CopyrightC Taylor and Francis Group, LLCISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903263200 Single-Phase Flow in Meso-Channel Compact Heat Exchangers for Air Conditio
Trang 2CopyrightC Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
Laboratory of Heat and Mass Transfer, Ecole Polytechnique F´ed´erale de Lausanne, Lausanne, Switzerland
As the Chairman of ALPEMA (Aluminum Plate-Fin Heat
Exchanger Manufacturers’ Association) since May 2008, I wish
to announce the new third edition of the ALPEMA Standards
for the construction of brazed aluminum plate-fin heat
ex-changers The development of the new third edition of the
ALPEMA Standards has involved a significant effort by the
former chairman of ALPEMA, David Butterworth, the
cur-rent secretariat (Simon Pugh of IHS, London), and the five
ALPEMA member companies [Chart Energy and Chemicals
Inc (USA), Fives Cryo (France), Kobe Steel, Ltd (Japan),
Linde AG (Germany), and Sumitomo Precision Products Co.,
Ltd (Japan)] I wish to acknowledge their many contributions to
help me update and extend this industrial standard for the safe
construction and operation of brazed aluminum plate-fin heat
exchangers
In brief, brazed aluminum plate-fin exchangers are the most
effective and energy-efficient heat exchangers for handling a
wide range of services, noted particularly for their compactness
and low weight This class of heat exchangers nearly always
pro-vides the lowest capital, installation, and operating cost
when-ever the application is within the operating range of these units,
in particular over a wide range of cryogenic and non-cryogenic
applications Where it is feasible to use a brazed aluminum
plate-fin heat exchanger, it is usually the most cost-effective solution,
often by a significant margin These units enjoy a very large
heat transfer surface area per unit volume of heat exchanger
They provide a total surface area of 1000 to 1500 m2/m3of
vol-Address correspondence to Prof John Thome, Laboratory of Heat and Mass
Transfer, EPFL-STI-IGM-LTCM, Mail 9, CH-1015 Lausanne, Switzerland.
E-mail: john.thome@epfl.ch
ume; this compares very well with the approximate range of 40
to 70 m2/m3for shell-and-tube units Plate-fin heat exchangerswith surface area per unit volume of 2000 m2/m3are sometimesemployed in the process industry!
Plate-fin heat exchangers find applications in aircraft, mobiles, rail transport, offshore platforms, etc However, themain applications are in the industrial gas processing, naturalgas processing, LNG (liquefied natural gas) facilities, refining
auto-of petrochemicals, and refrigeration services Their ability tocarry multiple streams, occasionally up to 12 or more (as op-posed to typically only two streams in a shell-and-tube heatexchanger), allows process integration all in one unit The verylarge surface area per unit volume is particularly advantageouswhen operating at low temperature differences between thehot and cold streams Such applications are typically found incryogenic systems and hydrocarbon dewpoint control systemswhere temperature difference is linked to compressor powerconsumption
The first edition of the ALPEMA Standards was published
in 1994, and it was extremely successful and popular The ond edition was published in 2000 New industrial develop-ments and applications, experience with using the ALPEMAStandards, and feedback from users have indicated that thetime was right for a third edition The new third edition isexpected to appear early in 2010 The most significant addi-tions and amendments that have been made are summarizedhere:
sec-1 A new Chapter 9 has been added to cover cold boxes andblock-in-shell heat exchangers
1
Trang 32 J R THORNE
2 Many figures have been redrawn to make them easier to
understand
3 Photographs of the most common types of fin geometries
have been added
4 Information has been provided on two-phase distributors
with diagrams
5 Guidance on flange design and transition joints is
included
6 Guidance on acceptable mercury levels is given
7 Allowable nozzle loadings have been updated
8 Many small changes have been made to improve
clarity
The new third edition can be purchased and downloaded
from the following website: http://engineers.ihs.com/products/
standards/petrochemical-standards.ht
John R Thome has been professor of heat and mass
transfer at the Swiss Federal Institute of ogy in Lausanne (EPFL), Switzerland, since 1998, where his primary interests of research are two-phase flow and heat transfer, covering both macro-scale and micro-scale heat transfer and enhanced heat transfer.
