Measurement of heat-transfer and friction coefficients for flow of air in noncircular ducts at high surface temperatures

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Material Information

Title:
Measurement of heat-transfer and friction coefficients for flow of air in noncircular ducts at high surface temperatures
Series Title:
NACA RM
Physical Description:
26 p. : ill. ; 28 cm.
Language:
English
Creator:
Lowdermilk, Warren H
Weiland, Walter F
Livingood, John N. B
Lewis Research Center
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Skin friction (Aerodynamics)   ( lcsh )
Nusselt number   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: Measurements of average heat-transfer and friction coefficients were obtained with air flowing through electrically heated ducts having square, rectangular (aspect ratio, 5), and triangular cross sections for a range of surface temperature from 540° to 1780° R and Reynolds number from 1000 to 330,000. The results indicated that the effect of heat flux on correlations of the average heat-transfer and friction coefficients is similar to that obtained for circular tubes in a previous investigation and was nearly eliminated by evaluating the physical properties and density of the air at a film temperature halfway between the average surface and fluid bulk temperatures. With the Nusselt and Reynolds numbers based on the hydraulic diameter of the ducts, the data for the noncircular ducts could be represented by the same equations obtained in the previous investigation for circular tubes. Correlation of the average difference between the surface corner and midwall temperatures for the square duct was in agreement with predicted values from a previous analysis. However, for the rectangular and triangular ducts, the measured corner temperature was greater by approximately 20 and 35 percent, respectively, than the values predicted by analysis.
Bibliography:
Includes bibliographic references (p. 11).
Statement of Responsibility:
by Warren H. Lowdermilk, Walter F. Weiland, Jr., and John N.B. Livingood.
General Note:
"Report date October 9, 1953."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003808103
oclc - 128345931
sobekcm - AA00006158_00001
System ID:
AA00006158:00001

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FOR FLOW OF AIR IN NONCIRCULAR DUCTS AT HIGH

SURFACE TEMPERATURES

By Warren H. Lowdermilk, Walter F. Weiland, Jr.,
and John N. B. Livingood

Lewis Flight Propulsion Laboratory
Cleveland, Ohio ,,,, ..,.. ...

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IVERSTIY OF FLORIDA
CUMENTS DEPARTMENT
SMARSTON SCIENCE UBRARY
i. BOX 117011
NESVILLE, FL 32611-7011 USA


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NATIONAL ADVISORY COMffITEE

FOR AERONAUTICS
WASHINGTON
January 25, 1954


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RESEARCH MEMORANDUM




". :'.:MEASUREMENT OF HEAT-TRANSFER AND FRICTION COEFFICIENTS


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NACA RM E53J07


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


MEASUREMENT OF HEAT-TRANSFER AND FRICTION COEFFICIENTS

FOR FLOW OF AIR IN NONCIRCULAR DUCTS AT HIGH

SURFACE TEMPERATURES

By Warren H. Lowdermilk, Walter F. Weiland, Jr.,
and John N. B. Livingood


SUMMARY

Measurements of average heat-transfer and friction coefficients
were obtained with air flowing through electrically heated ducts having
square, rectangular (aspect ratio, 5), and triangular cross sections for
a range of surface temperature from 5400 to 17800 R and Reynolds number
from 1000 to 330,000.

The results indicate that the effect of heat flux on correlations
of the average heat-transfer and friction coefficients is similar to
that obtained for circular tubes in a previous investigation and was
nearly eliminated by evaluating the physical properties and density
of the air at a film temperature halfway between the average surface
and fluid bulk temperatures. With the Nusselt and Reynolds numbers
based on the hydraulic diameter of the ducts, the data for the non-
circular ducts could be represented by the same equations obtained in
the previous investigation for circular tubes.

Correlation of the average difference between the surface corner
and midwall temperatures for the square duct was in agreement with
predicted values from a previous analysis. However, for the rectangu-
lar and triangular ducts, the measured corner temperature was greater
by approximately 20 and 35 percent, respectively, than the values pre-
dicted by analysis.


