Experimental determination of local and mean coefficients of heat transfer for turbulent flow in pipes

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Title:
Experimental determination of local and mean coefficients of heat transfer for turbulent flow in pipes
Series Title:
NACA TM
Physical Description:
18 p. : ill. ; 27 cm.
Language:
English
Creator:
Aladyev, I. T
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Heat -- Transmission   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Heat-transfer coefficients were determined for the flow of water through a heated pipe. The local heat-transfer coefficient was found to decrease along the length of the pipe up to a distance of about 40 diameters from the entrance. Equations are given for the local and mean heat-transfer coefficients as functions of the Reynolds number, Prandtl number, and length of the pipe in diameters.
Bibliography:
Includes bibliographic references (p. 11-12).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by I.T. Aladyev.
General Note:
"Report date February 1954."
General Note:
"Translation of "Eksperimental ̕noe Opredelenie Lokal ̕nykh i Srednikh Koeffitsientov Teplootdachi Pri Turbulentnom Techenii Zhidkosti v Trubakh." Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, no. 11, 1951."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778687
oclc - 86080835
sobekcm - AA00006148_00001
System ID:
AA00006148:00001

Full Text
Ack-!rVlo











NACA TM 1356


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1356


EXPERIMENTAL DETERMINATION OF LOCAL AND MEAN COEFFICIENTS

OF HEAT TRANSFER FOR TURBULENT FLOW IN PIPES*

By I. T. Aladyev

A large number of papers have been devoted to the investigation of
heat transfer in the flow through pipes. Many of these papers are
directly or indirectly concerned with the study of the dependence of
the heat-transfer coefficient on the length of the pipe. These investi-
gations have been analyzed in detail in a number of papers refss. 8, 9,
and others). A short resume of only the most important investigations
concerned with the study of the effect of the pipe length on the mean
and local heat-transfer coefficients is given here for the case of tur-
bulent flow of a fluid.

The earliest work is that of Stanton (ref. 1). He determined the
mean heat-transfer coefficient in vertical pipes of different diameters
when they were heated or cooled by water. As a result of this investi-
gation, it was found that the mean heat-transfer coefficients for pipes
with the ratios z/d = 31.6, 41.6, and 62.4 are practically the same
in magnitude; somewhat higher values were obtained only for l/d = 33.8
(I is the length and d the internal diameter of the pipe in meters).
The results of this investigation thus do not permit drawing conclusions
as to the effect of the pipe length on the heat-transfer coefficient.

In 1910, Rietschel (ref. 2) conducted an investigation of heat
transfer by heating the air flowing through horizontal pipes. A brass
pipe of 119-millimeter diameter and of 1985-millimeter length and five
different pipe boilers of different diameters were tested. The number
of pipes in the boiler varied from 3 to 53. In the tests, the over-all
flow of air through the boiler was measured, and the flow of air through
each pipe was determined by dividing the total flow by the number of
pipes. The correctness of the results obtained by such method of measur-
ing the quantity of air discharged is open to serious doubts.

On the basis of these and also some of his own data, Nusselt
(ref. 4) proposed a method of taking into account the effect of the pipe


*"Eksperimental'noe Opredelenie Lokal'nykh i Srednikh Koeffitsientov
Teplootdachi Pri Turbulentnom Techenii Zhidkosti v Trubakh." Izvestiya
Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, no. 11, 1951, pp. 1669-
1681.







NACA TM 1356


length by introducing in the basic equation, of the type Nu = cRenPrm,
the ratio (1/d)-0054. This method received wide application and is
still used at the present time.

In 1930, Burbach (ref. 3) undertook a special study of the effect
of the pipe length on the heat transfer. His method differed from the
method of the preceding investigators and permitted, according to the
author, the determination of the local values of ax. An analysis of
the method of Burbach showed, however, that it was erroneous. The
error consisted in the assumption that the temperature of the wall at
the center part of the pipe is the same as at the end of a pipe of half
the length. Because of the heat conductivity along the wall, however,
this assumption is far from being true. For this reason the results
of Burbach were in error.

