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Study of Mixing and

Shine-Zen (Joseph) Kuhn Augmentation of Natural

University of California, Berkeley,
Department of Nuclear Eng.,
Berkeley, CA 94720-1730
Convection Heat Transfer by a
e-mail: jkuhn1@txu.com
Forced Jet in a Large Enclosure
Hwan Kook Kang
Dae Hong Enterprise Co., This paper presents an experimental study of gas mixing processes and heat transfer
30 Yangpyung-Dong-4ga, augmentation by a forced jet in a large cylindrical enclosure with an isothermal bottom
Young Deung Po-Gu, heating/cooling surface. Cold/hot air was injected at several positions with varying pipe
Seoul, Korea, 150-104 diameters and injection orientations, and was removed from the top of the enclosure.
Under natural convection, the mean Nusselt number was correlated by the enclosure
Per F. Peterson Rayleigh number RaD 1/3
, and under strong forced convection, by the jet Reynolds number
University of California, Berkeley, 2/3
Rej . A combining rule for mixed convection was found for both the stabilizing (cooled
Department of Nuclear Eng., surface) and destabilizing (heated surface) density gradients. Using the ratio of forced
Berkeley, CA 94720-1730 and free convection Nusselt numbers, this correlation could be further reduced to predict
e-mail: peterson@nuc.berkeley.edu forced-jet augmentation as a function of the Archimedes number, with a correction factor
to account for jet orientation and enclosure aspect ratio. 关DOI: 10.1115/1.1482081兴

Keywords: Enclosure Flows, Fire, Heat Transfer, Impingement, Jets, Mass Transfer,
Mixed Convection, Natural Convection

Heat and mass transfer inside large enclosures is important in
Nu⫽k Rea Prb 冉 冊
␮b c
⫻ f 共 geometric factor兲 (1)
many applications, such as mixing process in the containment
where Re is evaluated as the jet Reynolds number ␳ ud j / ␮ or
structures of passive light water reactors and other large stratified
volumes, including high-level radioactive waste storage tanks, en- impeller rotational Reynolds number ␳ Nd i2 / ␮ . Despite differ-
closure fires, chemical processing, and pollutant dispersal. For the ences in mechanical impeller designs and tank geometry, the val-
AP-600 reactor containment design, the augmentation of conden- ues of K typically fall within a relative narrow range from 0.33 to
sation mass transfer caused by break-jet flows is of interest. The 0.74 for heat transfer to vessel walls, and from 0.02 to 0.08 for
transfer to helical coils.
present study can be used to quantify the augmentation of heat
The geometric correlations can depend on several factors in-
removal rate under natural convection, generated by injection of a
cluding the type of agitator 共jet, propeller, and various turbines兲,
forced convection as during reactor blow down in a loss-of-
heat transfer surfaces 共vessel wall, helical coils and tube baffles兲,
coolant accident. For high-level liquid waste tanks, this work pro-
and geometric locations of agitator systems. As shown in Fig. 1
vides the basis for the evaluation of the performance of purge and
for a jet-agitated tank and a mechanically agitated tank, the geo-
ventilation systems to remove flammable gases generated from metric factor could include the following dimensionless length
stored waste, and to maintain a well-mixed conditions in the gas ratios,
space. In liquid high-level waste tanks, natural convection from
the liquid surface to the gas space removes radiolysis-generated
flammable gases. Injected gas jets, used to inert such tanks, can
augment this mass transfer. When heavy gases such as benzene
f 冋 冉 冊冉 冊冉 冊冉 冊冉 冊冉 冊 册

are generated and stable stratification can occur, the forced- The recirculation of fluid in a vessel to generate uniform mixing
convection augmentation can reduce the liquid temperature re- can be performed using a pump, which draws fluid from the vessel
quired to maintain adequate mixing and mass transfer. and returns the fluid to a nozzle. The fluid jet from the nozzle
For chemical processing, heat transfer in agitated vessels has induces mixing and produces a large-scale circulating motion,
been extensively studied with comprehensive coverage summa- which can reduce concentration, property, and temperature gradi-
rized by Sterbacek and Tausk 关1兴, Uhl and Gray 关2兴, and Penney ents. Some research, reported for the helical-coil heat transfer in a
关3兴. Mechanical agitators are most often used to provide agitation. jet-agitated vessel, found that the ratio of nozzle diameter d j to the
However, gas sparging and liquid jets injected into vessels are tank diameter D has a weak effect on average heat transfer, where
also used in some applications to provide mixing. To evaluate heat the geometric factor could be correlated with (d j /D) 0.09. This
transfer in mechanically agitated and jet-agitated vessels, a gen- work further suggested that jet orientation into the vessel did not
eral correlation form similar to the forced convection correlation play an important role. In present study for enclosures with a
is usually employed. In addition to the Reynolds, Prandtl, and heated or cooled bottom surface, however, jet orientation is found
viscosity ratio numbers, the general equation also contains several to have a large effect on heat transfer.
ratios of linear dimensions to describe the effect of the complex Fossett and Prosser 关4兴 studied the mixing of an aqueous
geometry on the fluid motion. The general equation is formulated Na2 CO3 solution in tanks mixed by a jet. They carried out tests in
as a 1.50-m diameter, 0.92 m deep vessel in which a single jet was
introduced to create large-scale recirculation flow patterns. The
Contributed by the Heat Transfer Division for publication in the JOURNAL OF
results of their jet mixing data suggested that the mixing time will
HEAT TRANSFER. Manuscript received by the Heat Transfer Division July 9, 2001; be the same for jets nozzles of different diameters, if the effluent
revision received March 14, 2002. Associate Editor: K. S. Ball. jet velocities are adjusted to provide the same momentum flux.