Technol-He directs the Laboratory of Technol-Heat and Mass Transfer (LTCM) at the EPFL with a research staff of about 18–20 and is also director of the Doctoral School
in Energy He received his Ph.D at Oxford
Univer-sity, England, in 1978 He is the author of four books: Enhanced Boiling Heat Transfer (1990), Convective Boiling and Condensation, third edition (1994), Wolverine Engineering Databook III (2004), and Nucleate Boiling on Micro- Structured Surfaces (2008) He received the ASME Heat Transfer Division’s
Best Paper Award in 1998 for a three-part paper on two-phase flow and flow
boiling heat transfer published in the Journal of Heat Transfer He has received
the J&E Hall Gold Medal from the UK Institute of Refrigeration in February,
2008 for his extensive research contributions on refrigeration heat transfer Since
2008, he has been chairman of ALPEMA (the plate-fin heat exchanger facturers association) He has published widely on the fundamental aspects of
manu-micro-scale two-phase flow and heat transfer He is an associate editor of Heat Transfer Engineering.
heat transfer engineering vol 31 no 1 2010
Trang 4CopyrightC Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903263200
Single-Phase Flow in Meso-Channel
Compact Heat Exchangers for Air
Conditioning Applications
1School of Engineering and Computer Science, Washington State University–Vancouver, Vancouver, Washington, USA
2Mechanical and Nuclear Engineering Department, Kansas State University, Manhattan, Kansas, USA
Experimental study of the single-phase heat transfer and fluid flow in meso-channels, i.e., between micro-channels and
mini-channels, has received continued interest in recent years The studies have resulted in empirical correlations for various
geometries ranging from simple circular pipes to complicated enhanced noncircular channels However, it is still unclear
whether the correlations developed for conventional macro-channels are directly applicable for use in micro-/mini-channels,
i.e., hydraulic diameter less than 3 mm, with heat exchanger applications A few researchers have agreed that similar results
may be obtained for the laminar flow regime regardless of the channel size, but no general agreement has been reached for
the transitional and turbulent flow regimes yet In this study, different meso-channel air–liquid compact heat exchangers
were evaluated and the experimental results were compared with published empirical correlations A modified Wilson plot
technique was applied to obtain the heat transfer coefficients, and the Fanning equation was used to calculate the pressure
drop friction factors The uncertainty estimates for the measured and calculated parameters were also calculated The results
of this study showed that the well-established heat transfer and pressure drop correlations for the macro-channels are not
directly applicable for use in the compact heat exchangers with meso-channels.
INTRODUCTION
There have been many experimental studies conducted on
single-phase fluid flow within compact heat exchangers with
micro- and mini-channels, and new findings have been reported
for different applications However, the researchers offer
differ-ing opinion on the role of channel size, as classified by Kandlikar
and Grande [1], in correlating heat transfer and pressure drop,
especially at the transitional and turbulent regimes This issue
becomes more complicated when the heat exchanger channel
geometries are compact and enhanced, such as in automotive
compact heat exchangers, as described by Webb and Kim [2]
Some researchers reported the possibility of significant
dif-ferences between the macro- and micro-scale theories and
cor-relations, while others believe the differences are not significant
and that the same correlations can provide results that are
gener-ally in good agreement For example, Webb and Zhang [3] found
the existing correlations for conventional macro-channels can
Address correspondence to Dr Amir Jokar, School of Engineering and
Computer Science, Washington State University Vancouver, 14204 NE Salmon
Creek Ave, Vancouver, WA 98686, USA E-mail: Jokar@vancouver.wsu.edu
adequately predict the single-phase heat transfer and pressuredrop in multiport circular and rectangular mini-channels withhydraulic diameter ranging from 0.96 to 2.13 mm However,they mentioned their findings were in contrast to the resultsand conclusions that Wang and Peng [4] obtained in a simi-lar study Steinke and Kandlikar [5, 6] recently made extensivereviews of single-phase heat transfer and pressure drop in micro-channels They generated a database from the available litera-ture and compared the results obtained by different researchers
in order to answer this fundamental question of whether the sical macro-scale theories can be applied to micro- and mini-channels Subsequently, they concluded these theories are ingood agreement with smaller channel size provided all the flowfactors, such as development of flow, efficiency of fins, andexperimental uncertainties, are accurately taken into consid-eration We believe more applied research on micro- and mini-channel heat transfer and fluid flow with different industry appli-cations can further clarify the answers to this question, and mayresult in a set of general correlations for each scale and regime.The co-authors previously obtained heat transfer and pressuredrop for different type of air–liquid meso-channel compact heatexchangers and published the results in conference proceedings
