Experimental heat-transfer and friction coefficients for air flowing through stacks of parallel flat plates

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Material Information

Title:
Experimental heat-transfer and friction coefficients for air flowing through stacks of parallel flat plates
Series Title:
NACA RM
Physical Description:
33 p. : ill. ; 28 cm.
Language:
English
Creator:
Sams, Eldon W
Weiland, Walter F
Lewis Research Center
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

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

Notes

Abstract:
Abstract: Forced-convection heat-transfer and pressure-drop data were obtained for stacks of parallel flat plates of short length-to-effective-diameter ratio. Two such stacks were alined sic and misalined sic in the direction of air flow with gap spacings between stacks of 1/32, 1/8, and 1/4 inch. Data were obtained with heat addition to the downstream stack only over a range of Reynolds number from 15,000 to 80,000 and average surface temperatures of about 680° R. The average and local heat-transfer coefficients were only slightly higher than predicted values from established round tube data. The friction data for both stacks are compared.
Bibliography:
Includes bibliographic references (p. 19).
Statement of Responsibility:
by Eldon W. Sams and Walter F. Weiland, Jr.
General Note:
"Report date June 14, 1954."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003808501
oclc - 129617091
sobekcm - AA00006160_00001
System ID:
AA00006160:00001

Full Text

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RM E54F11


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RESEARCH MEMORANDUM




* : :: :
i" EXPERIMENTAL HEAT-TRANSFER AND FRICTION COEFFICIENTS

FOR AIR FLOWING THROUGH STACKS

OF PARALLEL FLAT PLATES

By Eldon W. Sams and Walter F. Weiland, Jr.

Lewis Flight Propulsion Laboratory
Cleveland, Ohio


UNIVERSrTY OF FLORIDA
:24 DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE UBRARY
PRO. BOX 117011
: GAINESVILLE, FL 32611-7011 USA


NATIONAL ADVISORY COMMITTEE


FOR AERONAUTICS


WASHINGTON
August 19, 1954


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NACA RM E54F11


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


EXPERIMENTAL HEAT-TRANSFER AND FRICTION COEFFICIENTS

FOR AIR FLOWING THROUGH STACKS

OF PARALLEL FLAT PLATES

By Eldon W. Sams and Walter F. Weiland, Jr.


SUMMARY

An investigation is being conducted at the NACA Lewis laboratory to
obtain forced-convection heat-transfer and pressure-drop data for flow
of air between electrically heated parallel flat plates stacked to form
passages of short length-to-effective-diameter ratio. Two such stacks
of plates were alined in series in the direction of air flow, with a
1/4-inch spacing between plates in each stack. Data were obtained for
three gap spacings between stacks of 1/32, 1/8, and 1/4 inch, as well
as for various degrees of plate misalinement between stacks, and also
with the upstream stack removed from the tunnel. The primary purpose
of the investigation is to determine the interference effects of the
upstream stack of plates on the heat-transfer and friction character-
istics of the downstream stack.

Data were obtained over a range of Reynolds number from 15,000 to
80,000, average surface temperatures from 6610 to 6830 R, heat fluxes
up to 8080 Btu per hour per square foot, inlet air temperature of about
5180 R, and inlet pressures up to 45 inches of mercury absolute, and
with heat addition to the downstream stack only.

The average and local heat-transfer coefficients obtained for the
downstream stack with the two stacks alined were slightly higher than
the values predicted from established data on round tubes for the
length-diameter ratio used herein. Also, the effects of changes in
plate misalinement and gap spacing between stacks were found to be neg-
ligible. The friction factors for the upstream stack, when fully cor-
rected for nonfiction losses and length-diameter ratio effects, were
in good agreement with data for smooth round tubes; for the downstream
stack, the friction factors were lower than those for smooth tubes when
a uniform velocity profile at the entrance was assumed, and higher than
those for smooth tubes when a fully developed velocity profile at the
entrance was assumed in the correction of the data for entrance effects.


p'







NACA RM E54F11


INTRODUCTION

The present investigation was undertaken to obtain forced-convection
heat-transfer and pressure-drop data for flow of air between electrically
heated parallel flat plates forming passages of short length-to-effective-
diameter ratio. More specifically, the program was designed to provide
these data for successive stacks of plates wherein the effects of such
configuration variables as distance between parallel plates, gap spacing
between successive stacks, and degree of misalinement between plates in
successive stacks are to be evaluated for a range of Reynolds numbers,
plate surface temperatures, and heat fluxes.

These data are of interest in the design of heat exchangers of sim-
ilar geometries (and plain or interrupted surfaces), where advantage can
be taken of the increased heat transfer associated with flow in passages
of short length-diameter ratio. In addition to average heat-transfer
values, the design of heat exchangers may require a more detailed knowl-
edge of local surface-temperature gradients or limiting local tempera-
tures, in which case recourse must be made to local heat-transfer values.
Both local and average heat-transfer coefficients are reported herein.