660 Õ Vol. 124, AUGUST 2002 Copyright © 2002 by ASME Transactions of the ASME

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Fig. 1 Mixing processes for „a… jet-agitated, and „b… impeller-
agitated vessels Fig. 2 Photo of the cylindrical enclosure experiment

key parameters governing the heat transfer augmentation by a

Compared with limited information available for jet-agitated forced jet, and to investigate the effect of geometric factors, in-
vessels, substantial research has been devoted to study the me- cluding jet diameter, jet injection orientation, enclosure geometry
chanically aided 共impeller兲 heat transfer in agitated vessels. Inves- 共aspect ratio兲, and the presence of flow obstructions. Flow veloc-
tigation includes the geometric effects of impeller diameter ratio ity measurements were also performed to provide a better under-
d i /D, impeller height ratio Z/d i and Z/H, thickness ratio w/D, standing of the flow patterns generated inside the enclosure, which
and number of impeller blades 共see Uhl and Gray 关2兴, and Penney have substantial effects on the effectiveness of enclosure mixing
关3兴兲. Others have studied the heat transfer phenomena in gas and heat transfer augmentation. Finally, a correlation is proposed
sparged vessels, such as Hart 关5兴, and Nagata et al.关6兴, and a heat using a weighted relation to balance the contributions of natural
transfer correlation for gas-sparsed mixing of vessels was recom- convection and forced convection to heat transfer.
mended by Hart 关5兴 in the form:
c p␷ s␳
⫽0.125 冉 冊
␷ s␳
Pr⫺0.6 (3)
Experimental Apparatus
In this experiment, a cylindrical enclosure was constructed with
a diameter of 2.29 m and a maximum height of 0.8 m, with a
where the heat transfer coefficient h is calculated by using the
vertically adjustable ceiling. The enclosure was designed as a
sparged-gas superficial velocity ␷ s .
1/10-scale model of a Savannah River Plant high-level liquid
For natural convection inside enclosures with a horizontal heat
waste tank. Figure 2 shows a picture of the experimental enclo-
transfer surface, some research work has been reported by Globe
sure. At the bottom of the enclosure, a 3.18 mm thick copper plate
and Dropkin 关7兴, and Catton and Edward 关8兴. Globe and Dropkin
共2.43 m long and 2.43 m wide兲 was installed. The copper plate
studied natural convection heat transfer in liquids confined by two
could be heated or cooled by hot or cold water running through a
horizontal plates and heated from below. Catton and Edward 关8兴
total of 23 tubes soldered under the plate. This method provided a
further investigated effects of side walls on natural convection
nearly isothermal heating/cooling surface for the experiment.
between horizontal plates, and found that, as the Rayleigh number
Nine sheathed J-type thermocouples with 0.508 mm diameter
was increased to 106 and above, the Nusselt numbers for various were embedded in the copper plate to measure the heating surface
aspect ratios asymptotically reached the limit corresponding to the temperature. Due to the high thermal conductivity of copper and
smallest aspect ratio used (H/D⫽1). That implies that for large the arrangement of the heating tubes, the plate was shown to be
Rayleigh numbers extending into the turbulent regime, the size nearly isothermal. Four T-type thermocouples were mounted in
and thermal conductivity of the horizontally confining surfaces the inlet of heating water and another four in the outlet, creating a
did not affect the mean heat transfer coefficient. The experiments thermopile to measure the temperature difference for the calcula-
performed by Globe and Droplin 关7兴 also demonstrated that there tion of total heat loss from the heating plate. Cold or hot air was
has no significant effect of a lateral surface with low aspect ratio injected into the enclosure with four different injection orienta-
at high Rayleigh numbers. Experimental data reported by Catton tions using long tubes of 15 mm, 26 mm, and 53 mm I.D., and
and Edward 关8兴 are limited to Ra⬍107 . Ulucakli 关9兴 investigated was removed at the top of the enclosure through two 53 mm
the natural-convection heat transfer in a enclosure, heated par- diameter openings. The heat removal rate by the injected air was
tially from below by a disk-shaped surface and cooled from the calculated from the temperature difference between the air inlet
top and the side, and the data were presented for 108 ⬍Ra and outlet, and the air flow rate measured by a bank of parallel
⬍1010. He found that, for large temperature differences between rotameters of different capacities. Temperatures inside the enclo-
the heating surface and bulk, the mean heat transfer rates can be sure were measured with a vertical array of thermocouples de-
expressed in terms of conventional correlations, and the results signed to be traversed horizontally, measuring the vertical tem-
were well correlated by the equation perature distribution as well as the mean enclosure temperature.
A micromanometer device is applied to measure the flow ve-
Nudc ⫽0.2Ra0.35
dc (4)
locity, using two gage liquids of small density difference. The
where Nudc and Radc are evaluated using the heating-disk diam- gage liquids 共water, and oil with a specific gravity of 0.9375兲 were
eter. immiscible. The heavier gage liquid filled the lower portion of the
After reviewing of the previous work for mixing in agitated manometer U tube up to the 0-0 level 共a half of the tube length兲,
vessels and for natural convection inside enclosures, an experi- and lighter gage liquid was added to both sides, filling the upper U
mental apparatus was constructed to measure the combined tube and two reservoirs placed on top of the U-tube. A large gage
natural-and forced-convection heat transfer in a cylindrical enclo- difference could be produced by a small pressure difference with
sure mixed by an injected jet. The tests were designed to study the this micromanometer device. A pitot tube was inserted into the