In 1931, Lawrence and Sherwood (ref. 5) conducted a new investi-
gation of the effect of the pipe length on the heat-transfer coefficient.
The heat transfer for heated water flowing through horizontal pipes of
four different lengths was studied. The ratio Z/d in these tests
changed with the shortening of the pipe. Notwithstanding the change in
1/d from 59 to 224, the authors did not find any appreciable effect of
the length of the pipe on the heat-transfer coefficient. In 1936, a
detailed investigation of the heat transfer of airplane radiators was
published by Maryamov (ref. 6). The author determined the coefficients
of heat transfer of radiators of various constructions. By computing
the coefficient of heat transfer from the water to the wall according
to the formula of Ten-Bosch (ref. 9) and by neglecting the thermal
resistance of the radiator wall, the author was able to determine the
average value for the entire radiator. Then, applying the assumption
of the hydrodynamic theory of heat exchange as regards the relation
between the heat transfer and the resistance and making a number of
assumptions, the author indirectly determined the effect of the depth
of the radiator, that is, the effect of the length of the pipe on the
heat-transfer coefficient. For taking account of this effect the fol-
lowing equation was proposed:

Nu = 0.25 d RePrl 0.96 exp 0.015 (1)


This equation shows a still greater effect of pipe length on the
heat-transfer coefficient than does the equation of Nusselt. The
latter equation and that of Maryamov show that the degree of the effect
of 1/d on the heat-transfer coefficient a does not depend on Re.

In 1937, Rubinshtein (ref. 7) proposed a new detailed investigation
of the variation of the coefficient of heat transfer with the length
of the pipe. This investigation was conducted by a method based on the







NACA TM 1356


analogy between the phenomena of heat exchange and diffusion. The tests
were conducted in both the presence and the absence of hydrodynamic
stabilization. As a result of the investigation, the change of the
local and mean heat-transfer coefficients with the pipe length was
established.

The preceding summary of the principal experimental investigations
on the change of the heat-transfer coefficient along the length of a
pipe shows that a variety of results have been obtained by different
authors. The available data do not permit formulation of a sufficiently
reliable method for taking into account the effect of the pipe length
on the heat-transfer coefficient; in recent years, a number of authors
(ref. 8 and others) have therefore recommended that this effect be
completely neglected. In view of the theoretical and practical sig-
nificance of the problem, the present author undertook an extensive
investigation of the changes of the local and mean heat-transfer coef-
ficients along the pipe length, some results of which are described
herein.


1. EXPERIMENTAL SETUP

The essential scheme of the test setup is shown in figure 1. Water
distillatee) from the tank (1) enters a high-pressure gear pump (2) driven
by a constant current motor. The pump discharges the water into a heat
exchanger indirectly or, for small discharge quantities, through the
intermediate tank (4) which together with the regulating tank (6) permits
a fine regulation of the discharge and the maintenance of a constant
value. On issuing from the heat exchanger the water enters a tank (6)
and then, depending on the position of the three-way cock'(7), enters
either a measuring vessel or returns to the collecting tank (1). The
heating is effected by slightly superheated steam.

The construction of the heat exchanger is shown at the top of fig-
ure 1, and its principal geometric characteristics are given in the fol-
lowing table:


TABLE I.
Segment 1 2 3 4 5 6 7 8 9 10 11 12 Total

Distance
to center
of seg-
ment, mm 2.45 7.25 11.95 19.00 28.40 48.10 78.10 117.70 177.70 248.10 323.20 488.10

Length
of seg-
ment, mm 4.9 4.7 4.7 9.4 9.4 30.0 30.0 49.2 70.7 70.2 80.0 235.0 599.0






NACA TM 1356


The heat exchanger consists of three coaxially arranged pipes: the
innermost one (5) of 15.0-10.2 millimeter diameter, the middle one (inner
jacket) (12) of 52-50 millimeter diameter, and the outer one (outer steam
sleeve) (13) of 72-70 millimeter diameter.

In the lower half of the inner jacket, at the distances shown in
table 1, are soldered 10 diaphragms (8) forming 12 segments for the con-
densate. The diaphragms at 0.5 millimeter along the height do not reach
the inner pipe. Plates of thin copper foil (of 0.2 mm thickness) are
soldered on the outer surface of the inner tube. This construction
assured the flow of all the condensate formed over a given part of the
pipe into the corresponding segment and did not require a constant
accurate setting of the heat exchanger in the horizontal position. From
the segments the condensate flowed into the measuring vessel. To guard
against the loss of heat to the outside, the annular space between the
middle and outer pipe was likewise heated with steam.