In the present investigation, heat-transfer and pressure-drop data
were obtained for two stacks of parallel flat plates with a 1/4-inch
spacing between plates in each stack, various degrees of plate misaline-
ment, and various gap spacings (1/32, 1/8, and 1/4 in.) between stacks,
as well as with the upstream stack removed. The data reported herein
were obtained with heat addition to the downstream stack only. Data
were obtained over a range of Reynolds numbers from 15,000 to 80,000,
average surface temperatures from 6610 to 6830 R, heat fluxes up to
8080 Btu per hour per square foot, an inlet air temperature of about
5180 R, and inlet pressures up to 45 inches of mercury absolute.

The results are presented herein in the form of curves of average
values of a Nusselt-Prandtl number relation (Nu/Pr0"4) against Reynolds
number, of local heat-transfer coefficients and plate temperatures
against distance from leading edge of plate, and of average friction
factors against Reynolds number.


APPARATUS

Air and Electrical Systems

A schematic diagram of the test section and related components of
the air and electrical systems used in this investigation is shown in
figure 1.







NACA RM E54F11


Air system. As indicated in figure 1, air at 110 pounds per
square inch gage passed through a pressure-regulating valve, a filter,
and an orifice run consisting of an air straightener and an A.S.M.E.
type flat-plate orifice where the air flow was measured before entering
the test section. The air then passed through the test section which
consisted of an approach section, the two stacks of electrically heated
flat plates, and a three-pass mixing tank having a thermally insulated
approach, after which the air discharged to the atmosphere. A window
and light source were provided upstream of the test section for visual
observation of the flat plates during testing.

The temperature of the air entering the test section was measured
by an iron-constantan thermocouple upstream of the approach section; the
temperature of the air leaving the test section was similarly measured
by two thermocouples just downstream of the mixing screens in the mixing
tank.

Electrical system. Provisions were made for electrically heating
both stacks of flat plates, although only the downstream stack was heated
for the data reported herein, the electrical system for each stack being
separate and independently controlled (see fig. 1). Electric power was
supplied to each stack through a variable transformer and a power trans-
former, the latter being connected by flexible cables to bus bars which
were fastened to the two outer plates (top and bottom) of each stack;
the plates in each stack were connected in series. The capacity of the
electrical equipment for each stack was 12 kilovolt-amperes at a maximum
of 12 volts across the stack.


Test Section

Installation. A schematic diagram of the test section is shown in
figure 2(a). The two stacks of flat plates were independently mounted
in a steel tunnel provided with micrometer screws to permit vertical
movement of the stacks for plate alinement and misalinement between
stacks. The two stacks could be separated by 1/32-, 1/8-, or 1/4-inch-
thick transit spacers having a 2- by 3-inch opening in the center.
Similar spacers of 3/4-inch thickness were provided before the upstream
stack and after the downstream stack, with a high-temperature rubber
gasket recessed into the spacer to provide an air seal between stack and
spacer. Transite plates (not shown) running the length of both stacks
were provided between the stack and tunnel side walls with sufficient
clearance to allow vertical movement of the stacks; these transit plates
were grooved vertically to allow plate instrumentation leads to be
brought up the side of the stack. A wooden approach section (24 in. long
with 2- by 3-in. opening) having a rounded entrance was located before
the upstream stack. A three-pass mixing tank was located after the
downstream stack; the mixing-tank approach section was thermally







NACA RM E54Fll


insulated and contacted a rubber gasket in the rear spacer. Two screens
were provided in the mixing tank center passage for thorough mixing of
air ahead of the exit air thermocouples.

The two stacks of plates tested were identical, one such stack
being shown in more detail in figure 2(b). The stack consisted of nine
flat plates 'stacked vertically with a 0.25-inch spacing between plates.
This spacing was provided by either one 1/4-inch or two 1/8-inch spacer
strips running the length of the plates along either side. The strips
(conductors or insulators) were stacked in such a manner as to provide
an electrical series connection between plates. An insulator plate and
a steel support plate were provided over the two outside plates in the
stack (fig. 2(a)), the entire stack being clamped together by four in-
sulated bolts through the stack. The bus bars supplying electrical pow-
er to the plates were silver-soldered to the conductor strips of the two
outside plates in the stack. Bosses which were provided on the bottom
support plate of the stack rested on micrometer screws fastened to the
bottom tunnel wall, thereby providing means for vertical movement to ob-
tain alinement or misalinement of the two stacks.

Instrumentation. The instrumentation for each stack of plates is
indicated in figure 2(b). The flat plates were 0.018 inch thick, 3
inches wide (exposed to flow), and 3.5 inches long. Iron-constantan
thermocouples were imbedded in the plates by cutting grooves 0.014 inch
wide and 0.008 inch deep into the plate at right angles to the direction
of flow. The iron-constantan wires (0.005-in. diameter including insu-
lation) were then laid in the groove and covered with an insulating ce-
ment. The thermocouple leads from the edge of the plate were secured to
the outer edge of the spacer strips (between plates) and then brought up
the sides of the stack. Thermocouples were placed in the various plates
of the stack at the locations indicated in figure 2(c). The number of
thermocouples was progressively decreased toward the outer plates, giv-
ing a total of 68 thermocouples in each stack.