Journal of Heat Transfer AUGUST 2002, Vol. 124 Õ 661


tank with the opening directed upstream to the flow path and the
other end connected to one of the micromanometer reservoir,
which then measured the total pressure 共static and dynamic兲. An-
other tube measuring static pressure is connected with the other
reservoir. The flow velocity was then be determined by the gage
difference in U tube, which represented the dynamic pressure
head at the pitot-tube opening. Streeter and Wylie 关10兴 discuss the
use micromanometers.

Experimental Procedure
Experiments were first performed to investigate natural-
convective heat transfer from the heated bottom surface. These
results were used as the reference for subsequent evaluation of the
Fig. 3 Injection orientations and resulting large-scale recircu-
heat transfer augmentation from combined natural and forced con-
lation patterns
vection. The bottom surface was heated and maintained at con-
stant temperatures ranging from 30° to 70°C, and heat was trans-
ferred by natural convection and removed from the enclosure top
and sidewalls by conduction to the ambient. Tests were performed D. Vertical/down injection near enclosure vertical wall: injected
with enclosure heights of 42, 61, and 80 cm, giving enclosure downward from the top at 10 cm from the enclosure vertical wall.
aspect ratios H/D of 0.18, 0.27, and 0.35, respectively. An em-
pirical heat transfer correlation for natural convection was ob- Error Analysis
tained by correlating these experimental data.
With this correlation for natural-convection heat transfer inside The uncertainty of the temperature measurement, including er-
the enclosure, combined natural- and forced-convection heat rors from instrumentation and data acquisition system, was esti-
transfer was then studied. For a forced jet injected into the enclo- mated to be ⫾0.1 °C. The uncertainty of the flow rate measure-
sure without internal obstructions, the geometric factors investi- ment for water was calibrated to be ⫾3 percent, and for air,
gated included the use of three different enclosure aspect ratios, depending on the range of flow measurement, was estimated be-
three different jet diameters and four different injection orienta- tween ⫾2 percent to ⫾5 percent. The corresponding maximum
tions. To study the effects of obstructions inside the enclosure, a error was 18 percent for the calculated Nusselt number NuD , and
total of 31 3.35 cm diameter 共O.D.兲 tubes were inserted extending 8 percent for the calculated Archimedes number Ar.
vertically from the ceiling to the bottom. Tests were performed to
evaluate the effect of the tube structures on the mixing process. Experimental Results and Analysis
When the jet was directly obstructed by a tube located at various
distances from the nozzle, momentum in the original flow direc- Part I Heat Transfer Augmentation with Destabilizing
tion was reduced due to a form drag. „Positive… Effect of Buoyancy Force.
The experiments investigated the effects of destabilizing den-
Natural-Convection Heat Transfer in the Enclosure. Figure 4
sity gradients by injecting cold air into the enclosure the bottom
shows the relationship between the Nusselt and Rayleigh numbers
surface heated. The injected cold air temperature was approxi-
from the present data and the data of Uluckli 关9兴. These data can
mately 21°C, and the mean temperature at the heating surface was
be well correlated by
controlled at around 60°C to minimize the effects of fluid property
changes and to reduce the complexity for the comparison of geo- Nunc ⫽0.15RaD
metric factors. Experiments were performed to investigate the ef-
fects of stabilizing density gradients, by injecting hot air into the where the 1/3 power and a constant value of 0.15 agrees with the
enclosure with the bottom surface cooled. Under this condition, natural-convection correlations for horizontal heating surface in
the buoyancy force stabilizes the flow close to the bottom surface open space. The results demonstrate that the lateral boundary and
and reduces heat transfer. The injected hot air temperature was height-to-diameter ratio have a negligible effect on the turbulent
approximately 52°C, and the mean temperature at the cooling sur- natural-convection heat transfer at high Rayleigh numbers. Equa-
face was held around 12°C. tion 共5兲 is used as the reference to determine the heat transfer
Temperature changes were carefully monitored inside the en- augmentation from combined natural and forced convection.