clas-3
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[7–9] The objective of this article is to review the previous
results and offer a conclusion on the single-phase flow in
meso-channel compact heat exchangers of this sort
The air–liquid heat exchangers under study were analyzed
on both air and water sides A 50% glycol–water mixture was
pumped into the enhanced circular and noncircular channels of
these heat exchangers while, on the other side, air was pushed
through the fin passages with louvered surfaces The goal was
to obtain semi-empirical heat transfer correlations for the flow
of the glycol–water mixture in the meso-channels and the flow
of air through the louvered fin surfaces For this purpose, a
modified version of the Wilson plot technique presented by
Briggs and Young [10] was applied to find the single-phase heat
transfer correlations The glycol–water pressure drop was also
analyzed and the Fanning equation was used to calculate the
friction factor
The compact heat exchangers in this study were operated
as components of a refrigeration system They were in turn
in-stalled within the secondary fluid loops connected to the main
refrigeration loop of a custom automotive air conditioning
sys-tem The main refrigeration loop included a compressor,
con-denser, evaporator, and expansion valve The secondary fluid
system included two loops that exchanged energy with the main
refrigeration loop In air conditioning (AC) mode, one of these
loops was formed between the evaporator and the cooler-core
compact heat exchanger to absorb thermal energy from the
pas-senger cabin and transfer it to the evaporator during summer
conditions The other loop was formed between the condenser
and the radiator of compact heat exchanger to transfer thermal
energy from the condenser to the surroundings In heat pump
(HP) mode, the two secondary loops were switched using a
four-way valve, so that one loop was formed between the condenser
and the heat-core compact heat exchanger, and the other loop
between the evaporator and radiator By changing the glycol–
water mixture flow rates through the secondary fluid loops and
controlling the temperatures, the required energy was
trans-ferred to/from the compact heat exchangers The experimental
data were used to calculate the heat transfer rate and pressure
drop of the heat exchangers
In this article, the experimental test facilities are first
de-scribed, followed by the geometry and size of the compact heat
exchangers The calculation method and data analysis to
de-termine heat transfer and pressure drop correlations from the
measured data are then explained The resulting single-phase
correlations are finally presented, discussed, and compared with
the relevant previous studies
EXPERIMENTAL TEST FACILITY
The air conditioning system under study consisted of a main
refrigeration loop using R-134a as the working fluid and two
secondary fluid loops using a 50% glycol–water mixture as the
secondary cooling/heating fluid Figure 1 shows a schematic
Figure 1 Schematic of the test facility.
diagram of the test facility, and the following subsections give
a brief description of the system components
Secondary Fluid Loops
Secondary glycol–water mixture loops were designed to change energy with the evaporator and condenser The temper-atures at the inlet/outlet ports of each device were measuredusing 0.2 m long type-K thermocouples probes The thermo-couple probes were inserted a minimum of 0.1 m into the flowlongitudinally and fixed in the center of the 0.02 m inner diam-eter tubes such that the bulk temperature could be measured.The pressure drop of the glycol–water mixture passing throughthe compact heat exchangers was measured by differential pres-sure transducers installed between the inlet and outlet ports Theglycol–water mixture flow rates in each loop were measured by
ex-a turbine-type flow meter
Conditioned Air
Two environmental chambers were used in this study Ineach chamber, the air temperature and humidity were controlledusing conditioned air from external heating/cooling and hu-midifying/dehumidifying systems In one of the environmentalchambers, conditioned air was circulating through either thecooler-core (AC mode) or heater-core (HP mode) compact heatexchanger to simulate cabin conditions In the other chamber,either hot (AC mode) or cold (HP mode) air was circulatingthrough the radiator compact heat exchanger to simulate ambi-ent conditions Two ducts were designed and built for meteringair flow through the compact heat exchangers The heat ex-changers were installed in the middle of the air ducts duringeach test Induction fans installed at the ducts’ entrance wereused to push air through the heat exchanger, while calibratedASME standard nozzles were used to measure the air flow rate.The mean inlet and outlet air temperatures were measured usingseveral type-K thermocouples distributed on imaginary verticalheat transfer engineering vol 31 no 1 2010
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Figure 2 Thermocouples installation on the front and back of the
meso-channel compact heat exchangers.