The construction of the inlet and the outlet of the heat exchanger
is clearly shown in figure 1. The inlet, which assured a uniform dis-
tribution of the temperature and inlet conditions, approximated the ap-
paratus generally used in most commercial heat exchangers. At the out-
let a mixer (15) was mounted which assured good mixing of the water be-
fore its temperature was measured. In order to reduce the loss of heat
through heat conductivity between the heat exchanger and the mounting,
thick textolite disks (8 mm thick) were used. The wall temperatures
were measured by means of thermocouples located in the wall at a dis-
tance of 0.3 millimeter from the inner surface. All the thermocouples
were placed along one generator line of the pipe at distances of 5, 50,
120, 210, 450, and 595 millimeters from the inlet section. The thermo-
couples were of doubly insulated (enamel and silk) copper and constan-
tan wire 0.1 millimeter in diameter.

The change of the temperature of the water in the heat exchanger
was measured by 10-junction differential thermocouples located in
thick-walled glass tubes filled with paraffin. The tubes were passed
through packing glands into the inlet and outlet sections of the heat
exchanger in such manner that the thermocouple junctions were located
at the inlet section and immediately behind the mixer. The junctions
of single thermometers that were used to measure the absolute magni-
tudes of the temperature of the fluid at the inlet and outlet of the
heat exchanger were also placed in the glass tubes.

The electromotive force of the thermocouples was measured by a
potentiometer ("Etalon" type PN-2) with a mirror galvanometer which
assured an accuracy of measurement up to 0.0005 millivolt corresponding
to about 0.0150. The cold junctions of the thermocouples were at all
times situated in melting ice.







NACA TM 1356


The test was conducted in the following manner: After the required
water flow was established in the inner jacket of the heat exchanger,
slightly superheated (by 20 50) steam was admitted. First, in order
to remove the air, the steam was admitted in large excess. The
amount was then decreased and was established such that only slight
vaporization occurred through the pipes that conducted away the con-
densate. At the same time the steam was admitted to the steam jacket
of the heat exchanger. The steady thermal state of the system was
attained after about 1 to 1.5 hours. The temperature of the water at
the inlet and outlet and the temperature of the steam varied during the
test by no more than 0.10. At the start of the test, the temperature
of the water, its change in the heat exchanger, and the temperature of
the steam were measured. Then, 12 flasks fixed to a single base were
simultaneously placed under all the condensate pipes through which the
condensate from the sections of the inner jacket flowed and the time
was noted. The flasks were provided with special cocks and were
located 10 to 20 millimeters from the ends of the condensate pipes so
as to permit complete collection of the condensate and conduction of
the steam issuing with the condensate to the atmosphere.

After 5 or 6 days of operation the run was discontinued, the water
was changed, and the inner surface of the investigated pipe of the heat
exchanger was cleaned. Generally, it was found to be entirely clean, but
nevertheless this procedure was regularly repeated.


2. PROCEDURE IN EVALUATING TEST DATA

In the experimental investigation of the processes of heat exchange,
the required magnitude is the heat-transfer coefficient,- which may be
determined from the heat-balance equation which for an element of the
pipe has the form

dQ = axAtxdF = Gfcpdtx (2)

where Q is the amount of heat absorbed or given out per unit time in
kilocalories per hour, a. is the local value of the heat-transfer
coefficient in kilocalories per square meter per C, Atx is the tempera-
ture difference in a section a distance x from the inlet section of
the pipe in C, F is the area of an element of the pipe in square
meters, Gf is the weight of fluid discharged in kilograms per hour,
Cp is the specific heat of the fluid at constant pressure in kilo-
calories per kilogram per oC, and dtx is the change of temperature of
the fluid over the length element dx at distance x from the inlet
section of the pipe in 0C.






NACA TM 1356


Equation (2) may be solved for the local values of ax

Gfcpdtx (3)
ax = txZdF

Similarly, from the heat-balance equation for a pipe of length 1,
the average values of the heat-transfer coefficient a may be obtained

Gfcpbt (4)


where At is the mean temperature difference for a pipe of length 1
in oC and bt is the change in the temperature of the fluid in the pipe
of length I in OC.

