two of which were made with
direct illumination, and two with a magnified projection picture. The
Gr values are formed in the case of the cylinder with the diameter, in
the case of the plate with the critical height as characteristic length.
By means of equation (74) for the plate and, with f(xkr), equation (84)
for the cylinder, the critical Reynolds numbers of the boundary appear-
ing in the last column were computed. Their mean values of 303 for the
plate and 285 for the cylinder are weighted according to the different
illimination times of the individual pictures. From the mean value
R5kr = 303 the value 1.0x109 is obtained for the mean critical Gr
for the plate.


380n the right of the figure; the pictures with direct illumination
are laterally interchanged.








NACA TM 1366


5. Discussion of Results

The mean value of 303 for the critical Reynolds number of the
boundary later for the plate and 285 for the cylinder can be regarded
in agreement, in view of the accuracy of determination of a critical
number in general and the small number of the schiieren pictures and
disturbances of the room air in particular. This is understandable
from the fact that the velocity profiles of both flow processes at any
distance from the lower edge or lower stagnation point are affine to
each other. The development along the initial stretch occurs in a
different manner, but this should have only a higher-order effect. As
was to be expected, the critical number of the free profile Bkr 300

lies considerably below the critical number of the profile in the case
of forced flow (plate), which, referred tc the displacement thickness
of Burgers and Hansen, amounts to 950 in the wind tunnel, and accord-
ing to more recent tests in the GCttingen water tunnel for very quiet
approach, it amounts to about 1400. It would be of interest to know
whether a theoretical stability investigation for the free profile, the
analog of the familiar computations in the forced-flow case, would have
to be conducted in order to give this increased instability.

With the determination of the critical number for the vertical
plate and the horizontal cylinder and reduction to the critical Reynolds
number of the boundary layer, the data of other investigators will be
compared with regard to the transition from laminar to turbulent flow.

E. Schmidt (ref. 26) notes that according to tests in air, the flow
at about 1000 C plate temperature remains laminar up to a plate depth
of about 50 centimeters, corresponding to a critical Gr of 8x10, in
good agreement with the values obtained herein.

M. Jakob and W. Linke (ref. 19) determined from a combination of
the measurements of various investigators on vertical plates, vertical
and horizontal cylinders, block, and sphere, as the transition point
from the Gr1/4 to the Grl/3 law, the critical number of
(Gr.Pr)kr 3.0xl07, that is, for diatomic gases Grkr m 4.0x107. The
difference as compared with the present determinations should in large
part be ascribed to the fact that hyvirodynamically unequivalent bodies
(Reynolds number of the boundary, see Part III, sec. 5a, eq. (88)) were
here treated together. If account is taken only of the tests on the
vertical plates and vertical cylinders (Part III, sec. 6a), which for
large cylinder diameters and large Gr (thin boundary layer) are at
least approximately equivalent hydrodynamically, there is obtained a
critical Gr of 4x108, which is therefore quite close to the present
determinations.







NACA TM 1366


E. Griffith and A. H. Davis (ref. 17) investigated the local
distribution of the heat transfer on a 270-centimeter-high wall, which
consisted of 25 individual elements. They obtained a minimum value of
the heat transfer at about 38-centimeter height (their own determina-
tion) which they regarded as the transition from the laminar to the
turbulent condition. This agrees in order of magnitude with the pres-
ent values (columns between the individual elements), where it should be
further clarified which point of their curve is to be regarded as the
"critical point" proper.


6. Travel of the Critical Azimuth for the Horizontal Cylinder

On the basis of the given theoretical solution of the flow about
the horizontal cylinder and the point of transition experimentally
determined for a Gr = i09, the travel of the critical azimuth for a
variable Gr can be computed if it is assumed that the turbulence
always occurs if the boundary-layer Re has the same value of 285.
This reasonable assumption will correspond all the more to fact since,
as has just been shown, even for the vertical plate with a quite differ-
ent boundary-layer development, the turbulence is set up at the same
critical number of the boundary layer.