INTRODUCTION

An experimental investigation was instituted at the NACA Lewis
laboratory to obtain heat-transfer and related pressure-drop informa-
tion for air flowing in tubes at high surface and fluid temperatures.
The effects of such variables as surface temperature, inlet-air








NACA RM E53J07


temperature, length-to-diameter ratio, and tube-entrance configuration
on heat transfer and pressure drop in smooth round tubes are summarized
in reference 1.

The scope of the general investigation is extended herein to in-
clude the effect of flow-passage shape on heat-transfer and friction
coefficients for air flowing through electrically heated square, tri-
angular, and rectangular tubes at high heat-flux conditions. Data
were obtained for a range of Reynolds number from 1000 to 330,000
and surface temperature from 5400 to 17800 R, and the results are
compared with those of reference 1 for circular tubes.


APPARATUS AND PROCEDURE

Arrangement of Apparatus

A schematic diagram of the heater tubes and associated equipment
is shown in figure 1. The experimental setup is the same as described
in reference 1. Compressed air is supplied through a pressure-
regulating valve, a cleaner, and a surge tank to a second pressure-
regulating valve where the flow rate is controlled. From this valve,
the air flows through a bank of rotameters into a three-pass mixing
tank, through the test section, and into a second mixing tank from
which it is discharged to the atmosphere.

Electric power is supplied to the heater tube from a 208-volt, 60-
cycle supply line through an autotransformer and a 14:1 power trans-
former. The low-voltage leads of the power transformer are connected
to the heater-tube flanges by copper cables. The capacity of the elec-
tric equipment is 15 kilovolt-amperes.


Test Sections

Three different 24-inch-long cross-sectional shapes, as shown in
figure 2 square, equilateral triangle, and rectangle were investi-
gated, the inner dimensions of which were as follows:

Shape Perimeter, Hydraulic Length to hydraulic-
in. diam., diam. ratio
in.
Square 1.80 0.45 53
Equilateral 2.31 .45 53
triangle
Rectangle 3.0 .42 57







NACA RM E53J07


The test sections were fabricated from Inconel sheet stock with a
thickness of 0.031 inch. The test-section sides were cut to the desired
dimensions and clamped on ground-steel forms; each corner was welded
with a heliarc welder. The excess weld material was ground off so that
the corner wall thickness did not exceed 1/32 inch. The inner corners
of the test sections were sharp and even along the entire length of the
test section. Steel flanges welded to the test sections at each end
provided electric contact with the transformer leads from the power
supply. Channels were milled in the outer faces of the flanges to
minimize end heat losses, and the test sections were thermally
insulated.

Outside-wall temperatures were measured at 13 stations along the
length of the test sections (fig. 2) with chromel-alumel thermocouples
and a self-balancing indicating potentiometer. At each station, ther-
mocouples were located at the center of each side, except for the rec-
tangular test section, where the thermocouples were omitted on the short
sides. Thermocouples were also located at each corner at the three sta-
tions located 3, 12, and 21 inches from the entrance.

Static-pressure taps were located 1/8 inch from the entrance and
exit of each test section, and each section was fitted with a long-
radius nozzle the throat dimensions of which matched the cross-sectional
dimensions of the test section.


Range of Conditions

Heat-transfer and associated pressure-drop data were obtained with
the square, the equilateral triangular, and the rectangular test sec-
tions with rounded entrances over a range of Reynolds number from 1000
to 330,000, average outside-wall temperatures from 5400 to 17800 R, and
heat-flux densities up to 120,000 Btu per hour per square foot of heat-
transfer area.


SYMBOLS

The following symbols are used in this report:

A cross-sectional area, sq ft

cp specific heat, Btu/(lb)(oF)

4A
D inside hydraulic diameter, perimeter ft
Perimeter-


E voltage drop across test section, v







NACA RM E53J07


f average friction coefficient

ff modified film friction coefficient

G mass velocity, lb/(hr)(sq ft)

g acceleration due to gravity, 4.17X108 ft/hr2

h average heat-transfer coefficient, Btu/(hr)(sq ft)(oF)