closure as well as on the heating surface. Data were collected only
when the testing condition reached steady state. The reference
temperature used to calculate the thermodynamic and transport
properties of air was taken as the mean value measured by the
vertical thermocouple array. Air was injected into the enclosure
with Re j 共jet Reynolds numbers兲 from 3⫻103 to 3⫻106 , and RaD
共Rayleigh number scaled with enclosure diameter兲 inside the en-
closure was in the order of 109 ⬃1010.
To investigate how different flow patterns can affect heat trans-
fer augmentation, air was injected into the enclosure with four
different injection orientations. Figure 3 illustrates the four orien-
tations, and the five basic large-scale flow patterns that result. The
orientations studied are:
A. Radial injection: injected horizontally across the enclosure
B. Azimuthal injection: injected horizontally at a 60 deg angle
from radial injection.
C. Vertical/down injection near center: injected downward from
the top at 23 cm from the enclosure center. Fig. 4 Natural-convection heat transfer data

662 Õ Vol. 124, AUGUST 2002 Transactions of the ASME


Fig. 6 Horizontal velocities at positions near the heated bot-
tom surface
Fig. 5 Results for combined natural and forced-convection
heat transfer

of the enclosure兲. The data, including four injection orientations

and three jet diameters, are well correlated using Eq. 共8兲 with a
Combined Natural-and Forced-Convection Heat Transfer in the different constant C 2 for each injection orientation. It can be seen
Enclosure. A general correlation form is employed to combine that changing the injection orientation changes the heat transfer
the natural- and forced-convection heat transfer in the enclosure, augmentation significantly, while the geometric factor d j /D has a
given by very small effect. For the destabilizing density gradient the data
NuD ⫽Nunf c ⫾Nunnc (6) for all injection orientations asymptotically approach 1 at low
Archimedes number. This agrees with the prediction by Eq. 共8兲
This combining rule was used by Churchill 关11兴 to evaluate the when heat transfer is dominated by natural convection.
mixed-convection heat transfer in external boundary layer flows The magnitude of augmentation is largest for vertical/down in-
as well as by Chen et al. 关12兴 to evaluate transition heat transfer jection near the center 共C兲, and followed by vertical/down injec-
from laminar to turbulent film flows. This formulation uses a tion near the wall 共D兲, radial injection 共A兲, and azimuthal injection
weighted relationship to balance the contributions from the sepa- 共B兲. The constants C 2 correlated for radial 共A兲 and azimuthal 共B兲
rate correlations representing their extreme conditions. In the injection, in general, fall within the expected range, while that for
present study, a positive sign in Eq. 共6兲 represents a destabilizing vertical/down injection 共C兲 augmentation is clearly higher than
density gradient, and a negative sign a stabilizing gradient 共with expected.
the minimum Nu value constrained to be zero兲. Following the To further understand the flow pattern, the air velocity was
common practice, an exponent of n⫽3 is adopted, and Eq. 共5兲 is measured by the micromanometer device. Figure 6 presents the
employed as the natural-convection part Nunc , and the correlation velocity in the horizontal direction measured near the bottom of
form of Eq. 共1兲 as the forced-convection part Nu f c . the heating surface. For comparison, air was injected at Re j⬇2.8
It is found that the Nusselt numbers NuD , plotted against jet ⫻105 through a tube of 15 mm I.D. Three sets of data are in-
Reynolds number Re j , are clustered into groups of trend lines in cluded in this figure, including azimuthal injection with H/D
accordance with their injection orientations. At high Reynolds ⫽0.35, and 0.18, and vertical/down injection 共C兲 with H/D
numbers the data are fitted best with Reynolds number to the 2/3 ⫽0.35. For azimuthal injection, the data is plotted at the positions
power (⬀Re j2/3). The present data do not include a wide range of measured from the vertical geometric center of the enclosure, and
Pr, but from research summarized by Penney 关3兴 for agitated ves- for vertical/down injection at the positions measured from the
sels, a 1/3-power relation is used for the effect of Pr. Based on vertical jet center.
these arguments, Eq. 共6兲 can be recast in the form As illustrated in Fig. 3共II兲 air injection in the azimuthal direc-