From equation (84). the following relation is obtained, with the
value Rkr = 285:


(Grd) =4 (102)
kr f xkrT

which is evaluated in table VI and plotted on semilogarithmic grid in
figure 26. In accordance with the basic considerations, for a Gr = 109
the start of turbulence occurs for an azimuth of about 1200. Complete-
ly laminar fl.j about the cylinder, for extension of the theory up to
the upper stagnation point, occurs for Gr<3X108. The transition point
reaches the equator for Gr = 3x109.

For greater clarity, the Gr will be expressed for cylinders of
varying diameters, which are in air of 20 C with a surface temperature
of 1000 C. From the definition of Gr, for 6 = 80 (C),
P = 1/293 (OC-1), v(l00o) = 0.231 (cm2/sec), the following relation
obtains:
(Grd 3\
d[c = 15 X 103)1/3 (103)

which with the aid of the critical Grd values from equation (102) has
similarly been evaluated in table VI and plotted in figure 27. Corre-
sponding to the basic considerations, the critical azimuth for the







NACA TM 1366


relations considered for a cylinder of 60-centimeter diameter is 1200.
For cylinders below 41 centimeters, no turbulence occurs at all. The
transition point reaches the equator for a cylinder of 84-centimeter
diameter.

For practical applications, the result is thus obtained that for
heated horizontal conducting pipes up to 40 centimeters in diameter and
surface temperatures of 1000 C in room-temperature air the previously
given formulas are valid for the heat transfer in air (in particular,
eqs. (60) and (65)). For accurate computaticn, the nondimensional
formulation of figure 26 is naturally to be applied. The formulas can,
in the absence of other suitable data, still be applied up diameters of
60 centimeters, because two thirds of the cylinder periphery is still
in laminar flow. There should not likewise he much of a change for
considerably higher surface temperatures (2000 or 3000 C).

The experimental determination of the variation of the critical
azimuth for the cylinder with the Gr, through varying the temperature,
remained unsuccessful. For temperature difference up to about 500 C
the outer caustic curve is still not sufficiently far from the central
shadow in order to be perceived separately from the latter, as is re-
quired for determining the critical number. For this, the distance
between the cylinder and the screen would have to be considerably in-
creased, which is not possible on account of space requirements. Going
beyond the usually employed temperature differences of 800 C, because
of the strong increase in v, no longer gives an increase in Gr and
moreover gives increasingly stronger deviations from the assumption of
moderate temperature differences, which is at the basis of the theory.
The travel of the critical azimuth with Gr is best determined by
varying the cylinder diameter, as follows from figure 27.


V. SUMMARY

From the numerous tests already available on the heat transfer
from horizontal pipes and wires in diatomic gases, the dependence of
the nondimensional heat-transfer number Nu on the Grashof number Gr
and the temperature coefficient Te, which enters as a further non-
dimensional factor at large temperature differences, is determined. The
effect of Te on the heat transfer is quantitatively determined for the
first time. It is particularly large in the region of small Gr (10-4
to 10), where on the average a decrease in Nu by 22 percent for in-
crease of Te from Te = 0 to Te -= 1 is obtained. The experimentally
found dependence of Nu on Gr, Pr, and Te can then obtain a qualita-
tive theoretical explanation.







NACA TM 1366


For the region of Gr (about 104 to 3xlO8) in which the heat trans-
fer is limited to a thin (in comparison with the cylinder diameter),
heated layer with laminar flow, the velocity and temperature fields and
the heat transfer are quantitatively computed from the boundary-layer
differential equations without any additional empirical values; these
computations are found in good agreement with the available measure-
ments. In particular, the one-fourth power law of the heat transfer is
obtained theoretically as Nu = 0.37 Grl/4.

The flow and heat-transfer relations thus computed (Re of the
boundary-layer flow, variation of the local heat-transfer coefficient
along the cylinder periphery, mean heat-transfer coefficient, and total
heat given off by the cylinder) are compared with the already known
relations for the vertical plate. Among other results, the depth of a
vertical plate is determined which for free convection shows equal flow
condition and equal total heat transfer at a given cylinder, so that a
simple computation is made possible for converting the heat-transfer
data for a plate to those of a cylinder and conversely.