I current flow through test section, amp

k thermal conductivity, Btu/(hr)(sq ft)(oF/ft)

k* ratio of thermal conductivity of wall material to coolant

L length of test section, ft

Nu Nusselt number, hD/k

Pr Prandtl number, Cpp/k

p static pressure, lb/sq ft abs

Ap over-all static-pressure drop across test section, lb/sq ft

Qg heat loss from test section to surroundings, Btu/hr

R gas constant, 53.35 ft-lb/(lb)(oR)

Re Reynolds number, pVD/P

S heat-transfer area of test section, sq ft

s* ratio of wall thickness to hydraulic diameter

T total temperature, OR

Tl + T2
-o average fluid bulk temperature, 2 oR

Ts + Tb
Tf average fluid film temperature, s2 OR

Tc Tm
Ts average surface temperature, Tm + 2 R







NACA RM E53J07 5


t static temperature, OR

V velocity, ft/hr

W flow rate, lb/hr

T ratio of specific heats

p. absolute viscosity, lb/(hr)(ft)

p density, lb/cu ft

Subscripts:

av average

b bulk (when applied to properties, indicates evaluation at average
bulk temperature, Tb)

c peripheral location at corner of test section

f film (when applied to properties, indicates evaluation at average
film temperature, Tf)

fr friction

m peripheral location midway between corners of test section

s surface (when applied to properties, indicates evaluation at
average surface temperature, Ts)

1 test-section entrance

2 test-section exit


RESULTS AND DISCUSSION

Heat Balances

Heat balances for each test section are shown in figure 5, where
the electric heat input minus the heat loss determined for condition
of no air flow through the test section is plotted against the rate
of heat transferred to the air as determined by air-flow rate and
temperature measurements. The heat balances obtained at low heat in-
puts and correspondingly low flow rates were very poor. In this re-
gion the air temperature at the exit of the test section could not







NACA RM E53J07


be measured accurately, because the very low velocities in the outlet
temperature mixing tank prevented the attainment of equilibrium condi-
tions in the mixing tank.

The heat balances improved rapidly with increase in flow rate, and
for values corresponding to turbulent flow in the test sections the
data are in agreement with the match line (solid). The heat-transfer
coefficients presented in reference 1 for round tubes, with which the
present data are compared, were calculated from the flow rate and tem-
perature rise of the air measurements; hence, for consistency the pre-
sent data are calculated in a similar manner for flow rates correspond-
ing to Reynolds numbers of 10,000 or greater. For lower flow rates,
the electric heat-input and heat-loss measurements were believed to
be more accurate than the measured outlet-air temperature. Therefore,
the heat-transfer coefficients for Reynolds numbers less than 10,000
are calculated from the electric heat-input and heat-loss measurements.


Correlation of Heat-Transfer Coefficients

The average heat-transfer coefficient h was computed from the
experimental data by the relation

h Wp,b(T2 T)
S(Ts Tb) ()

where

(3.415EI Qj)
T2 = T1 +
Cp,b

for Reynolds numbers less than 10,000. The bulk temperature of the air
Tb was taken as the arithmetic mean of the temperatures at the entrance
T1 and the exit T2 of the test sections. The average surface tem-
perature Ts was taken as the arithmetic mean of the average outside
corner temperature and the average outside midwall temperature. The
temperature drop through the wall was neglected.

The physical properties of air used in calculating the Nusselt,
Reynolds, and Prandtl numbers are the same as those used in reference 1,
wherein the viscosity and specific heat were based on values reported
in reference 2, and the thermal conductivity was assumed to vary as the
square root of temperature.

The results presented in reference 1 for turbulent flow in circular
tubes indicate that the average Nusselt number decreases progressively







NACA RM E53J07


as the ratio of surface to fluid bulk temperature increases when the
fluid properties are evaluated at the fluid bulk temperature. The
effect of the ratio of surface to bulk temperature was eliminated by
evaluating the properties of the air, including the density term in
the Reynolds number at the film temperature, defined as the arithmetic
average of the surface and bulk temperatures. The data for Reynolds
numbers greater than 10,000 were well represented by the following
relation:

hD fVbD0.8 cfP 0.4 -0.1
hD 0.034 (P ( kf)P
kf \ k \ kf (D)

which, for values of L/D between 53 and 57, becomes

hD .03 fVb0.8 (cpf 0.4
kf = t0.023 ) (2)