再冋 冉 冊 册 冎
tion induces circumferentially circulating motion. The flow veloc-
NuD Re2/3
j Pr
␮b 0.14 3 1/3
ity distribution in Fig. 6 shows approximately equal velocities for
⫽ C1 f 共 geometric兲 ⫾1 (7)
Nunc RaD 1/3
␮w positions at the same radius from the enclosure center. It is also
where ␮ b is the bulk air viscosity, and a 0.14 power for the vis- reveals that while there is strong horizontally circulating motion
cosity property ratio is adopted. Equation 共7兲 can then be further in the regions close to the wall, regions near the center experience
little influence from forced flow motion, reaching an almost stag-
simplified using the Archimedes number Ar⫽Re2j /GrD and RaD nant condition at the center. The delineation, separating the
⫽GrD Pr, and by combining the constant C 1 and the geometric forced-convection dominated region near the wall and natural-
function, convection dominated region near the center, makes the whole-
⫽ 再冋 C 2 Ar1/3 冉 冊 册 冎
0.14 3
volume mixing less effective, and thus reduces the average heat
transfer augmentation. For H/D⫽0.35, approximately 25 percent
of the bottom surface area was not subjected to the influence of
As discussed previously, the constant k in Eq. 共1兲 usually falls forced convection under azimuthal injection, which explains the
within the range of 0.35 to 0.74 for heat transfer to enclosure relatively low augmentation in this injection mode.
walls 共not helical coil or tubes兲 in agitated tanks. Combining the For radial injection, velocity measurement shows that after the
constant in Eq. 共5兲 for natural convection, C 2 is expected to fall in expanded jet reaches the enclosure vertical wall, the jet undergoes
the range from 2.3 to 4.93. Figure 5 presents the heat transfer a boundary layer transition and spreads as a wall jet both in hori-
augmentation NuD /Nunc versus Archimedes number under a de- zontally circulating flow pattern 共Fig. 3共I兲兲 and also vertically
stabilizing density gradient 共with a heating surface at the bottom downward pattern 共Fig. 3共III兲兲. The downward flow then turns and