In order to know the upper limit of validity of the laminar-flow
and heat-transfer computations and the laminar heat-transfer formulas,
the start of turbulence was determined by schlieren photographs on a
vertical plate and a horizontal cylinder of sufficient size. This
occurs for different values of Gr, namely, Gr = l.OX109 for the plate
and Gr = 3.5xi0O for the cylinder, but for equal Re of the boundary-
layer flow, namely, for Re = 300. In this way, the critical Reynolds
number of the velocity profile of the free flow was determined for the
first time; as compared with other velocity profiles, this profile is
marked by the presence of a maximum value and a point of inflection.
Likewise computed with the Grashof number was the 'critical azimuth' on
the cylinder at which the laminar flow passes into the turbulent flow.


REFERENCES

1. Hermann, E.: Phys. Z., Bd. 33, 1932, p. 425.

2. Nusselt, W.: Gesundh. Ing., Bd. 38, 1915, p. 477.

3. Davis, A. H.: Coll. Res. Nat. Phys. Lab., vol. 19, 1926, p. 193.
(See also Phil. Mag., vol. 43, 1922, p. 329.)

4. Gr6ber, H., und Erk, S.: Die Grundgesetze der Warmeiibertragung.
Julius Springer Berlin, 1933.

5. Schmidt, H.: Ergebn. exakt. Naturwiss., Bd. 7, 1928, p. 342.







NACA TM 1366


6. Gr6ber, H.: Warmeibertragung, Julius Springer (Berlin), 1926.

7. Ayrton, W. E., and Kilgour, H.: Phil. Trans. Roy. Soc. (London),
vol. A183, 1892, p. 371.

8. Langmuir, I.: Phys. Rev., vol. 54, 1912, p. 401.

9. Langmuir, I.: Trans. Am. Inst. Elec. Eng., vol. 31, 1912, p. 1229.
(See also Proc. Am. Inst. Elec. Eng., vol. 32, 1913, p. 407.)

10. v. Bijlevelt, J. S.: Die kiinstliche Konvektion an elektrischen
Hitzdrahte. Diss., T. H. Dresden, 1915.

11. Kennelly, A. E., Wright, C. A., and v. Bijlevelt, J. S.: Trans.
Am. Inst. Elec. Eng., vol. 28, 1909, p. 365.

12. Petavel, J. E.: Phil. Trans., Bd. (A)197, 1901, p. 229.

13. Wamsler, F.: VDI Forschungsheft 98/99 (Berlin), 1911.

14. Nusselt, W.: Diss. T. H. Minchen, 1907.

15. Koch, W.: Beih. z. Gesundh.-Ing. Reihe 1, Heft 22 (Berlin und
Munchen), Oldenbourg 1927.

16. Jodlbauer, K.: Forsch. Ing.-Wes., Bd. 4, 1933, p. 157.

17. Griffiths, E., and Davis, A. H.: Transmission of heat by radiation
and convection. (London), 1931.

18. King, W. J.: Mech. Eng., vol. 54, 1932, p. 347.

19. Jakob, M., and Linke, W.: Forsch. Ing.-Wes., Bd. 4, 1933, p. 75.

20. Davis, A. H.: Phil. Mag., vol. 44, 1922, p. 920.

21. Schmidt, Ernst, und Beckmann, Wilhelm: Das Temperatur- und
Geschwindigkeitfeld vor einer Warme abgebenden senkrechter Platte
bei naturlicher Konvektion. Tech. Mech. u. Thermodynamik, Bd. 1,
Nr. 10, Okt. 1930, pp. 341-349; cont., Bd. 1, Nr. 11, Nov. 1930,
pp. 391-406.

22. Schmidt, E.: Forsch. Ing.-Wes., Bd. 3, 1932, p. 181.

23. Hermann, R.: Phys. Z., Bd. 34, 1933, p. 211.







NACA TM 1366 55


24. Beckmann, W.: Forsch. Ing.-Wes., Bd. 2, 1931, p. 165.

25. Weise, R.: Forsch. Ing.-Wes., Bd. 6, 1935, p. 281.

26. Jakob, M., und Linke, W.: Phys. Z., Bd. 36, 1935, p. 275.

27. Schmidt, E.: 1. VDI, Bd. 76, 1932, p. 1025.


Translated by S. Reiss
National Advisory Committee
for Aeronautics











NACA TM 1366


TABLE I. SUMMARY OF TESTS BY DIFFERENT INVESTIGATORS

Invest igator Ayrxton, Langmuir BiJlevelt Kennelly, Petavel WamEler Koch
KLigour Wright,
ErBlevelt