plains both in the front and back of the heat exchangers, as
typically shown in Figure 2 A chilled-mirror dew-point sensor
measured the dew-point temperature at the ducts’ inlets The
wet bulb temperature was calculated from psychrometrics using
the measured dry bulb and dew-point temperatures
Refrigeration Loop
The refrigeration loop included an evaporator, a condenser, a
compressor, and an expansion valve Thermocouples and
pres-sure transducers were installed at the inlet and outlet ports of
all components for temperature and pressure measurements A
Coriolis-effect flow meter was used to measure the refrigerant
mass flow rate, which was controlled by varying the compressor
speed using a frequency-controlled AC motor The refrigerant
charge was varied for each test condition to control the
sub-cooled and superheated temperatures at the condenser and
evap-orator exits, respectively These temperatures were controlled at
about 5◦C for a stable system operation
Test Procedure
A range of test conditions was used to obtain adequate data
for analyzing the performance of the heat exchangers All the
system variables such as temperatures, pressures, and flow rates
were recorded every 10 s as raw data Once the fluctuations
in glycol–water mean temperature within the heat exchangers
became stable (within±1◦C), the system was considered to be
at a steady-state condition The data collection then began and
continued for at least 10 min for each test condition The
time-averaged data were then used to analyze the heat exchangers’
performance
THE COMPACT HEAT EXCHANGERS
Five different meso-channel compact heat exchangers, asparts of the secondary fluid system, were tested and analyzed
in this study Theses heat exchangers, which were used as thecooler-core, heater-core, or radiator of the automotive air con-ditioning system, are described next
Cooler-Core
This heat exchanger is installed in cars to cool the cabin
in warm conditions (AC mode) Three compact heat ers, which were manufactured in different sizes and internalflow-passage configurations, were tested as cooler-cores in thisstudy Air flowed over the fin passages and the glycol–water mix-ture passed through the rectangular meso-channels, as shown inFigure 3
exchang-The glycol–water rectangular channels had small ments (bumps) on the top and bottom surfaces These enhance-ments contacted each other in the middle of channels creat-ing two-dimensional flow passages The geometry and size ofthe glycol–water flow rectangular channels and their internalenhancements are presented in Figure 4 This figure showedthat the glycol–water was flowing, perpendicular to the page,through cavities separated by these enhancements The enhance-ments created a pattern, as shown in Figure 4, and this patternwas repeated along the length of rectangular tube
enhance-On the air side, the three meso-channel compact heat changers had louvered thin-plate fins, as described by Kays andLondon [11] The interconnecting thin-plate fins were sand-wiched between the two parallel rectangular glycol–water chan-nels, as shown in Figure 3 The louvers on the thin-plate fin sur-faces were used to promote turbulence and reduce the boundary
ex-Figure 3 Cutaways of the three meso-channel compact heat exchangers used
as the cooler-core.
heat transfer engineering vol 31 no 1 2010
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Figure 4 Geometry and size of the glycol–water flow channels in the three
meso-channel compact heat exchangers used as the cooler-core.
layer thickness of the air flowing across the compact heat
ex-changers The geometry and size of these thin-plate fins are
presented in Figure 5
Heater-Core
This heat exchanger is installed in cars to warm the cabin in
cold conditions (HP mode) The heater-core used in this study
was a finned-tube cross-flow compact heat exchanger, which
was run in heat pump mode to heat the passenger cabin A
cutaway of this heat exchanger is shown in Figure 6
Air flowed through the finned passages, while the mixture of
glycol–water passed through the circular tubes Figure 6 showed
the cross-sectional area of the circular tubes through which the
glycol–water mixture flowed This figure also showed the
heli-cal springs that were inserted into the circular tubes to promote
turbulent flow and increase heat transfer This finned-tube
com-pact heat exchanger included eight circular tubes (two-passes)
with continuous fins, as described by Kays and London [11]
The fin surfaces were parallel continuous thin plates with 16
holes through which 16 circular tubes were inserted and fitted
to the plates tightly, as shown in Figure 6 The parallel
continu-ous thin plates were not simply flat plates In fact, part of the fin
surfaces between the circular tubes was sliced vertically along
the air flow passages creating louvers These louvers between
the fin surfaces promote the flow turbulence even at low air
flow rates The geometry and size of the circular tubes with the
helical-spring inserts and the fin surfaces is shown in Figure 7
Figure 5 Geometry and size of the louvered thin-plate fins in the three channel compact heat exchangers used as the cooler-core.
meso-The flow of glycol–water mixture within the heater-core wasanalyzed in the same way as the circular tubes For the airside, an analysis similar to the air flow across a compact heatexchanger with continuous parallel fins was applied
rectangu-Figure 6 A cutaway of the meso-channel compact heat exchanger used as the heater-core.
heat transfer engineering vol 31 no 1 2010
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Figure 7 Geometry and size of the glycol–water flow channels and the
lou-vered thin-plate fins in the meso-channel compact heat exchanger used as the
heater-core.