From equations (3) and (4) it follows that for determining the
local and mean values of the heat-transfer coefficients it is necessary
and sufficient to know, in addition to the weight of fluid discharged,
the change in the temperature of the pipe wall and of the fluid along
the length of the pipe. As previously mentioned, the temperature varia-
tion of the pipe wall was measured directly in the tests with the aid
of thermocouples. A number of typical curves, representing the change
in wall temperature along the pipe, are shown in figure 2. In computing
the mean temperature difference, the temperature of the pipe wall was
taken to be the mean integrated value for the element or section of the
pipe under consideration.

The variation of the temperature of the water along the length of
the pipe could be determined by computation because the quantity of heat
given off and absorbed by the individual segments was known. These com-
putations permitted constructing curves showing the variation of the
temperature of the water along the length of the pipe for each test.
The most typical curves are shown in figure 3. In making these compu-
tations, the quantity of heat qj, in kilocalories per hour, given by
the steam to an element of the pipe of length Zi was determined from
the amount of the condensate gki, in kilograms per hour, formed in the
corresponding segment according to the following equation:

qi = gkir (5)

where r is the latent heat of steam formation in kilocalories per
kilogram. The total quantity of heat imparted by the steam to the
water was computed as the sum of the heats imparted to all 12 segments
of the pipe according to the equation

12
Qk = gklir + Qs (6)







NACA TM 1356


where Qs is the superheat of the steam. (The superheat of the steam
was several tenths of a percent of the total quantity of heat taken up
by the pipe.) On the other hand, the same quantity of heat could be
determined from the change in the heat content of the water in the heat
exchanger

Qf = GfcpSt (7)

The second method of determining Q was the more accurate one
since the amount of water discharged and the variation of its tempera-
ture were determined with great accuracy. Although the difference
between Qk and Qf did not exceed 3 percent, the heat value in the
evaluation of the test data was that given by Qf. In correspondence
with this, the heat quantities computed by equation (5) were corrected
by the value Qf where the difference between Qf and Qk was distri-
buted between the parts in proportion to their areas.

After the curves for the temperature variation of the fluid and of
the wall along the pipe were obtained, the values of the beat-transfer
coefficients were computed by equation (3) for individual short ele-
ments of the pipe of length Ali, and these were assumed to be the
local coefficients. The chosen lengths of the elements Ali varied
from 2 millimeters at the inlet of the pipe to 40 millimeters at its
end. Depending on requirements, the heat-transfer coefficients were
computed for 14 7 elements at different distances from the inlet
of the pipe, the distances being reckoned from the inlet to the mid-
point of the element. In the same manner, the mean values of the
heat-transfer coefficients for pipes of different lengths. were
determined.

In accordance with the most recent data in the literature refss. 8
and 10), the determining temperature was taken to be the arithmetical
mean of the temperature of the fluid with the Pr characteristic raised
to the 0.4 power.


3. LOCAL VALUES OF HEAT-TRANSFER COEFFICIENT

The test data on the local values of the heat-transfer coefficient
are shown in figure 4, which shows that the heat-transfer coefficient
drops with increasing distance from the inlet section until x/d
attains a definite value which is a function of Rex and decreases with
increase in the latter. For x 40d, however, the heat-transfer coef-
ficient attains a practically constant value for all Rex. It is seen







NACA TM 1356


also from figure 4 that the effect of x/d on ax decreases with
increasing Rex.

It is of interest to compare these results with those obtained by
other investigators. From the brief review given at the beginning of
this paper, it follows that such comparison is possible only with the
results of Y. M. Rubinshtein (ref. 7). The results of the latter,
evaluated to correspond with the procedure here assumed, are shown in
figure 5. Comparison of figure 5 with figure 4 shows the identity of
character and the degree of dependence of ax on x/d and Rex
obtained in these two investigations. This agreement in the results
obtained by the method of the diffusion analogy and by the direct
method confirms the possibility of studying the phenomena of heat trans-
fer by the method of analogy with diffusion in those cases where the
temperature difference differs considerably from an infinitely small
value. The succeeding tests showed, however, the limitations of the
method of diffusion analogy. The latter gives considerable error if
the forces of gravity are comparable with those of inertia, which is
the case for the laminar flow of a fluid.