Voltage-drop leads were also silver-soldered to the conductor
strips at the edges of each plate (the conductor strips being spot
welded to the individual plates) in such a manner as to obtain the
voltage drop across each plate. Over-all voltage-drop readings were
also obtained for the stack.

Static-pressure taps were located in each of the three transit
spacers as indicated in figure 2(a), the static taps being placed at
the center of the bottom and side surfaces of the opening in each case.
The various pairs of static Iaps across each stack were read separately
on U-tube water manometers to obtain the pressure drop across each
stack.








NACA RM E54Fll


A photograph of the complete test section is shown in figure 3.
All thermocouple and voltage drop leads were brought out between two
rubber gaskets under an access plate on top of the tunnel. The access
plate covered both stacks of heated plates; it also provided seals for
the bus bars leaving the tunnel and fittings for insertion of probes
into the spacer openings.


METHOD OF CALCULATION

Evaluation of average plate and stack temperature. As previously
indicated in figure 2(c), thermocouples were located down the center
line of the plate at various distances from the plate leading edge with
a similar row of thermocouples along a line 3/4 inch from the center
line toward either side of the plate. The plate temperature gradients
normal to the direction of air flow were found to be small compared to
gradients in the flow direction; therefore, the temperature gradient for
the plate can be well represented by plotting the plate center line tem-
perature against distance from the leading edge. The gradients for each
plate in the stack were plotted in this manner; where center line ther-
mocouples were not available (see fig. 2(c)), the average of the two
side row thermocouples was used. The average temperature for each plate
was then obtained by dividing the measured area under the curve of the
plate center line temperature against distance from leading edge by the
plate length. The average surface temperature for the entire stack was
then taken as the arithmetic average of the various average plate tem-
peratures weighted according to the surface area exposed to flow. (The
two outside plates were exposed to flow on one side only.)

Average heat-transfer coefficients. The average heat-transfer co-
efficient for the stack was computed from the relation (symbols are de-
fined in appendix A):

avt (Q/S)avst WCp(T4 Tl)st/Sst
(Ts Tb)av,st (T Tb)av,st

The average bulk temperature Tb,av was taken as the arithmetic average
of the inlet-air and exit-air total temperatures, T1 and T4, respec-
tively. The heat-transfer surface area S was taken as the total sur-
face area of the plates exposed to flow. The exposed surface area con-
tributed by the spacer strips between plates is not included, but would
have a negligible effect on h. The values for physical properties of
air used herein are presented in figure 4.






NACA RM E54Fll


Local heat-transfer coefficients. Local heat-transfer .:effi-
cients for the center plate (plate 5) in the stack were evaluated in
the following manner. The local heat-transfer coefficient at any dis-
tance from the leading edge was taken as

(Q/S)z (Q/S)av P/Pav
S (Ts Tb) (Ts Tb) )

where all terms in the equation apply to the number 5 plate only.
For the number 5 plate, (Q/S)av was obtained by the relation

(Q/S)av,P = (Q/S)av,st Ep/Est (3)

Tb, was assumed to vary linearly with X (distance from plate leading
edge) between TI and T4; Ts was obtained from the plot of local
plate temperature against X as explained in the previous section.
This curve was also used to evaluate PZ/Pav in equation (2) in the
following manner: Inasmuch as the voltage drop across the plate is es-
sentially constant at any distance from leading edge X, and since heat
conduction along the plate can be neglected, the local power generation
at any point X is given by

P, = Ep2/RM = C/r1 (4)

That is, the local power generation at any point (distance X from
leading edge) in the plate is inversely proportional to the electrical
resistivity r at the point; hence from a curve of electrical resis-
tivity r against temperature for the plate material and the curve of
Ts,Z against X, a curve of l/rz against X was plotted for the
plate. By graphically integrating this curve to obtain 1/ray, the
ratio of l/r, to 1/ra (which also represents PZ/Pav) was then
plotted against distance from leading edge X. The local heat-transfer
coefficients hz for the number 5 plate could then be computed from
equation (2) and plotted against X.

Average friction coefficients. The method used for computing the
average friction coefficients for the individual stacks is as follows:
The friction .-:.,ff1icients are based on measured static-pressure drops
which were corrected for entrance, exit, vena contract, and momentum
losses. The equations used in calculating these losses are derived in
asindix B. The values of measured pressure drop, as previously ex-
plained, were obtained from several pairs of static taps located in the
transit spacers on either side of and between the stacks (fig. 2(a)).
The pressure drop across the upstream stack was taken as the average of
the pressure drops measured by the taps in the two sides of the spacer







NACA RM E54F11


opening. Because of slight surface interruptions between stacks, static
probes were used in measuring the drop across the downstream stack. The
equations derived in appendix B result in the following evaluation of
friction factor: The fully corrected average half-friction factor is
defined as


(f/2) corr corr2 (5)
L G
De 2gpav

where p was evaluated with the total temperature and the static
pressure in the center window opening (negligible error for range of
conditions investigated); and

Pcorr = meanss (en + ex + Apmom + v (6)

where:

G22
en = (1 F) (7)

APex = G12 1 + 1)()