Journal of Heat Transfer AUGUST 2002, Vol. 124 Õ 663


Fig. 7 Vertical temperature distributions in the enclosure

spreads across the bottom floor with decreasing velocity. The flow Fig. 8 Effect of aspect ratio on heat transfer augmentation
pattern generated by radial injection is found to be more produc-
tive in mixing the whole volume than is azimuthal injection, and
gives a higher magnitude of heat transfer augmentation. likely due to the different configurations of heat transfer surfaces
For vertical/down injection near enclosure vertical wall, Figure in previous investigations. For an agitated vessel with the entire
3共V兲 shows the primary flow pattern. After the jet impinges on the inner wall used as a heat transfer surface, the transport problem is
floor, it transforms to a wall-jet flow spreading with decreasing different from a partially heated wall. The flow patterns of the
velocity across the heating floor. This flow pattern mixes the cold wall jets generated by the injected fluid 共or mechanical impeller兲
injected air with hot air near bottom surface more effectively than in different orientations does not significantly change the mean
in the cases discussed earlier. This explains why this orientation heat transfer rate if the entire inner tank wall is involved. How-
induces more effective augmentation than the radial and azimuthal ever, if the wall is only partially heated or the heating source is
injections. localized in the enclosure, the effectiveness of heat transfer will be
The most effective injection orientation 共Fig. 3共IV兲兲 is vertical/ more strongly affected by the flow patterns, depending on which
down injection near the enclosure center. At high Re the augmen- can mix better the cold injected fluid with localized hot fluid near
tation is more than twice of that for azimuthal injection. After the heat transfer surface. Therefore, injection orientation should
impinging on the floor surface, the vertical velocity component in not be neglected with localized heat transfer surfaces in enclo-
jet core is decelerated and transformed into an accelerated hori- sures.
zontal flow. The horizontal wall jet spreads evenly outward with a
velocity decreasing uniformly radially from the stagnant point, Effect of Enclosure Aspect Ratio. Figure 8 shows high-Re re-
and then turns upward after reaching vertical walls. For this cy- sults for three different aspect ratios. In general, the heat transfer
lindrical enclosure this flow pattern proves to be most effective for coefficient increases with decreasing aspect ratio for all three ori-
uniform mixing. Figure 7 illustrates the vertical temperature dis- entations presented. While the increase is small with decreasing
tribution for three injection orientations, and shows that the enclo- aspect ratio for the radial 共A兲 and vertical/down injections 共C兲, the
sure volume is well mixed under vertical/down injection 共C兲, and magnitude of the increase is relatively large for azimuthal injec-
strongly stratified under azimuthal injection. tion 共B兲. The solid trend lines in the upper part of the figure
The heat transfer augmentation in vertical/down injection 共C兲 represent the increment due to the aspect ratio effect for vertical/
occurs due both to the effective large-scale mixing flow pattern down injection 共C兲, and the dashed lines represent the large incre-
and the substantially augmented local heat transfer coefficient ment for azimuthal injection 共B兲. From Figure 6 with enclosures
near jet impingement point. Jambunathan et al. 关13兴 performed a of aspect ratios of 0.35 and 0.18, the circulating velocity induced
detailed review of heat transfer data for single circular jet im- by azimuthal injection increases with reducing aspect ratio, and
pingement. They concluded that the local heat transfer coefficient the area affected by forced flow also extends further into the cy-
can be predicated as a function of jet Reynolds number, height to lindrical center. This explains the larger increase of heat transfer
jet diameter ratio (L/d j ), and the ratio of radial distance from augmentation for azimuthal injection with decreasing aspect ratio.
stagnation point to jet diameter (r/d j ). Following the results of Because the aspect ratio effect is not considered to be an impor-
Boguslawski and Popiel 关14兴, the potential core radius affected by tant geometric factor, as shown in Eq. 共2兲 for correlations for most
jet impingement is approximately four times the nozzle diameter. of agitated vessels, here the same argument, as for jet orientation,
Several correlations are available for the prediction of local and applies regarding the greater importance of geometry localized
average heat transfer coefficients in this region. The equation by heat transfer augmentation. Particularly for azimuthal injection a
Lee 关15兴 can be used to estimate the average Nusselt number in large increase of heat transfer is observed when the aspect ratio is
this region, reduced by half.
Table 1 summarizes the results for the correlated constants C 2
Nur/d j ⫽4 ⫽0.083 Re0.708
j 共 L/d j 兲 0.144 (9) with positive buoyancy effect on the mixed convection inside the
Based on the present experimental data, it is estimated that the
average heat transfer coefficient inside the present enclosure will Effect of Obstructions Inside the Enclosure. To investigate the
be increased by less than 13 percent due to the localized enhance- effect of flow obstructions, 31 tubes with 3.35 cm O.D. were
ment in the potential core region. Therefore, the majority of heat inserted vertically from the top to the bottom of the enclosure.
transfer augmentation by vertical/down impingement 共C兲 is con- Experiments were performed to measure the variation of heat
tributed by the effective large-scale mixing flow pattern. transfer under the conditions where the forced jet impinged di-
The effect of jet orientation has not typically been included in rectly on a tube at 10, 20, 38, and 63.5 cm from the injection
agitated vessel correlations 共e.g., Eq. 共2兲兲. The discrepancy is nozzle. It is found that if the jet does not impinge on the tube

664 Õ Vol. 124, AUGUST 2002 Transactions of the ASME


Table 1 C 2 values for injection orientation and aspect ratio

before reaching the enclosure wall, the change of mean heat trans-
fer coefficient can be small as long as the total cross-section area
of tubes is much less than that of the flow area. If the jet hits a
tube before reaching the enclosure wall, the loss of jet momentum Fig. 10 Effect of stabilizing density gradient on mixed convec-
in the flow direction can significantly affect the effectiveness of tion
mixing the whole volume, and thus reduce the mean heat transfer
coefficient. Figure 9 shows the results of jet impingement on tubes

at certain distances from the injection nozzle.
C D␳ mU m
dt &b
The jet velocity profile downstream of the injection nozzle has e ⫺ 共 r/b 兲 dr
been discussed by List 关16兴, and is used here to estimate the M loss 2 ⫺&b dt

momentum loss in the flow direction due to form drag with the ⫽ &b ⫽0.327C D (11)
M 4 b
impinged tube. Assuming that the tube diameter d t is much 2 ␲␳ m U m
e ⫺ 共 r/b 兲 rdr
smaller than expanded jet radius R, the fraction of momentum loss 0
can be evaluated by
The drag coefficient C D ⬇1.1 and the experimental results from