Substance 9 Pt wiree 5 Pt wires 6 wires: 3 Cu wires 1 Pt w-re Pipes: 4 steel
Ta, Pt, Fe, 6 wrought pipes
Cu, Ag, Ni Fe; 6 Cu,
1 cast Fe

Cylinder aiam., 0.0031- 0.00404- 0.0043- 0.01143- 0.1106 2.05-8.9 1.4-
Scm 0.0556 0.0510 0.1000 0.06907 5.9 10.05
5.9

temperature of 40-00 227-1027 46-239 ------- ------ --- 27.6-188.6
body, ty. .-

Excess tempera- ------ -------- 6------ 015-1 200-1000 36-243 13-174.4
ture, 6, C

Room tempera- 10. 5--15.9 i-?-22 20 (approx) 16 10-29 14.2-22
tare, t o:

Medium Air Air Air Air Air, 2, Air Air
0,

Preeure 760 750 50 70 120-1 900 0.1i- 715 715.7-722.3
mm fig 1 : atm

Cylinder lerntn, 32.5 100 35 150 (approx) 45 300 138-198
cm

S-rround ing BoraiHintal Bas", ,10C': Height, Vertical Horizontal Ba e, ".35 Height,
space pipe 15 im:; 300 cm tanr: pipe: zq meters; 400 cm
lengthr, beigrit, length, Length, height,
2.5 cm; 30 cm 152 cm; 45 cm, 210 cm
d am. height, diam.
5.O cm 660 cm .0 cm

Ratio of ;pace 1640:143 7450:5886 0,000:j30.0 58,0C0:9640 18.6 102:24 286:40
neieht to
ylirider diam.

le number 0.70-1.00 0.67-3..3 0.09-0..58 0.C.'-0.Ei 0.69-3.4r. 0.13-0.82 0.045-
0.606

Relative 7.6-16.6 7.2-2.2 9.4-45.6 25..2 Irreguiar None None
decrease of
Nu for
z'e = 1,
percent

Mean value S'temat i: 25.6 25.2 --------- ---- ----
of ru for
-'', = 1i,
percent

Fige re 1 2 4 6 7










NACA TM 1366 57





co 0
aj O




-0 0 w
.0 3 0
,-i




H LO r. ')
-- O
C.I C

Wo 0 j ED-
rx S- 0 Lb
-- --- CJ 0 o -O O N
F-) 0 CO 0 p o Om
H n r ci w a j
49 o1- .o

'di.. ur OO -H L LO 00
u H
H -r LO S 1- *) r J-- C 0 o 0 1O
II r- C ) -1r 0 1-1 Lj ftj c r- N C I) n CDO T
S" '1 O I O o C to
O-i *'-" a at -- t -1 H V) ( *

0 H- ,.u t o

S -"] ,: 0 ,O '4- 0- 0
U 4 ," ,:D ,9I., C,
o 0 1- 0 O 0
Oa4LO O o

a0 5 x o -C-o -a o x 0
S- '1' rj- -al 0 if) b 0 00 C D
S, r-. ,- .l

n1 ,--I H ,-. LiL. ,o ,O 0 D 0 r-4 O H

.rH ) v l CO H .


.i W H 0 0.
S t 'O 'C0 D uo Oo O O o Fri-


7o ,, ZH oD ,' 0 E. uC. P Pq
0.. ^ f, v *




H" E
H--- ,i- u p "rL 0 I o0 H-



uW .- r- W "-' H> 0 -, 4C
l C: H HQ H


D) H 'C U. 0 L, G) 0- 0 H r- Cf


o 0 .


.--H -* 0_ P4
Z T. '4-4 .-4

,, i :

S3 .- ,- .- m > B -<



.-I .-- f I H-I X C b-





i II
Li' '1'







NACA TM 1366


TABLE V. EVALUATION OF SCHLIEREI] PHOTOGRAPHS FOR OCCURRENCE

.JF TURBULENCE

(a) Vertical plate.