2
G22 P + a (9)
A"Pmom gPI (9)
pmam= gPl 1 a

and:

G22
Pvc = K2 (10)
2go1

where Kc is a function of free-flow factor F for the stack, the
value of which was obtained from reference 1. The measured average
half friction factor, also used herein, is similarly defined as:

S means (11)
2/meas L G2
D8 2p
De 2gpav






NACA RM E54Fll


PROCEDURE

As previously noted, two stacks of flat plates were used in this
investigation. The plates were spaced 1/4 inch apart in each stack with
either a 1/32-, 1/8-, or 1/4-inch gap between stacks. The two stacks of
plates were first alined in the direction of air flow, and data were ob-
tained at various Reynolds numbers. The average stack temperature level
reported herein was that which resulted from limiting the highest local
temperatures (trailing edge of outside plates) to a safe operating value
for the iron-constantan thermocouples used. Data were similarly ob-
tained for two degrees of plate misalinement, which were obtained by mov-
ing the upstream stack vertically. For the case of slight misalinement,
the upstream stack was moved a distance equal to one plate thickness
(top of plate in upstream stack level with bottom of plate in downstream
stack); while for complete misalinement, the upstream stack was moved
by one-half the distance between plates.


RESULTS AND DISCUSSION

Plate Temperature Gradients

Typical plate temperature gradients obtained for the downstream
stack are presented in figure 5, where local surface temperature Tsz
is plotted against distance from plate leading edge X.

The plates are numbered from top to bottom of the stack, but are
listed in an order of symmetry starting from the center plate and moving
toward the outer top and bottom plates. The curves for the three cen-
termost plates are essentially coincident, while the next two plates on
either side of the three centermost plates show slightly higher tempera-
tures at the trailing edges. The curves for the two outside plates fall
considerably above the other curves, particularly at the trailing edge;
this would be expected inasmuch as the top and bottom plates are cooled
on one side only (see fig. 2(a)). The dashed lines, for the four outer-
most plates, represent an approximation of the temperature gradient,
since thermocouples were located only at the leading and trailing edges
of these plates. The average plate and stack temperatures were obtained
from these curves as indicated in the section METHOD OF CALCULATION.


Heat-Transfer Data

Average heat-transfer coefficients for stack.-- The average heat-
transfer coefficients obtained for the downstream stack are [resented
in figure 6 where the Nusselt-Prandtl number relation Nub/PrbO.4 is
plotted against Reynolds number DeG/4b. Included for comparison is






NACA RM E54F11


the McAdams line (solid) which was found to best represent the data of
various investigators (ref. 1) for fully developed turbulent flow in
smooth tubes; the equation is:

Nub DeG 0.8
N"b =0.023 (. (12)
Prb0.4 b (b

The predicted line (dashed) in figure 6 includes a correction to
the conventional McAdams equation (eq. (12)) to account for the increase
in average heat-transfer coefficient to be expected with passages of
short L/De (7.6). This correction was made in reference 2 by inclusion
of a power function of L/De in equation (12), the equation being:

Nub 0.034 ( 0.8 L-0.1 (13)
=0.034 (13)
Pr 0.4 \ ) De


which, for L/De = 60, becomes equation (12) and, for L/De = 7.6,
becomes

Nub /DeG\0"8
b 0.028 (e\) (14)
Prb0.4 4b I

represented by the dashed line in figure 6. The magnitude of this in-
crease in the heat-transfer coefficient is further verified in reference
3 (investigation of average and local heat-transfer coefficients as
functions of Re and X/De for smooth entrance flow through tubes
and between parallel flat plates) for the case of uniform heat flux
with uniform initial temperature and velocity distributions.

The average heat-transfer coefficients for the downstream stack
(fig. 6) are shown for the case of stacks (plates) alined in the di-
rection of air flow with various gap spacings (1/32, 1/8, and 1/4 in.)
between stacks, as well as with the upstream stack removed from the tun-
nel. The data for all configurations are in reasonably good agreement
(about 15 percent high) with the predicted line. The effect of gap
spacing and removal of the upstream stack is seen to be negligible.

The effect of plate misalinement is shown in figure 7. With the
two stacks completely misalined as described in PROCEDURE, the top in-
sulator plate of the upstream stack (see fig. 2(a)) partly blocks off
the top passage of the downstream stack. Hence, in order to compare
the data for various degrees of misalinement, the average heat-transfer
coefficients for the three center plates of the downstream stack were






NACA RM E54F11


used in each case. The results are shown in figure 7, with the same co-
ordinates as in figure 6. The data for the alined condition fall
slightly higher than in figure 6 inasmuch as the electrical heat input
was used in evaluating Q for the three center plates. These data are
compared (fig. 7) with those for the completely misalined and slightly
misalined conditions for the various gap spacings investigated. The
effect of plate misalinement also appears to be negligible.

Visual observation of the plates during testing indicated that the
center seven plates of the heated stack remained flat and perfectly
alined with the first stack; the two outside plates, which were cooled
on one side only, showed slight bending.

Local heat-transfer coefficients for center plate. The local
heat-transfer coefficients obtained for the center plate of the down-
stream stack are presented in figures 8 to 11.