The rectangular meso-channels had small enhancements, i.e.,
bumps, which were raised from the bottom and top surfaces to
promote flow transition from laminar to turbulent and to increase
the heat transfer effectiveness, as shown in Figure 9
The interconnecting thin-plate fins were sandwiched between
two neighboring meso-channels, as shown in Figure 8 These
fins were not simply flat plates, and in fact, the fin surfaces
were louvered along the flow passes These louvers promoted
turbulence and reduced the boundary layer thickness of the air
flowing through the radiator The geometry and size of the
lou-vered thin-plate fins on the radiator are shown in Figure 9
Figure 8 A cutaway of the meso-channel compact heat exchanger used as
the radiator.
Figure 9 Geometry and size of the glycol–water flow channels and the vered thin-plate fins in the meso-channel compact heat exchanger used as the radiator.
lou-DATA REDUCTION AND CALCULATION METHOD
A multi-channel data acquisition system allowed continuousdata collection and monitoring of the experimental test facility.Heat transfer and pressure drop correlations within the meso-channel compact heat exchangers were obtained from extensivedata sets gathered from multiple experimental test runs Thissection reviews in detail the equations used for the heat ex-changer analysis
Heat Transfer Calculation Method
A set of experiments was performed to analyze the hydrodynamic performance of each heat exchanger The Wilsonplot technique was then applied to find the heat transfer corre-lations for both the glycol–water mixture and air The first stepheat transfer engineering vol 31 no 1 2010
Trang 9thermo-8 A JOKAR ET AL.
was to calculate the overall heat transfer coefficient for each
data point The experimental data and measured dimensions of
the heat exchangers were used to obtain the overall heat transfer
coefficient based on the glycol–water side This coefficient was
calculated from the following heat transfer equations:
exchangers This factor was empirically estimated using the
inlet/outlet temperatures of fluids passing through the heat
ex-changer, as described in Incropera and DeWitt [12] The
correc-tion factor was estimated as 0.9 for most condicorrec-tions in this study
Combining Eqs (1) and (2), the overall heat transfer coefficient
was calculated as:
Ug= ˙mg,totCP,g(Tg,out– Tg,in) / AgTLMF (3)
where the log-mean temperature difference was defined as:
The fin surfaces on the compact heat exchangers were
ef-fective in transferring heat between glycol–water mixture and
air This effect was taken into account by adding the fin
ther-mal efficiency to the energy balance equation The fin therther-mal
efficiency, presented in Incropera and DeWitt [12], is shown as:
ηfin= Q˙fin
˙
Qmax = Q˙fin
hAfin(Ta– Tb) (5)where “b” denotes the fin base This equation implies the maxi-
mum heat transfer rate is attainable only if the entire fin surface
is at the base temperature, which is generally not the case The
fin efficiencies for different fin geometries have been presented
graphically in many references as a function of the heat transfer
coefficient One simple case is the straight fin with a uniform
cross-sectional area As shown in Incropera and DeWitt [12],
the fin efficiency in this case is analytically calculated as
Since the fin plates on the compact heat exchangers in this
study had a uniform cross-sectional area, Eq (6) was applied to
approximate the fin efficiency
The fin overall surface efficiency to characterize an array of
fins, as presented in Incropera and DeWitt [12], is defined as:
hgAg,tot(Twall– Tg)= ηohaAa,tot(Ta– Twall) (10)Knowing the overall heat transfer coefficient and overall finsurface efficiency, the modified Wilson plot technique, intro-duced by Briggs and Young [10], was applied to find a heattransfer correlation for both sides of the heat exchangers As-suming the areas on both sides of the channel are not the same,and neglecting the wall thermal resistance due to the small wallthickness and high thermal conductivity of the wall’s material,the overall heat transfer coefficient was calculated as:
1
UgAg = 1
hgAg + 1
haAaηo (11)The heat transfer coefficient in this study was assumed to
be in the form of the Dittus-Boelter equation, as described inIncropera and DeWitt [12], and presented as:
It should be noted that the Dittus–Boelter equation is valid for
a single-phase fluid with the Prandtl number greater than 0.7;however, that correlation was not exactly used in this study.Equation (12) was applied to both glycol–water and air flows inthe heat exchangers knowing the Prandtl numbers of air and theglycol–water mixture were greater than 0.7 The parameters ofthe Wilson plot technique were then defined as:
prop-ASHRAE Handbook of Fundamentals [13].