Number b, t,, Exposure tw, Critical height, h cm Grhkr R5kr
mm oC time, 0C Left Pight Middle
se.id

24 74d 16.5 20 78 76.0 55.9 F6.0 13.42 x10 324
8
25 748 16.5 20 113 53.8 47.3 50.h 6.83 x10 274
26 748 16.5 25 100 66.2 60.8 633.5 13.13 x1O0 323
27 748 16.5 20 100 60.0 47.3 54.1 8.12 x108 286

Mean: 303



(b) Horizontal cylinder. Diameter, d, 58.45 centimeters.

lumber b, t. Exposure ty, Critical azimuth, X Gr kr f(xkr) RBkr
am oC time, 0oC Left Ri, ht Middle
sec

6 760 19 10 102 125.50 123.5. 124.5 10.26 x 10 4.55 291
10 760 19 20 102 125.8 110.00 117.90 10.26 w 108 4.35 278
11 760 19 5 102 120.90 116.0 11,.50 10.26 x 10' 4.37 279
12 760 19' 210 102 126.7 117. 'H 122.1' 10.261 x 10 4.48 286
17 752 18 20. 102 13. 1.0 11.40 10.23x X 10 4.39 281
1i 752 1. 5 102 144.5'' 12.5 37 .00I' 10.23 x 1o 4.90 513

Mean: 285


TABLE VI. TRAVEL OF TURBULENCE IRAI'JSITIOI] POC1nT (CRITICAL AZIMUTH xkr)

WITH GRASHOF I UMBER OF C YLTDER Crd OR WITH CYLINDER DIAMETER d

[ty, 100i'' C; te, 200 C; b, 760 mm.]

x. 1 0 1 65 150 120r' 90' '. 3 50C 150 10 5

Grd 3.52 4.30 5.42 1.07 2.96 1.37 2.10 3.35 1.69 2.72
xl08 xO18 x108 x109 x109 xl'0 xio11 xi012 x1013 xl014
d, cm 41.3 44.1 47.7 59.9 84.0 140 347 890 1500 3790








NACA TM 1366 59


0.fPr


oor rP i
00 Te- ___ ____


.3 /./

S.... ,----; J-, (Extrapolated) > /"
S Adju-tment ----- I O ,


0 IC
0









^7.7 --- -- -- j^ ^ ^ ---- --- -------












o +' e-O,
a 1,3 ,
Tests x 1 1 ',0
-7 0
4
19 Grf










A JJ3 .

v --Te-0 (Extrapolated) "-
Adjustment ---. f,33


G-3 -- -2
log Gr

Figure 2. Convection tests of Langmuir on five platinum wires.






60 NACA TM 1566




pr-Lv r
at-- + le 'k to r "
e -=isIto. / \ /
Tests x -0oZto2 Z' / N
=Cito.e5 /
S_ -4"Zto47F ____ ____
r----Tep- (Extrapolated)
Adjustmeati -0.J
.65 7,/* ,






a,7-1






log Gr
Figure 3. Convection tests of Bijlevelt on six wires o.f tantalum, platinum
iron, copper, silver, and nickel.
4z
I r I -,

Ci A---4. A f0
.-Wire diaom, I, Tests / "
---Wire diam,.. c. em .Tes
0o -0 1xt iupw li
O,./- ^ --- -- -,--- ---







479-1


'-4 ---- _______ ---_____


log Gr
Figure 4. Convection tests of Kennelly, Wrsht, ar.d Bijlevelt on three
copper wires.








NACA TM 1366


log Gr
Figure 5. Convection tests of Petavel on one platinum wire.



P-a0 Te
S3- ~ aJ to 026
Tests : 0 .X to JS9
S +z to 0.50 9
2 054O to O. _

Adjustment :- All
values
of'Te/




y
,--i


Figure 6. Convection tests of Wamsler on one copper, one cast iron, and six
wrought iron pipes.