M loss

␳u共 r 兲2
d t dr
Fisher et al. 关17兴 give b/x⫽0.107. Then the fraction of momen-
tum loss can be estimated in terms of the distance from the jet
origin and tube diameter by

⫽ (9)
M R ␳u共 r 兲2
2 ␲ rdr M loss dt
0 2 ⫽3.36 (12)
M x
where C D is the drag coefficient, and u the jet velocity. The dis- By Eq. 共12兲, 29.6 percent of momentum is lost in the flow
tribution of velocity across a jet can be evaluated by 共List 关16兴兲 direction after impingement on a 3.35 cm tube at 38 cm from the
u 共 r 兲 ⫽U m e ⫺ 共 r/b 兲
(10) jet nozzle, and 17.7 percent of momentum is lost after impinge-
ment on a tube at 63.5 cm from the jet origin. By accounting for
The evaluation of how the mean velocity U m and width parameter the momentum loss and the corresponding reduction of the jet
b vary with the distance from the jet origin x, depends upon the Reynolds number, the resulting data shifts much closer to the
evolution of the momentum flux, and can be determined from correlations without tubes inside the enclosure. This information
experimental measurements. For a given self-similar distribution implies that the reduction of heat transfer augmentation by struc-
of mean velocity and pressure, the width parameter is related to tures can be predicted by the loss of momentum in the jet. The
the jet radius by b⫽R/&. Assuming ␳ ⬇ ␳ m , Eq. 共9兲 can be fur- large-scale motion induced by the jet dominates the large scale
ther simplified as enclosure flow, and the augmentation of the heat transfer inside
the enclosure.
Part II Heat Transfer Augmentation with Stabilizing „Nega-
tive… Effect of Buoyancy Force. For mixed convection with a
stabilizing buoyancy force from a cooled bottom surface, a nega-
tive sign is used in Eqs. 共6兲 and 共8兲. Figure 10 shows the com-
parison with the experimental data. The dashed line represents the
destabilizing 共positive兲 effect of buoyancy force, and solid line
represents the stabilizing 共negative兲 buoyancy effect. As shown by
the data, the augmentation ratio approaches close to 0 at low
Archimedes numbers, with the weakening of forced convection
and stabilizing effect of negative buoyancy force. Even though
this combining rule to address stabilizing density gradients is
rather crude, it generally fits the data well, and should be adequate
for most engineering purposes.

The present experimental study has investigated heat transfer
under combined natural convection and forced convection due to
jet injection. Under natural convection, the average Nusselt num-
ber inside the large enclosure with heating surface at the bottom
can be correlated by the Rayleigh number to the 1/3 power in the
Fig. 9 Effect of tube obstructions inside the enclosure high-Rayleigh-number flow regime (RaD ⬎108 ). Under forced