It was necessary to evaluate the heat exchangers’ fin acteristics to complete the Wilson plot technique Based on theheat exchanger geometries presented in previous section, thecharacteristics of the fin surfaces of the five meso-channel com-pact heat exchangers are summarized in Table 1
char-In order to apply the Wilson plot technique, Eq (14), it wasnecessary to calculate the hydraulic diameter on both glycol–water and air sides The hydraulic diameter on the glycol–waterside was calculated based on the flow cross-sectional area andheat transfer engineering vol 31 no 1 2010
Trang 10A JOKAR ET AL 9
Table 1 Measured and calculated parameters for the five meso-channel compact heat exchangers
Parameter Cooler-core (42 mm) Cooler-core (58 mm) Cooler-core (78 mm) Heater-core Radiator
Area of single fin plate, A single −fin
(m 2 )
Total unfinned area (pipe base), Ab,tot
Total glycol–water heat transfer area
(based on the internal wall
projected area), Ag,tot(m 2 )
wetted perimeter The hydraulic diameter on the air side was
calculated from the following equation, as defined by Kays and
London [11]:
Dh
L = 4AC
where Ac is the minimum cross-sectional area of the air flow
(i.e., Aminin Table 1), and A is the total heat transfer area on the
air side (i.e., Aa,totin Table 1) In Eq (15), “L” is an equivalent
flow length measured from the leading edge of the first channel
to the leading edge of the second channel
The Reynolds number in Eq (12) was defined as a function
of mass flux:
ReD= GDh
The mass flux of the glycol–water mixture was calculated
based on the measured flow rate and the minimum free-flow
area of the meso-channels The mass flux of the air flowing
through the compact heat exchangers was also calculated based
on the measured air flow rate and the minimum free-flow area
of the fin plates, as presented in Table 1
The parameters X and Y used in the Wilson plot technique
were calculated for each single experiment These data points
were then curve-fitted linearly using the least-squares method
The slope and intercept of the fitted line would thus be the inverse
of coefficients Caand Cg, respectively, as presented in Eq (14)
The heat transfer coefficients of the glycol–water mixture (hg)
and air (ha) were then obtained by Eq (12)
For the air side, the new value of “ha” was used to
recalcu-late the parameter “m” in Eq (7) The fin efficiency and overall
surface efficiency were then calculated from Eqs (6) and (8),
respectively, based on the calculated parameter “m.” These
cal-culations created a trial–error loop between “m” and hain the
Wilson plot technique The procedure was repeated until the
difference between the new and old value of ha became lessthan 0.1%
Friction Factor Calculation Method
The frictional pressure drop for the glycol–water mixturewithin the meso-channel compact heat exchangers was calcu-lated by subtracting the pressure drop across the inlet/outletmanifolds and gravitational pressure drop from the total pres-sure drop:
The pressure drop from the inlet port to the outlet port wasmeasured by a differential pressure transducer The gravitationalpressure drop was considered to be zero for this analysis sincethe inlet and outlet ports were at the same height The inlet/outletmanifolds pressure drop was approximated as a function of theinlet head velocity as:
ρu2 m
approxi-The Fanning friction factor for the glycol–water flow wasthen defined as:
Cf,g= f
4 =
PfDhL
Trang 11Cooler-core (58 mm)
Cooler-core (78 mm) Heater-core Radiator
Air volumetric flow rate (m3/s) × 10 −2 ±2% 4.601–17.35 4.851–17.74 4.837–17.21 2.742–13.84 14.85–79.38 Glycol–water volumetric flow rate (m 3 /s) × 10 −4 ±1% 1.346–3.710 1.301–3.690 1.350–4.031 1.415–3.966 1.365–4.104
For each individual test point, the frictional pressure drop
and the Fanning friction factor were calculated An appropriate
correlation was then developed for the Fanning friction factor
based on the experimental data, as shown in the Results section
The Measurement Uncertainty
The uncertainty estimates for the measured parameters were
obtained from the relevant manufacturer’s literature and through
calibration, as summarized in Table 2 An uncertainty analysis
method, introduced by Coleman and Steele [15], was then
ap-plied to estimate the overall uncertainties for the Nusselt number
and Fanning friction factor The last three uncertainty estimates
in Table 2 were the most important quantities The uncertainties
of Nusselt numbers in this table were estimated by applying the
propagation of uncertainty estimates in the parameter Y of the
Wilson plot technique As presented in Eq (14), the intercept
and slope of the X–Y plot were equal to the inverse of
lead-ing coefficient “C” in the heat transfer correlation given by Eq