62 NACA TM 1366


7.4

Pr 0. Te
7.3 OfO ro .10
I0 0 to 025
Te tS
7.2 0. 5 to 0.6

AdjuEtment :-----Al
v values
of Te _





--/-
1.0
















I- I- -- ---- ---
-9







/
rO













5I
.5 4 5 6 7
log '-r
Fiv-pure 7. ic.rn.e.tion tests of FKocr orn four teel pipes.


1 Ay,Tt.:,n arina Kliloour
2 Bijleve
3 Kennelly et ai. 7
O-- longmuir --

.,_---^ --




r-J _
O j7------^ ^--------


-J log ,or 0
Figure 8. CompariZon iof teit result of, different investigators for Te = 0
(extrapolster) Limitminr la of smill temperature differences.








NACA TM 1366 63


I I
Pr=0.74
S- Ayrton and Kilgour ____
2 Bijlevelt
J Kenrell:, et al.
0 Languruir


.9-1
a -
b D f -------- >- -----


Figure 9. Comparison of test results of various investigators
for Te = 0.65.


4 3 2 1 0 f 2 3 4 5 6 7
DAo" log Gr

Figure 10. Summary of all test results with diatomic gases (Pr = 0.74) with
the adjustment curve \vi-ing the most probable heat-transfer law based on
present irU'ormation.









64 NACA TM 1566



co
S- I( 0 U
























** N 0O
S r-4 _0 r-4
o -
a) ci) 0c. o \










..-. I










Sm --
>-- l -- r-
-*M U) C m 0
A rd u
E,\ i / yn u
















a r /-1 1 -1 O
o +2 Ar









PO ?D CD il1 03










+) O















LQ V4









TIACA TM 1366


___ "
.1 o1 a 0 tS-S9,I C,- 18.10 C
O G Or-J3. 6




o


UO
0 ., ,

r0 +










O & 1/4 S .5




11e,1-1 en .,),lt uer ir ir = : r ,:ntimeters; t, = 99.20 C; to= 18.10 C;


A.z uatr t deg


0 JO

IZO 7
0 Or3. 6 VF

















J 1 ...
Fijurc L3. '..'Vloc it: Frofi'1,r r -'r .yrirer referred to nondimensional coordinates.
;:.' nruj 'e :r''6 r'ipre.-eriT. th4i.:.r.frci le :ls.:.1in. Points according to
r-i ,,-:miieT.; :i ...lts'je.r icr -r *:TrITimetiers; tW, = 99.20 C; ty,= 18.10 C;



SA;i ut n / deg



.8- 90 t -9S2! t -18.1 c --

o 30 Gr-j376x10'



5 -__ 0


Lr*r'l nGri/4
"r. .: i l ; d i; .;n.c : m r ill, -"- $ 1 gl )
r8 6 1/4-
,t re 14. i.,pe'ture Tjro frilc. near -"lin.er referred to nondimensional
oI'd I;Ljtes. t 'rnt inuor, -'ur.e rieprie-nt: heoretical solution. Points
acc-rirric rt me's.uremerint of .TdlbIsu!er f:', 2r = 9 centimeters; tw = 99.20 C;
':, = i i.1 ,I ; ,r = ,'.. xi,?' .








NACA TM 1366


S2r 5cm



00




o
00
o

0
0- ./

I I*" .,^ -\ ----
,4 0








0 0




W^ n Gr4
Dimensionless distance from wall, rL- .

Figure 15. Velocity profiles near cylinder. Continuou: :ar-. c.'e
theoretical solution. Points according to measurement- o: .*._1. i.er _:r
2r = 5 centimeters; t, = 104.60 C; to3= 18.10 C G = r Y .


de,-

Zr- Scmn

t,- 6.54, t-l.
Gr-b.54.i0 5i


-.N. i

0 2 3 .*
'A n6r%
Dimensionless distance from wall, 8 gtl

Figure 16. Temperature profiles near cylinder. ConT-u;,. cr.- v .-.ve
theoretical solution. Points according to measureme.- :.: .o.icauer for
2r = 5 centimeters; t, = 1:4..- C; to = 18.10 C; Gr -: .-4xi'.







TIACA TM 1366


Iv
EWISW


Azimuth, x, deg


.9


.6



.2




0


Figure 17. Azimuth function g(x) giving variation of local heat-transfer
coefficient or of a reciprocal distance from the wall which is characteristic
for profile (for example, boundary-layer thickness) along cylinder perimeter.