The average heat-transfer coefficients obtained herein for square,
rectangular, and triangular ducts for a range of Reynolds number from
1000 to 200,000 and ratios of surface to bulk temperature from 1.2 to
2.3 are correlated accordingly in figure 4. A solid line representing
data obtained in reference 1 with a circular test section having a
length-to-diameter ratio of 60 for similar conditions is included for
comparison. For Reynolds numbers greater than 10,000, the reference
line represents equation (2); for smaller Reynolds numbers the refer-
ence line represents the data of reference 1 recomputed on the basis
of the electric heat-input and heat-loss measurements for purposes of
comparison. The data for the square tube (fig. 4(a)) agree well with
the reference line for all surface to bulk temperature ratios and
Reynolds numbers. For the rectangular duct having an aspect ratio of
5, the data (fig. 4(b)) are considerably higher than the reference line
for Reynolds numbers from 1000 to 10,000 and are represented by equa-
tion (2) for Reynolds numbers above 2500. The higher values are in
agreement with data obtained for noncircular ducts at lower heat fluxes
by other investigators. These data indicate that use of the hydraulic
diameter does not result in correlation of data for various passage
shapes in the laminar and transition flow regions. For Reynolds num-
bers above 40,000 the data fall slightly below the reference line.

In figure 4(c), the average heat-transfer coefficients for the
triangular duct vary similarly to those obtained for the square duct.
For Reynolds numbers above 10,000 the data were lower than the reference
line by 5 to 15 percent and could best be represented by a line having
a slope of 0.78 rather than 0.8. This difference in slope is also
noticeable in figure 4(b) for the data of the rectangular duct for
high Reynolds numbers. This variation in slope could be eliminated








NACA EM E53J07


by defining the average surface temperature as the midwall temperature
instead of as the arithmetic average of the corner and midwall tempera-
tures. For example (as is shown in fig. 7), the difference between the
average surface and the average midwall temperatures for the triangular
tube is 3 percent of the difference between the average midwall tempera-
ture and the fluid bulk temperature for a Reynolds number of 10,000,
and 11 percent for a Reynolds number of 100,000. Evaluating the average
heat-transfer coefficient on the basis of the difference between the
midwall and fluid bulk temperatures would result in a corresponding
increase in the average heat-transfer coefficient and, hence, would
bring the data into agreement with the reference line for circular
tubes. Measurements of the variation in rate of heat transfer around
the periphery of noncircular ducts are required in order to define the
average surface temperature for evaluating the average heat-transfer
coefficient.


Correlation of Friction Coefficients

The method of calculating the average friction coefficient is es-
sentially the same as described in reference 1, wherein

APfr
f = (3)
L PfVb
4
D 2g

where

G2R (t2 t
Apfr = (pl ) g 22 )


(Pl + P2
Pf (-f4- )

and


t = (T 2 rg 22 + 2T R 2

The subscripts 1 and 2 refer to positions within the ducts, located
1/8 inch from the entrance and exit ends of the ducts, respectively.
For Reynolds numbers less than 10,000 the exit static temperature t2
is based on the value of the exit total temperature T2 determined
from the electric heat-input and heat-loss measurements, as was men-
tioned in the preceding section.







NACA RM E53J07


The average friction coefficients as calculated above are shown in
figure 5 correlated by the method summarized in reference 1 for high
heat-flux conditions. Included for comparison is the line representing
the KArmAn-Nikuradse relation for turbulent flow in pipes modified for
the effect of heat flux on the friction coefficient, which is

1 = 2 logloP 0.8 (4)




In figure 5(a), the average friction coefficients for the square
duct for Reynolds numbers above 10,000 to 20,000 agree reasonably well
with the reference line for circular tubes, although the friction coef-
ficient increases with an increase in heat flux or surface to fluid
bulk temperature ratio. In the transition region for low Reynolds num-
bers, the friction coefficient varies considerably with heat flux.
Similar variations were obtained in reference 1 for circular tubes.