(12) Thus, the uncertainty estimate for the Nusselt correlations
might be approximated by the corresponding uncertainties of
the parameter Y
RESULTS AND DISCUSSION
The five different meso-channel compact heat exchangers
were installed in the system, and the experimental data were
taken for each of them These data were then used to developheat transfer correlation and the Fanning friction factor for eachseparately, as discussed next
Heat Transfer Correlations
The Wilson plot procedure, as discussed in the previous tion, was applied to each data set to determine the heat transfercorrelation for each heat exchanger Table 3 summarizes thecoefficients of these correlations based on the general form of
sec-Eq (12)
The Prantdl number exponent “n” is estimated as 0.3 forcooling and 0.4 for heating, based on the Dittus–Boelter equa-tion introduced in Eq (12) The optimized Reynolds numberexponent “p” gave the minimum deviation in the Wilson plot
It should be noted that in the conventional Wilson plot method,the constant coefficients for the heat transfer correlations onboth sides (Cg and Ca) are obtained for the specific geome-tries of a heat exchanger, while the Reynolds exponents (pg
and pa) are assumed known (e.g., 0.8 in fully turbulent flow as
in Dittus–Boelter correlation) However, in the modified Wilsonplot method, introduced by Briggs and Young [10], the Reynoldsexponent, along with the constant coefficients, is optimized andobtained through a trail-and-error calculation process The opti-mized Reynolds components for the specific geometries of eachheat exchanger in this study are presented in Table 3
If one compares the correlations just discussed with the onesfor smooth macro-channels, the proposed correlations provide
Table 3 Heat transfer correlations for the five meso-channel compact heat exchangers Coefficients of Nu equation:
Nu = CRe p Pr n a
Cooler-core (42 mm)
Cooler-core (58 mm)
Cooler-core (78 mm) Heater-core Radiator
a nis equal to 0.3 for cooling and 0.4 for heating the fluid.
bDeviation between experiment and correlation = | Experiment—Correlation| /Experiment.
heat transfer engineering vol 31 no 1 2010
Trang 12A JOKAR ET AL 11
Figure 10 The Wilson plot X and Y parameters for the five meso-channel
compact heat exchangers.
higher heat transfer rates under the same flow conditions This
difference could be explained by (1) the surface enhancements
on both glycol–water and air sides, and (2) the scale of
chan-nels On the glycol–water side, the flow cross-sectional areas
in rectangular meso-channels had enhancements (bumps) on
the top and bottom surfaces On the other side, air passed
through the heat exchangers’ openings, which were small
rect-angular channels with louvers on the sides These corrugations
made the flow turbulent and enhanced the heat transfer rate
Figure 10 shows the results for the parameters X and Y in
the Wilson plot technique for the meso-channel compact heat
exchangers
Figure 10 shows satisfactory agreement between the
experi-mental results and the proposed correlations The deviation
be-tween the experimental and predicted Y value versus Reynolds
numbers of glycol–water mixture and air are shown in Figure
11 for the five meso-channel compact heat exchangers
Figure 11 shows that the deviations of parameter Y in
Wil-son plot method were randomly distributed (not correlated) with
Reynolds number around zero for both the glycol–water and air
flows The average deviation between experiment and
correla-tion in Wilson plot for the five heat exchangers under study was
from 3 to 15%, as presented in Table 3, while the uncertainty on
the heat transfer was 6%, as presented in Table 2
These are air–liquid heat exchangers that were tested under
real operating conditions for an automotive air conditioning
system The temperature and flow rate of both air and liquid
sides were controlled to meet the desired summer and winter
conditions During the testing of the cooler-cores for summer
conditions, there were a few test points at which condensation
took place, although the amount of condensation was estimated
and taken into account This two-phase flow on the air side could
be a source of some of the larger deviations in Figure 11, and may
affect the energy balance on the air side but not on the liquid
side It is important to note that the energy balance on the liquid
side was used to calculate the overall heat transfer coefficient
in Wilson plot The measurement on the liquid side, which was
highly accurate, was the base of calculations for the heat transfer
and pressure drop in meso-channels of heat exchangers
Figure 11 Deviation of Y parameter from the curve-fitted line in Wilson plot for the five meso-channel compact heat exchangers.