-- Local heat-transfer coefficient
- -- .lean heat-transfer coefficient, as 1:0.777
///. Total heat-transfer coefficient, as 1:1.22


.- / t i. '
0 .2 .6 .8 0-
Bei.ht H of vertical plate


0 M0 6M0 0 x a 5 fII W
*---Azimuth, x deg---
Diameter d = 2r of horizontal cylinder


Figure 18. Cornparis.uon of heat trianEier for rectangular plate and horizontal
cylinder. For H = 1, under otheriiz-e equal conditions, the two diagrams
give dimen-i.anal hest-tarinsfer coefficients a to scale.







NACA TM 1366


---157


1Z00 : : 1.00 1 Z5


Figure 19. Ratio of total heat transfer of horizontal cylinder and vertical
plates (on both sides) of different heights under otherwise equal conditions.
Ratio of boundary-layer Re at upper stagnation point of cylinder and upper
edge of plate. Abscissa, heat transfer and Re; ordinate, plate height.


Figure 20. Vertical plate 100x100x1 centimeters for determining occurrence
of turbulence for free convection.








IACA TM 1366


Figure 21. Borizontal cylinder (58.5-cm diam., 100-cm length) for determining
occurrence .:,f turt.ulenc.e for free *:n-.'.ec-tion.


E ,ir'e L'. -:rlterer ph t.3grap- h ji re.r transfer at the vertical plate. Time
prct.-,-gapi E i -'- .: .Iti L:;. t r.. .I-: .. Surface temperature, t, = 1000 C;
t = 1 .:.- '.; L. T4- milli.eter.; -.rit ial height (arrows), left 66.2
center re-p ri,-rr ... :Ertmet r: critical Grashof number, 13.1x108;
'rit i :'j1 '_. uijn' r':,'-- ." : 'e; ] e, _" ..








NACA TM 1366


.'- -." 253. Sel-ie:-e:: 7.-- --e nee-- --sser -- -he -ariz aal cyl-ider.

b -:- '_-_-e.r ; .'-:" he 7:yLi-ie-. '. "- s~_i l -- ar-ows)
"- _








NACA TM 1366 7]



















b d
































-- .. ----' -2 .er .
07








NACA TM 1366


Figure 25. Schlieren photographs of heat transfer
Simultaneous instantaneous photographs (1/20 sec)
tw = 1020 C; tc= 180 C; b = 752 millimeters; Gr,
nation point, laminar; (b) azimuth of about 1200,
point, upflow of warm air with current lead wires


at the horizontal cylinder.
with indirect illumination.
10.2x108. (a) lower stag-
turbulent; (c) upper stagnation
and so forth.








NACA TM 1366


a,




I "--- J-






0 I15 JO W0 3 iZO 15 W 10
(1 Critical cylinder azimuth, deg
Figijre 26. Variation of position o.f ,-tart of turbulence with Gr
number of cylinder. Computed for critical boundary-layer Re, 285.









3)
'~
















L ----- -- I-------
SI\








0 O 80 2# 720 1_ 0 ___ _
iM jCriticil cylinder a:Lmuth. deg

Figure V7. r, nation of Ionition of -tart of turbulence Vith the cylinder
diameter for surface temperature of l,'C0 iLn air at 200 C and 760 millimeters
of mercur-,'. Computed for critical boundary-layer Re, 285.
0) \


s

-\ ^
'
n <
a ,**
*- 15 ^
^ 0 -- -\-- ----


so U 'i.-^














of mrercur-,'. Computed I'.^r critical sijoundary-lay:er Re, 285.


NACA-Lingley 11-3-54 1000
























j







~ C:c

U~i
L:

CS : L. ; -
43 F.


r b


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0n C, M




0 d = <
-c




eOi
.