Journal of Heat Transfer AUGUST 2002, Vol. 124 Õ 665


convection with a strong jet, the data is well represented as a RaD ⫽ enclosure Rayleigh number (⫽GrD Pr)
function of jet Reynolds number to the 2/3 power. The combining Re j ⫽ jet Reynolds number (⫽ ␳ ud j / ␮ )
form of Eq. 共6兲 is adequate to evaluate the mixed-convection heat T ⫽ temperature
transfer regime inside the enclosure. As shown by Eq. 共8兲, the ␮ ⫽ dynamic viscosity
correlation can be formulated as a function of Archimedes num- v ⫽ kinematic viscosity
ber, fluid property factor, and geometric correlation, and correlates u ⫽ velocity
data well under both stabilizing and destabilizing buoyancy ef- Um ⫽ mean velocity
fects. The most important geometric factor is the jet injection vs ⫽ superficial velocity
orientation. w ⫽ width of impeller
Augmentation is affected strongly by the flow patterns gener- x ⫽ distance from jet origin
ated by different injection orientations, which, in turn, govern the Subscripts
effectiveness in large-scale mixing in the enclosure. As shown in
Fig. 5, the heat transfer augmentation for vertical/down injection b ⫽ bulk value
near the enclosure center can be two times the augmentation for D ⫽ diameter of cylindrical enclosure
azimuthal injection. The improved mixing is reflected in the ver- dc ⫽ disk
tical temperature distributions shown in Fig. 8. Depending on the fc ⫽ forced convection
injection orientation, the aspect ratio can also play an important j ⫽ jet
role at low aspect ratios, particularly for azimuthal injection. In m ⫽ mean value
addition, the change of heat transfer augmentation, caused by ob- nc ⫽ natural convection
structions inside enclosure, can also be important, and the reduc- w ⫽ wall
tion can be by considering the momentum loss from form drag.
Nomenclature 关1兴 Sterbacek, Z., and Tausk, P., 1965, Mixing in the Chemical Industry, Perga-
mon, London.
Ar ⫽ Archimedes number 共⫽Re2j /GrD 兲 关2兴 Uhl, V. W., and Gary, J. B., 1966, Mixing-Theory and Processing, Vols. I and
b ⫽ width parameter of velocity profile II, Academic, New York.
关3兴 Penney, W. R., 1983, Agitated Vessels, Heat Exchanger Design Handbook,
␤ ⫽ coefficient of thermal expansion E.U., Schlunder, ed.,, Hemisphere Publishing Corp., New York.
cp ⫽ specific heat at constant pressure 关4兴 Fossett, H., and Prosser, L. E., 1951, Trans. Inst. Chem. Engr. 共London兲 29, pp.
C1 ⫽ constant 322.
C2 ⫽ constant 关5兴 Hart, W. F., 1976, ‘‘Heat Transfer in Bubble-Agitated Systems—A General
Correlation,’’ Ind. Eng. Chem. Process Des. Dev., 15共1兲, pp. 109–114.
CD ⫽ drag coefficient 关6兴 Nagata, S., et al., 1976, Study of Heat Transfer for Aerated Mixing Vessel and
di ⫽ impeller diameter Aerated Tower, Heat Transfer Jpn. Res., 5, No. 4, pp. 75– 83.
dj ⫽ jet diameter 关7兴 Globe, S., and Dropkin, D., 1959, ‘‘Natural Convection Heat Transfer in Liq-
dt ⫽ tube outer diameter 共O.D.兲 uids Confined by Two Horizontal Plates and Heated from Below,’’ J. Heat
Transfer , 81, pp. 24 –27.
D ⫽ diameter of cylindrical enclosure 关8兴 Catton, I., and Edwards, D. K., 1967, ‘‘Effect of Side Walls on Natural Con-
h ⫽ heat transfer coefficient vection Between Horizontal Plates and Heated from Below,’’ J. Heat Transfer,
f ⫽ geometric factor 89, pp. 295–299.
g ⫽ standard gravitational acceleration 关9兴 Ulucakli, M. E., 1996, ‘‘Turbulent Natural Convection in an Enclosure with
Localized Heating from Below,’’ Exp. Heat Transfer, 9, pp. 305–321.
GrD ⫽ enclosure Grashof number (⫽g ␤ (T w ⫺T b )D 3 / v 2 ) 关10兴 Streeter, V. L., and Wylie, E. B., 1985, Fluid Mechanics, McGraw Hill, New
H ⫽ height of enclosure York.
I.D. ⫽ inner diameter 关11兴 Churchill, S. W., 1978, ‘‘A Comprehensive Correlating Equation for Laminar,
Assisting, Forced and Free Convection,’’ AIChE J., 23, pp. 10–16.
K ⫽ constant 关12兴 Chen, S. L., Gerner, F. M., and Tien, C. L., 1987, ‘‘General Film Condensation
k ⫽ thermal conductivity Correlations,’’ Exp. Heat Transfer, 1, pp. 93–107.
L ⫽ distance between jet exit and impingement plate 关13兴 Jambunathan, K., Kai, E., Moss, M. A., and Button, B. L., 1992, ‘‘A Review of
n ⫽ constant Heat Transfer Data for Single Circular Jet Impingement,’’ Int. J. Heat Fluid
Flow, 13, pp. 106 –115.
N ⫽ impeller rotational speed 关14兴 Boguslawski, L., and Popiel, C. O., 1979, ‘‘Flow Structure of the Free Round
NuD ⫽ Nusselt number inside the enclosure (⫽hD/k) Turbulent Jet in the Initial Region,’’ J. Fluid Mech., 90, pp. 663– 685.
O.D. ⫽ outer diameter 关15兴 Lee, J., and Lee, S. J., 1999, ‘‘Stagnation Region Heat Transfer of Turbulent
Pr ⫽ Prandtl number Axisymmetric Jet Impingement,’’ Exp. Heat Transfer, 12, pp. 137–156.
关16兴 List, E. J., 1982, ‘‘Turbulent Jets and Plumes,’’ Annu. Rev. Fluid Mech., 14,
r ⫽ radial distance from stagnation point pp. 189–212.
␳ ⫽ density 关17兴 Fisher, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H., 1979,
R ⫽ radius of jet Mixing in Inland and Coastal Waters, New York Academic.

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