In figure 8, hi is plotted against X for the case of plates
alined with various gap spacings between stacks, as well as with the
upstream stack removed; this comparison is shown for two values of
Reynolds number. As seen in figure 8, gap spacing and removal of the
upstream stack have no appreciable effect on the local coefficients,
as was also shown for the average coefficients in figure 6.

As explained in METHOD OF CALCULATION, the local heat-transfer co-
efficients were calculated with the assumption of no heat conduction
along the plate. The effect of heat conduction can be taken into ac-
count by use of the following equation (all terms are for the center
plate):


(+ bkm 2T (15)
S )Z = (g dX2

where the first term (right side, eq. (15)) is defined in equation (2),
and the second term accounts for heat conduction along the plate. The
corrected values of h, can then be computed; the uncorrected values
of hz obtained at the highest Reynolds number with the upstream stack
removed from the tunnel are compared with the corresponding corrected
values of h1 in figure 9. The effect of conduction along the plate
is seen to be small; hence, the effect of conduction is neglected in
subsequent discussion.

A comparison of the local heat-transfer coefficients ob-ailed with
the upstream stack removed, and the predicted values of reference 3 is
shown in figure 10 as a variation of local Nusselt number Nuj with







NACA RM E54F1l


X/De. The values obtained from reference 3 (dashed lines) are for the
case of a gas (Pr = 0.73) flowing between parallel flat plates with the
conditions of uniform heat flux, constant properties, and uniform tem-
perature and velocity distribution at the entrance. The predicted
curves are shown for essentially the same values of inlet Reynolds num-
ber Rei and heat-flux parameter as obtained in the present investiga-
tion. At the lowest Reynolds number, the experimental values (solid
lines) are in good agreement with those predicted, although slightly
higher near the leading edge. With increase in Reynolds number, however,
the experimental values are somewhat higher throughout as also indicated
by the average coefficients (downstream stack only) in figure 6.

The effect of plate misalinement on the variation of local heat-
transfer coefficient along the plate is shown in figure 11 where hZ/hav
is plotted against X. The data are shown for the cases of plates
alined, slightly misalined, and completely misalined at a Reynolds number
of about 40,000 and a 1/8-inch gap between stacks. Figure 11 indicates
that plate misalinement results in slightly higher values of hj/ha at
the leading edge and slightly lower values at the trailing edge. The
average heat-transfer coefficient for the center plate was essentially
the same for all degrees of misalinement, as was also shown in figure 7
for the three center plates.


Friction Data

Average friction factors for upstream stack. The average half-
friction factors obtained for the upstream stack are presented in fig-
ure 12 wherein half-friction factor f/2 is plotted against Reynolds
number DeG/ib. Included, for comparison, is the Karman-Nikuradse
line (solid), representing the relation between friction factor and
Reynolds number for turbulent flow in smooth pipes, which is

S1 = 2 log (Re V f/) 0.8 (16)


In comparing the data presented herein with those for fully devel-
oped turbulent flow (reference line), the effect of L/De must be taken
into account; this effect consists of (1) a momentum loss in the stack
associated with transition from a flat velocity profile at the entrance
to some degree of fully developed turbulent velocity profile at the
exit, and (2) an increase in average friction factor associated with
short L/De wherein the entrance effect (region of high local shear
stress and friction) may exist over a considerable portion of the pas-
sage length. These effects are accounted for in the analysis of ref-
erence 3 which predicts friction factors for flow between parallel flat







NACA RM E54F11


plates with uniform velocity distribution at the entrance as a function
of X/De and Reynolds number. Reference 3 indicates that the ratio of
average friction factor for X/De = 7.6 to the friction factor for fully
developed turbulent flow is about 1.54 for the range of Reynolds numbers
investigated herein. The predicted line (dashed, fig. 12) represents
the reference line (solid) corrected by this ratio.

Two different values of friction factor are presented in figure 12
for the present data: (1) friction factor based on measured static-
pressure drop, and (2) friction factor based on measured pressure drop
fully corrected for entrance, exit, momentum, and vena contract losses
as explained in the section METHOD OF CALCULATION. The measured values
of f/2 fall slightly above the predicted line. When the various
losses are taken into account, the data fall within 8 percent of the
predicted line.

Average friction factors for downstream stack. The average half-
friction factors for the downstream stack are presented in figure 13 in
the same manner as that used for the upstream stack in figure 12. The
friction data presented are for a 1/8-inch gap between stacks. The
friction factors based on measured pressure drops fall near the pre-
dicted line; but when the various losses and the same L/De correc-
tion used for the upstream stack is taken into account, the data for the
downstream stack fall considerably below the predicted line. The L/De
correction, as previously indicated, is for the case of uniform velocity
distribution at the stack entrance. This condition is satisfied for the
upstream stack (fig. 12), but it is questionable as to whether the
1/8-inch gap between stacks is adequate to completely flatten out (be-
fore it enters the downstream stack) what is essentially a fully devel-
oped velocity distribution at the exit of the upstream stack (indicated
by ref. 3). The results of figure 13 overcorrectionn of the data) indi-
cate that complete flattening out does not occur.