Analysis of the test data established the fact that the effect of
the ratio x/d and of Rex can be taken into account with the aid of
an exponential function with negative exponent. The results of this
analysis are presented in figure 6. The equation of the curve passing
through the mean values of the test data is of the following form:

2.25
0.8 0.4 Re0.3
Nux = 0.044 Rex Prx ) (8)

This equation holds for x/d < 40. For the sections of the pipe
which are at a distance x > 40d from the inlet section, the local
values of the heat-transfer coefficient practically do not depend on
x/d and according to figure 4 may be determined from the equation


Nux = 0.0156 Re 6 Pr4 (9)

The values of the numerical factors appearing before Rex in
equations (8) and (9) hold only for the conditions applying to the
present investigation.

The method described for taking into account the effect of x/d
and Rex on the local values of the heat-transfer coefficient is
inconvenient and complicated for practical purposes. In order to







NACA TM 1356


simplify the computation of this effect, another method was applied; the
method is based on the proportionality of a( for any section of the
pipe to the value of the local heat-transfer coefficient a0,x for a sec-
tion at a distance x > 40d from the inlet section of the pipe; that is,
it was assumed that


cax = kxC,x (10)

where kx is a factor taking into account the change in ax along the
pipe and equal to unity for x/d > 40.

The results of the computations of the factor kx from the test
data of the present investigation are given in the following table:

TABLE II.

e x/d 0.5 1.0 2.0 5.0 10 20 30 40

104 2.04 1.65 1.46 1.29 1.18 1.10 1.04 1.0
2X104 1.78 1.45 1.36 1.23 1.15 1 1.8 1.03 1.0
5x104 1.50 1.34 1.26 1.17 1.11 1.06 1.02 1.0
105 1.28 1.20 1.15 1.10 6 16 1.2 1.01 1.0
106 1.12 1.10 1.08 1.05 1.03 1.01 1.00 1.0


MEAN VALUES OF HEAT-TRANSFER COEFFICIENT

The test data on the mean values of the heat-transfer coefficient
are presented in figure 7 from which it is seen that, with decrease in
the relative length of the pipe, the mean value x of the heat-transfer
coefficient a increases. For pipes of length I > 50d, however, a
practically ceases to depend on the length of the pipe. This result
agrees qualitatively with the results obtained in most of the previous
investigations, while quantitatively it agrees with the results of
Lawrence and Sherwood (ref. 5) and these of Y. M. Rubinshtein (ref. 7).
It is also seen from figure 7 that the effect of Z/d on a decreases
with increasing Re. This result does not agree with the results obtained
by Nusselt (ref. 4) and Maryamov (ref. 6), according to whom the degree of
the effect of 1/d on the heat-transfer coefficient a does not depend
on Re. The result agrees, however, with that of Rubinshtein (ref. 7).
The latter's data on the mean heat-transfer coefficients were evaluated
according to the procedure assumed by us. It was then found that the
character of the dependence of a on Re and 1/d is similar to that
obtained in the author's tests. This is particularly well seen in fig-
ure 8, which shows the dependence of the exponent of Re on ./d accord-
ing to the tests of the author and of Rubinshtein.







NACA TM 1356


For the purpose of reducing the obtained results to a unique rela-
tion, the test data on the mean values of the heat-transfer coefficient
were evaluated by a procedure similar to that employed in Section 3 for
the local values of ax. The results of this evaluation are shown in
figure 9. The curve passing through the mean values of the test data
has the following equation:

3.1
%- Re0.355
Nu = 0.124 Re0.7 Pr0.4 (1) (Re3


This equation holds for pipes of length I <_50d. For pipes of
length Z<50d the mean value of the heat-transfer coefficient for any
Re is practically independent of the length of the pipe and may be
determined from the equation


lu = 0.031 ReO.8 prO.4 (12)

The numerical values of the factors in front of Re in equa-
tions (11) and (12) are valid only under the conditions of the present
investigation. For the purpose of simplifying the computation of the
effect of the length of the pipe on the mean value of the heat-transfer
coefficient, the same method was used as for the case of the local
values ao; that is, it was assumed that

a = ka (13)

where k is a factor that takes into account the effect of the length of
the pipe on the mean values of the heat-transfer coefficients and is
equal to unity for 1/d >_ 50 and aO is the mean value of the heat-
transfer coefficient for pipes of length Z > 503.