The data for the rectangular and triangular ducts (figs. 5(b) and
(c), respectively) indicate that the friction coefficient increases
more with increasing heat flux than in the square duct (fig. 5(a)),
which may possibly be caused by the development of secondary flows in
the rectangular and triangular ducts.


Correlation of Peripheral Temperature Variations

Corner surface temperatures were measured at three stations along
the length of each test section (3, 12, and 21 in. from entrance). The
average differences between the corner and midwall temperatures are
plotted against the mass velocity in figure 6. The average difference
plotted herein was taken as the arithmetic average of the differences
between the local average corner and midwall temperatures at each sta-
tion. In general, the local average peripheral temperature difference
increased along the length of the test section; however, in several
instances the local values near the exit were less than those measured
at the center station, which indicates the possibility of locally de-
veloped secondary flow near the corners. The average peripheral tem-
perature differences increased at a decreasing rate with increases in
mass velocity and heat flux. Maximum values of 350, 1270, and 1100 R
were obtained for the square, the rectangular, and the triangular
ducts, respectively.

In reference 3, a method was developed for predicting peripheral
wall-temperature variations for flow in noncircular-tube heat exchangers
with internal heat generation in the tube walls based on shear-stress








NACA RM E53J07


distributions measured by Nikuradse for flow in noncircular ducts. The
average peripheral temperature differences are correlated accordingly
in figure 7, where the ratio of the difference between the corner and
the midwall temperatures to the difference between the surface (which
was taken as the average midwall temperature) and fluid bulk tempera-
tures is plotted against the dimensionless parameter, Nusselt number
divided by the ratio of wall thickness to hydraulic diameter of the
flow passage and the ratio of thermal conductivity of the wall mate-
rial to the fluid (wherein the average heat-transfer coefficient is
based on the difference between the average midwall and fluid bulk
temperature). Included for comparison are the reference lines cal-
culated for turbulent flow in the ducts from reference 3 for the case
of no flow over the outer surface of the duct; hence for the heat-
transfer coefficient for the outer surface equal to zero. The data
for the square duct agree fairly well with the predicted values in
the turbulent-flow range. The measured values for the rectangular and
triangular ducts were greater on the average by about 20 and 35 per-
cent, respectively, than the predicted values. This difference in
measured and predicted values results in part from the assumption of
similarity between variation of rate of heat-transfer and shear-stress
distribution around the periphery of a noncircular duct and in part
from the uncertainty in estimating the shear-stress distribution for
a duct with aspect ratio of 5 from measurements obtained for a duct
with aspect ratio of 3.5. Similar results are indicated for the tri-
angular duct. Maximum values of the ratio of the average difference
between the corner and midwall temperatures to the difference between
the average surface and bulk temperatures obtained were 0.05, 0.25, and
0.20 for the square, the rectangular, and the triangular ducts,
respectively.


SUMMARY OF RESULTS

The results of this investigation of heat transfer and pressure
drop for air flowing through noncircular ducts having square, rectan-
gular (aspect ratio, 5), and equilateral triangular cross sections for
a range of surface temperature from 5400 to 17800 R, corresponding
surface to bulk temperature ratio from 1.2 to 2.3, and Reynolds num-
bers from 1000 to 330,000 may be summarized as follows:

1. The effect of the ratio of surface to bulk temperature on cor-
relations of the average heat-transfer and friction coefficients was
the same as that obtained in a previous investigation for similar
ranges of conditions for flow in circular tubes and was nearly elim-
inated by evaluating the physical properties and density of the air
at a film temperature halfway between the bulk and surface temperatures.







NACA RM E53J07


2. Correlations of the average heat-transfer and friction coeffi-
cients were in reasonable agreement with the results obtained for the
circular tubes with the Nusselt number and Reynolds number based on
the hydraulic diameter of the duct.

3. Correlation of the average peripheral temperature difference
for the square duct was in agreement with values predicted by a pre-
vious analysis. However, the measured values for the rectangular and
triangular ducts were greater by approximately 20 and 35 percent,
respectively, than the predicted values.