Average friction factors for both stacks combined. It might be of
interest to compare the results in figure 13 with those for the case
where no flattening out of the flow at the entrance to the downstream
stack is assumed to occur, that is, where the two stacks are considered
as a continuous passage. The data for the latter case are presented in
figure 14, where the fully corrected friction factors are based on meas-
ured pressure drops across both stacks corrected for entrance, exit, mo-
mentum, and vena contract losses, and with the reference line corrected
for an L/De = 15.2 (dashed line). Friction factors based on measured
pressure drops are again included. The fully corrected half-friction
factors now fall slightly above the predicted line (data undercorrected).
Hence, a comparison of figures 13 and 14 indicates that the velocity
distribution at the entrance of the downstream stack is neither uniform
nor substantially fully developed as at the exit of the upstream stack.







NACA RM E54Fll


SUMMARY OF RESULTS

The results of tests to obtain heat-transfer and pressure-drop data
for flow of air between electrically heated parallel flat plates, with
various gap spacings and degrees of misalinement between stacks, can be
summarized as follows:

1. Average heat-transfer coefficients obtained for the downstream
stack fell considerably above the conventional McAdams line representing
average coefficients for fully developed turbulent flow in smooth tubes
of length-diameter ratio L/De approximately equal to 60. When the
McAdams line was corrected to account for L/De effects (L/De = 7.6),
the data gave reasonably good agreement with the predicted line (about
15 percent high). The effects of gap spacing and plate misalinement
between stacks were found to be negligible.

2. Local heat-transfer coefficients obtained for the center plate
of the downstream stack (with upstream stack removed) were compared with
those predicted by analytical methods for flow of a gas (Pr = 0.73) be-
tween parallel flat plates for the case of uniform heat flux and initial
temperature distribution, and uniform velocity distribution at the en-
trance. The experimental and predicted values were in fairly good
agreement at the lowest Reynolds number, the experimental values becom-
ing somewhat higher with increase in Reynolds number.

3. Average half-friction factors for the upstream stack, when fully
corrected for entrance, exit, momentum, and vena contract losses were
in good agreement with a predicted line; which was based on the Karmin-
Nikuradse line representing average friction factors for fully devel-
oped turbulent flow in smooth pipes with suitable correction for L/De
effects. The data fell within +8 percent of the reference line.

4. Average half-friction factors for the downstream stack fell be-
low the predicted line when a uniform velocity profile at entrance was
assumed and above the predicted line when a fully developed velocity
profile at the entrance was assumed.


Lewis Flight Propulsion Laboratory
National Advisory Committee for Aeronautics
Cleveland, Ohio, June 14, 1954







1ACA RM E54F11


APPENDIX A


SYMBOLS

The following symbols are used in this report:

A free-flow area, sq ft

b plate thickness, ft

C constant

c specific heat of air at constant pressure, Btu/(lb)(OF)

D effective diameter of stack passage, 4A/Z, ft

E potential difference, volts

F free-flow factor (free flow area/total frontal area)

(f/2) average half-friction factor based on measured static-
meas
pressure drop

(f/2) average half-friction factor based on measured static-
corr
pressure drop corrected for entrance, exit, momentum, and
vena-contracta losses

G mass velocity (mass flow per unit cross-sectional free-flow
area) W/A, lb/(hr)(sq ft)

g acceleration due to gravity, 4.17x108 ft/hr2

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

Kc vena contract pressure-loss coefficient

k thermal conductivity of air, Btu/(hr)(sq ft)(F/ft)

-~ thermal conductivity of plate material,
Btu/(hr)(sq ft)(0F/ft)

L length of stack passage, ft

P power .-erLe ration, watts

p static pressure, lb/sq ft abs






NACA RM E54Fll 15


Pmeas measured static-pressure drop across stack, lb/sq ft

4corr measured static-pressure drop across stack corrected for
entrance, exit, momentum, and vena-contracta losses

APen entrance pressure drop, Ib/sq ft

Apex exit pressure drop, lb/sq ft

Pmomn momentum pressure drop, Ib/sq ft

APVc vena contract pressure drop, lb/sq ft

Q rate of heat transfer to air, Btu/hr

Q' rate of heat transfer to air (corrected for heat conduction
along plate), Btu/hr

R gas constant for air, ft-lb/(lb)(oF)

R' electrical resistance, ohms

r electrical resistivity, ohm-cm

S heat-transfer surface area, sq ft

T air total temperature, OR

Tb air bulk temperature, OR

Ts surface temperature, OR

t static temperature, OR

At static-temperature difference between entrance and exit of
stack, OR

W air flow, lb/hr

X distance from plate leading edge, in.