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30 '
u 60





u C,



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z5Ec
U i C


S!LM3 "
u 2.C M u 5 B
203







2 zw0 o> <



z z = : tNzo
4; "->4
^ or'*- ~ -"'
^*1.5|ii^
Z Z B NZ


N


U u a > ip S u
N c< c: 0 c 43" >
o c c w

lwEL iS'.t
E 4" Ei





oa~ N.o 545g5>
Cu.




o d : i CW 06-
5 L- C' 0 041N.
cca ZE





LC ME 5 M G 05.( ,




r& -. V 43 ss'S C.'^
43a0
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us t Sa t i-
a54a

C w VCu4 C .
Z1 Ea ;a 0639-






c d c c cbBal
S~t 2 ? nl a
M'a
w m- a-

M= E2 ;QO bdc it .


0

V. 0

z 0O g
c u




4;4
U Zb 4 E|
0s0Q0 -a


t^ P4 bDg
*Og cd


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L) b
aM


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u iil l


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ilg" ll
zzaaSMaoa


I 3 I -^^^
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0 > k 43,









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006 4








" vo c "45 aos
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43*
F44)tsV -RI
w 0 a)-Y~' E
0,81 3 tr a
> r,~


I I
0


s- .. z


c. u C0
2 z *~ >
^ e 1.03


0
< u E .5






c

-. rs 6 -


-00 L .. -
i aE




2 it |
E 0 w -







ZOz=tjy^"
<-a *1Sz
2zaaM z ,z


" G WC 0 0. e
> > Cu 06-0
" EL 0 0 .- 0
0 4







00 ia c


ec a C- u -


'0 0 -* PC 3 C u
Iliillll


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ArJ%
g".
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iM r-s
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r; i.4B



























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witn
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in fc in
0 inSl



in CU 0
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s>...s
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i 00


























tw
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in-. in -






o P























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S..i..00







U >


































U,*

in&
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w' o u '0
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'0 ?* i n1














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in f.*

01 0
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0. "


a













0


0

0a
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001


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s2 ,cs 2
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SI Bz > .5e
-; -a


L" ,
0z S

|ga .l"o

O0" o'P.
.0 u La.
a) zwC e.5e



Co 5La**.*J *
(UZ w
z S Ei 54
001 MS0 V.^



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14 u. 2"
0^f.ajc o
s^ zf si
z-rf-Oa ^ _
Z.0-P-R<
I25C"
E- H 0 a'W > .uE
< 5 -O > 8. < 0 o
2 Q, L)S LS"!


141


0eo w


o> > Ga ct
0 ba LE B a





1-~ .-0 C
a.' C -. ( 'S



.2 5 .~I* 0
.6.S o3-l DD





a.0
30 r 0 S .!2

a) Ti= 0 m .
*- 8 ^ .s&0 a)I



6- .- an Z


a)CU -0 Uc.w .
C bb v 0 0'~a
C C = w lb
s a)~ ** a 5^e
a)0 Wa j a)a)


C'

001



r-.

La a) C~~
G'f l w


S 2>
^ g,
zS^!^


C


a)'T -.

80U

L>E. U to

o 0 0-a0 c
L. w E
cc





L L. -0
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U bb .-
S 0 B 1
a h U0 cK b
z
i0z e.e
Co OPo S *
- U-t, 1: Z -


54i


- 10 .

0 0;'.2 'd J 0a)


E C, 06
0" >- 00 ) a)a


ou 0 o- 0






u cu 0 I 1,r I .
^s sCCa)F-^





ul ~ ~ F. z-.- -
C: S o s^ rs 0





," 0 c


v .
'9 Ei L.'n c L)^



C ,'- .- C a)
t; sl.




v


C: CD .o a) 40 ) W~ 0
I -, a. a> C.= I.


CIO

K^ I '-


~14 10
0 >
Cd cu"



f- Z
a rtsz N -
) a, 0 t





0 D. -. 0 .6 A
w t w 9 1k 2F a E .a
*< -;dB



.. ~ S I ..0-1 ^
0 >

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V ,E F10 4


9.. -4 I' =0 .0~ a .-- .'W
k~ _O
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ga).
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Z w m 0! ho -s g ~ l












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0 aw2o .r u s 411. 00.A (






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4r., ;;,

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0 d 0 ) -4 1t 0 ;
X F. 0.

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a ris 2N~ at-2 a^ S 410 19d>

























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UNIVERSITY OF FLORIDA
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