The results of the computations of the factor k from test data
obtained in the present investigation are given in the following table:

TABLE TIT.
/d 0.5 1.0 .0 5.0 10 15 20 30 40 50

104 1.81 1.65 1.50 1.34 1.23 1.17 1.13 1.07 1.03 1.0
2x104 1.63 1.51 1.40 1.27 1.18 1.13 1.10 1.05 1.02 1.0
5>104 1.42 1.34 1.27 1.18 1.13 1.10 1.08 1.04 1.02 1.0
105 1.34 1.28 1.22 1.15 1.10 1.075 1.06 1.03 1.02 1.0
106 1.17 1.14 1.11 1.08 1.05 1.13 1.03 1.02 1.01 1.0







NACA TM 1356


The factor k may also be determined from the equation

2

k = 5.22 Re-1/8 ( Re0.3 (14)



CONCLUS I ONS

1. In the steady turbulent flow of a fluid in a pipe, the local
values of the heat-transfer coefficient decrease along the length of the
pipe practically up to a section at the distance x = 40d from the inlet
section of the pipe. This distance is a function of the Reynolds num-
ber Rex and decreases with increase in the latter, but starting from
x 40d the local heat-transfer coefficients for any Rex do not depend
on x and may be computed by equation (9).

The values of the local heat-transfer coefficients at a distance
x <40d may be determined by equation (8) or by multiplying the local
values a determined by equation (9), by the factor kx, taken from
table II.

2. For a steady turbulent flow of a fluid in a pipe, the mean values
of the heat-transfer coefficient decrease with increase in the relative
length of the pipe. The length of the thermally stabilized part for mean
values of the heat-transfer coefficient is a function of Re and
decreases with increase in the latter; but starting from I = 50d for
all Re, the mean values of the heat-transfer coefficient practically
do not depend on 2/d and may be computed by equation (12). The values
of the mean heat-transfer coefficients for pipes with i/d < 50 may be
determined by equation (11) or by multiplying a, determined from equa-
tion (12), by the factor k, taken from table III.

The author wishes to acknowledge the valuable suggestions he received
in writing this paper from Academician M. V. Kirpichev and Correspondent
member of the Soviet Academy of Science M. A. Mikheev.


REFERENCES

1. Stanton, W.: Proc. Roy. Soc. (London), 1897.

2. Rietschel, H.: Mitt. Prifungsanstalt f. Heizung, Berlin, Bd. 3,
Sept. 1910.

3. Harmon, R., und Burbach, Th.: Stromangswiderstand und Warmeubergang
in RPhren. Leipzig, 1930.







NACA TM 1356


4. Nusselt, W.: Der Wgrmeubergang in Rohr. VDI, 61, 1917.

5. Lawrence, A. E.', and Sherwood, T. K.: Heat Transmission to Water
Flowing in Pipes. Ind. and Eng. Chem., vol. 23, 1931.

6. Maryamov, N. B.: Drag and Heat Transfer of Aircraft Radiators.
Trudy TSAGI (CAHI Report), no. 280, 1936.

7. Rubinshtein, Y. M.: Investigation of the Processes of Regulating the
Heat Transfer and Inverse Cooling. GONTI, 1938.

8. McAdams, William H.: Heat Transmission. Second ed., McGraw-Hill Book
Co., Inc., 1942.

9. Ten-Bosch.: Heat Transfer. Neftyanoe Izd., 1930.

10. Mikheev, M. A.: Fundamentals of Heat Transmission. Gosenergoizdat,
1949.








NACA TM 1356



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14 NACA TI 1356



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Figure 3. Curves of variation of temperature of fluid along pipe.








NACA TM 1356


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Figure 5. Local values of heat-transfer coefficient. Data from reference 7.







NACA TM 1356


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NACA TM 1356


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2 l7 1 19 #9 6 I 71 7


Figure 7. Mean values of heat-transfer coefficient.


Figure 8. Dependence of exponent of Re on I/d according to results of Rubinshtein and
of author.


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Figure 9. Mean values of beat-transfer coefficient for pipes of length I 1 60 d


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