Z wetted perimeter of stack passage, ft

Pmeas/P1
3 At/t1






16 NACA RM E54F11


. absolute viscosity of air, lb/(hr)(ft)

p density of air, lb/cu ft


Dimensionless parameters:

Nu Nusselt number, hDe/k

Pr Prandtl number, cp_/k

Re Reynolds number, DeG/p


Subscripts:

av average value

b physical properties evaluated at bulk temperature

i physical properties evaluated at inlet temperature

1 local value

P plate

st stack

The following diagrammatic sketch of the stack defines the subscripts
pertaining to various stations:

12 34

I
Air
flow _' I -Plates






NACA RM E54F1l


APPENDIX B


DERIVATION OF PRESSURE LOSS EQUATIONS USED HEREIN

A schematic diagram of the test section (stack) showing the various
stations is given below:


,rF -Pmeas

Air
flow
!--Ii
~i'l !


Entrance pressure loss APen" A pressure loss occurs in the flow
stream in the entrance region between stations 1 and 2. Assuming no
friction and fluid incompressibility (pl = p2), the general energy equa-
tion between these stations can be written:


PlV12
Pl + 2g


PlV22
= 2 2g


Also,


Al
V2 A2 V


hence,


2g A2 1


Apen = PI P2 =


Rewriting equation (B2) in terms of the mass velocity G
flow factor F (defined as A2/Al) results in:


lpen 2 (1 F2)
,1


and the free-


(B3)


(Bl)


(B2)






NACA RM E54Fll


Exit pressure drop Apex. The momentum change between stations 3
and 4, where an expansion occurs, can be stated as follows:


W V3 + 3A4 W V4 + 4A4 (B4)
g g

Assuming incompressibility, equation (B4) becomes:

3 3 A A_
pex = P3 P = g(5)


As before, A2/A1 = F and A3/A4 = F; therefore,

2
Apex= (F2 F) (B6)
APex gp3

In order to write equation (B6) in terms of entrance conditions at sta-
tion 1, the following was assumed:

P3 P A7)
3= Rt3 R(t + At) (B7)

The Ap used in equation (B7) was approximated by using APmeas to
avoid a trial-and-error solution. The error incurred by this assumption
is negligible.

From equation (B7), equation (B6) can now be written:

G2
Ap = G 1 + (1 1/F) (B8)
ex gp 1 a

where

a = Ap/pl (B9)

and


P = At/t


(B10)







NACA RM E54F11


Vena-contracta pressure loss Apvc. The vena-contracta pressure
loss was taken as a factor Kc times the dynamic pressure in the test
section as given by
2
G2
pv = Kc (Bll)
e 2gp1

where Ke is a function of the free-flow factor of the test section;
values of K were obtained from reference 1.

Momentum pressure loss APmom. Using the general momentum equa-
tion and assuming pl = p2 result in the following equation

G2 /1 1\
pmo = ( (B12)
;nom g \Pg P3

Rewriting equation (B12) with the aid of equations (B7), (B9), and (B1O)
gives

G22 3 +
Pmom g gpl 1-



REFERENCES

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

2. 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 E50H25.)

3. Deissler, Robert G.: Analysis of Turbulent Heat Transfer and Flow
in the Entrance Pegicns of Smooth Passages. NACA TN 3016, 1953.






























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NACA RM E54F11


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p












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22 NACA EM E54F11






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NACA RM E54F11


o
0) 0)


p p4

0








C,
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o
E-

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CoT------ ------ --t-


& k-Mf







24 NACA lM E54FII
















g0
r-

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NACA RM E54F11


.026




.022









.014




.010 -

.070




.0601




.050 -




.040 _______
400 500 600 700 800 900 1000
Temperature, 'R
.060 --- -- -- -- -- ^ -- -




. 0 5 0 --- -- -- -- -- -- -- -- -- --




.040 *-----------------------



Figure 4. Physical properties of air over temperature range
4000 to 1000' R.







NACA RM E54F11


- --_ __ /- -


Plate Tsav'
---- --- --- --- --- -- ^ / --OR
0 5 581
__ 0 4 649
0 6 650
V 3 660
___ 7 654
/ 2 662
S 8 656
___1 748
17 9 728
S---Thermocouples located
_only at leading and
trailing edges of
plate





1// --___

2 o --








/K -


1 2 3
Distance from plate leading edge, X, in.


Figure 5. Typical temperature gradient curves for plates in downstream
stack. Average surface temperature for stack, Ts,av, 6650 R; air bulk
temperature, Tb, 5280 R; heat-transfer rate to air, Q, 5890 Btu per
hour; Reynolds number, Re, 25,800; stacks alined.

















Tb,av
(stack),
OR

524-532
526-532
526-534
524-532


Condition



Stacks alined; 1/8-in. gap
Stacks alined; 1/32-in. gap
Stacks alined; 1/4-in. gap
Second stack only


McAdams reference line
- Predicted line (for L/D = 7.6)


40
DeG/o


400x103


Figure 6. Average heat-transfer coefficients for downstream
stack as affected by gap between stacks.


NACA RM E54F11


s,av
(stack),
OR

665-672
664-669
668-675
670-681


,pI







NACA RM E54F11


s,av
(stack),
OR


0 665-672
0, 667-673
,0 666-683
o 664-669
0% 661-669
)3 662-674
0 668-675
0(. 668-674
,, 665-668


Tb, av
(stack),
OR
524-532
526-532
525-531
526-532
524-531
524-530
526-534
526-532
526-532


----Predicted line (for


8 10


40
DeG/io


Condition



Alined
Slightly misalined
Completely misalined
Alined
Slightly misalined
Completely misalined
Alined
Slightly misalined
Completely misalined
L/De = 7.6)


100


Figure 7. Average heat-transfer coefficients for three
center plates of downstream stack as affected by plate
misalinement and gap between stacks.