4. Maximum values of the ratio of the average difference between
the corner and midwall temperatures to the difference between the
average surface and bulk temperatures obtained were 0.05, 0.25, and
0.20 for the square, the rectangular, and the triangular ducts,
respectively.


Lewis Flight Propulsion Laboratory
National Advisory Committee for Aeronautics
Cleveland, Ohio, October 9, 1953


REFERENCES

1. Humble, Leroy V., Lowdermilk, Warren H., and Desmon, Leland G.:
Measurements of Average Heat-Transfer and Friction Coefficients
for Subsonic Flow of Air in Smooth Tubes at High Surface and
Fluid Temperatures. NACA Rep. 1020, 1951. (Supersedes NACA
RM's E7L31, E8L03, E50E23, and E50H23.)

2. Keenan, Joseph H., and Kaye, Joseph: Thermodynamic Properties of
Air. John Wiley & Sons, Inc., 1945.

3. Eckert, E. R. G., and Low, George M.: Temperature Distribution in
Internally Heated Walls of Heat Exchangers Composed of Noncircular
Flow Passages. NACA RM E50J25, 1950.













12 NACA RM E53J07















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Average surface
temperature,
Ts,,
OR
A 680
V 890
0 1325
D 1780


Ratio of surface
to bulk fluid
temperature,
Ts! b
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1.9
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Rate of heat transferred to air, Wcp(T2 T1), Btu/hr

(a) Square duct.


Figure 3. Heat balance.


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NACA RM E53J07


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Average surface
temperature,
Ts'
OR
A 670
V 890
0 1320
0 1710


Ratio of surface
to bulk fluid
temperature,
Ts/Tb
1.2
1.4
1.8
2.1


-
Rate of heat transferred to air, Wep(T2 T1), Btu/hr
(b) Rectangular duct.
Figure 3. Continued. Heat balance.


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680
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1175
1740


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Ratio of surface
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Ts/,


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Rate of heat transferred to air, -'T T1), Btu/hr
(c) Triangular duct.


Figure 3. Concluded. Heat balance.


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\ Average surface Ratio of surface
t 40 temperature, to bulk fluid
Ts, temperature,
oR Ts/Tb

S, A ,680 1.2
20 V 890 1.5
0 1325 1.9
S1780 2.3
Circular duct (ref. 1)
10
8


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4 6 8 10


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20 40X10'


PfVbD
Reynolds number, Re,
f

(a) Square duct.

Figure 4. Correlation of heat-transfer coefficients for air flow in ducts with
variable heat flux. Bellmouth entrance; inlet temperature, 5350 R; properties
of air evaluated at film temperature.


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NACA RM E53J07


Nusselt number, Nu = 0.023Re0.8Pr04


Average surface Ratio of surface
temperature, to bulk fluid
__ Ts, temperature,
OR Ts/Tb

A 670 1.2
\ V 890 1.4
/ 0 1320 1.8
D 1710 2.1
S-- Circular duct (ref. 1)


1 .2 .4 .6 .8 1 2 4 6 8 10 20 40>)


pfVbD
Reynolds number, Re,
Pf

(b) Rectangular duct.

Figure 4. Continued. Correlation of heat-transfer coefficients for air flow in
ducts with variable heat flux. Bellmouth entrance; inlet temperature, 5350 R;
properties of air evaluated at film temperature.


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NACA RM E53J07


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100
80




Nusselt number, Nu= 0.023Re FrO' .P 1


40 Average surface Ratio of surface
O temperature, to bulk fluid
A T Ts, temperature,
I OR Ts/Tb
20 680 1.2
V 870 1.4
S0 1175 1.9
l 1740 2.3


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PfVbD
Reynolds number, Re,
f


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6 8 10


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20 40X10'


(c) Triangular duct.

Figure 4. Concluded. Correlation of heat-transfer coefficients for air flow in
ducts with variable heat flux. Bellmouth entrance; inlet temperature, 5350 R;
properties of air evaluated at film temperature.


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UNIVERSITY OF Fl'lllRlilllDA









UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
P.O. BOX 117011
GAINESVILLE, FL 32611-7011 USA


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