Gap
spacing,
in.
1/8
1/8
1/8
1/32
1/32
1/32
1/4
1/4
1/4


600

400


200


z -


100
80

60


406
6


/001
- -7





EE-- __ _- -


200x106











s,av
(plate),
oR


Re
(stack)


41,400
41,400
41,500
41,600
25,800
26,000
26,100
26,100


Condition



Alined; 1/8-in. gap
Alined; 1/32-in. gap
Alined; 1/4-in. gap
Downstream stack only
Alined; 1/8-in. gap
Alined; 1/32-in. gap
Alined; 1/4-in. gap
Downstream stack only


1 2 3 4
Distance from plate leading edge, X, in.


Figure 8. Local heat-transfer coefficients for center plate
of downstream stack as affected by gap between stacks.


NACA BM E54F11


Ts,av
(stack),
OR

665
667
669
670
665
664
668
673


648
651
649
649
651
649
654
656


O
0

A
0-
D-
0-

A-


200


160


120







NACA RM E54F11


O Uncorrected
O Corrected for
180 conduction






- 140


O-4O
0 3
















Distance from leading edge, X, in.
U 4-

















Figure 9. Local heat-transfer coefficients for

center plate of downstream stack, with upstream
stack removed from tunnel, as affected by heat
conduction along plate. Average surface temper-
ature, Tsa for stack, 6700 R; for plate 5,
uav>
6490 R; average bulk temperature for stack,
5240 R; Reynolds number, Re, 41,600.
0C





Distance from leading edge, X, in.

Figure 9. Local heat-transfer coefficients for
center plate of downstream stack, with upstream
stack removed from tunnel, as affected by heat
conduction along plate. Average surface temper-
ature, T Sav ; for stack, 6700 R; for plate 5,
6490 R; average bulk temperature for stack,
5240 R; Reynolds number, Re, 41,600.







NACA RM E54F11


500





400


X/De


Figure 10. Local values of
plate of downstream stack,
as compared with predicted


Nusselt number for center
with upstream stack removed,
curves of reference 3.


Re s,av s,av b,av
(stack) (stack), (plate), (stack),
R OR OR
0 41,600 670 649 524
0 26,100 673 656 527
A 16,400 681 673 532
















__Rei

--040,000

25,000


3001





200







NACA RM E54F11


Re Tsav
(stack) (stack)
OR


0 41,400
O 41,500
A 41,700


665
669
666


Ts,av
(plate)
OR

648
653
640


b, av
(stack)
oR


Condition


524 Alined
526 Slightly misalined
524 Completely misalined


Distance from leading edge, X, in.

Figure 11. Ratio of local to average heat-transfer coefficients for
center plate of downstream stack for various degrees of plate mis-
alinement and 1/8-inch gap between stacks.


35.0




-p
0
S2.6
1-8
0
0
0
o

2.2

s-p





o



-4
a!



0
,-
aS









NACA RM E54F11


Karmin-Nikuradse line
----Predicted line (for L/De = 7.6)
[] Based on measured pressure drop
.01 O Based on measured pressure drop
corrected for entrance, exit,
Z- momentum and vena-contracta
0 losses
006






002 -



rn ^ __ ____ -- ---- -- -- ____ __ _


2 4 6 10 20 40 60 100 200
DeG/ub


Figure 12. Average half-friction factors for upstream stack, based on measured
values of static-pressure drop, and measured pressure drops corrected for entrance,
exit, momentum, and vena-contracta losses.
--- Karmgn-Nikuradse line
---- Predicted line (for L/De = 7.6)
0 Based on measured pressure drop
.01 0 Based on measured pressure drop
corrected for entrance, exit,
momentum and vena-contracta
S- losses
.006 -

.004- -



.002



nmni- -- --


2


3 4(
DeG/Hb


200 400X103


Figure 13. Average half-friction factors for downstream stack, based on measured
values of static pressure drop, and measured pressure drops corrected for entrance,
exit, momentum, and vena-contracta losses.


4 6 10 20
DeGG/1


40 60 100 200


Figure 14. Average half-friction factor for upstream and downstream stacks com-
bined, based on measured values of static-pressure drop, and measured pressure
drops corrected for entrance, exit, momentum, and vena-contracta losses.


NACA-Lagley 6-9-55 75


.UuF
rfAI


Ka'rman-Nikuradse line
---- Predicted line (for L/De = 15.2)
Q Based on measured pressure drop
O Based on measured pressure drop
corrected for entrance, exit,
momentum and vena-contracta
-losses

-- I I;- IB I I I I -I I 1 I I I


2


UL1


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UNIVERSITY OF FLORIDA

3 1262 08106 565 7



UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE UBRARY
P.O. BOX 117011
fAINESVII.LE, FL 32611-7011 USA


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