The correlations obtained for the glycol–water mixture inthis study were qualitatively compared to other correlations forlaminar and turbulent flows in macro-channels, as shown inFigure 12 Kays and Crawford [16] showed that the Nusseltnumber for the laminar flow inside a smooth rectangular channel
is constant For an aspect ratio greater than 8, such as parallelplates, the correlation is obtained from the following equations:L
Trang 1312 A JOKAR ET AL.
Figure 12 Comparison of the single-phase heat transfer correlations for the
five meso-channel compact heat exchangers with other relevant correlations.
where the friction factor is given by:
f= 4Cf = 1
[0.79lnReD− 1.64]2 (22)
It should be noted that the last two correlations are basically
used for the flow inside a smooth circular tube However, as long
as the Prandtl number is greater than 0.5, these correlations can
be applied accurately for noncircular cross sections, provided
that the tube diameter is replaced by the hydraulic diameter
defined by Eq (15); see Kays and Crawford [16]
The differences among the correlations shown in Figure 12
might be explained by the difference in geometric patterns of
the flow passages and the size of channels However, it was
hard to compare closely the channels with different geometries
and enhancement configurations For example, Figure 12 shows
the higher Nu numbers belonged in turn to 42 mm, 78 mm,
and 58 mm cooler cores, although the numbers for 78 mm and
58 mm are close These results are consistent with Figure 10 for
the Wilson plot, with Eq (14) for the overall heat transfer
coeffi-cient and with Table 3 for the correlations Looking carefully at
the enhancements inside the cooler cores in Figure 4, it appears
the higher number of enhancements belonged in turn to 58 mm,
78 mm, and 42 mm It looks like these enhancements might add
more turbulence to the flow but at the same time decreased the
direct surface heat transfer area between the liquid inside and
air outside
It should also be noted that in this study, the overall heat
transfer coefficient on the glycol–water side, as in Eq (1), was
calculated based on the internal wall projected area of the
chan-nels, as presented in Table 1 In case of the heater-core, however,the helical-spring inserts could have greater effect on the inter-nal heat transfer surface enhancement This might explain therelatively higher values of heater-core Nusselt number in Figure
12, compared to other rectangular meso-channels
It is clear from Figure 12 that the Nusselt numbers of the flowinside the meso-channels of the compact heat exchangers arenot constant but increasing with Reynolds number, although theReynolds numbers are less than the nominal critical Reynoldsnumber (2300) for the entire testing range In fact, the slopes ofthe curve-fitted lines for all the meso-channels are quite similar
to those of the Dittus–Boelter and Gnielinski correlations forfully turbulent flow
This mismatch indicates that the flow regimes inside themeso-channels could not be laminar but probably are in a tran-sition from laminar to turbulent The laminar theory was orig-inally implemented to the obtained experimental data, whichresulted in much higher errors After carefully analyzing theheat exchangers and their complicated internal geometries, itwas realized that the flow might not be laminar (Reynolds expo-nent equal to zero), as anticipated, but might be near turbulent
A modified Wilson plot method was thus implemented to count for the Reynolds exponents larger than zero The resultswere much more promising and the errors dropped significantly.The Reynolds exponents were optimized in order to obtain theminimum possible error through the experimental data As canbeen seen from Table 3, all these exponents were found to beless than 0.8, which can belong to transition flow The theory ofcritical Reynolds number 2300, known as the border of laminarflow, is quite well established for simple geometries, such as thesmooth circular tubes However, it is believed this theory cannotnecessarily be true for confined complex micro-channels, such
ac-as the three-dimensional meso-channels of this study It wac-as ferred in this study to rely on the experimental data, which werecollected under operating conditions of a real thermal system,rather than referring to the theories that are applicable to otherconfigurations
pre-Furthermore, the entrance-length effects were assumed quitesmall and negligible for the channel configuration and arrange-ment in this study As seen in Figures 3 through 9, the parallelmeso-channels of the cooler-cores and radiator are not sim-ply smooth and straight channels The enhanced plate surfaces
of these heat exchangers, once brazed together, make smalland two-/three-dimensional confined spaces that are distributedalong the channels These patterns, as shown in Figures 4 and
9, are much more complex than assuming parallel and straightchannels These surface enhancements are also intended to pro-mote turbulence and to reduce the boundary layer thicknessalong the flow
The results of the work by Olsson and Sunden [17] were alsoreviewed and included in Figure 12 for comparison They stud-ied heat transfer through single rectangular tubes that were used
in automotive radiators The results were somewhat ble with the results of this study, and interestingly their Nusseltnumber did not stay constant even at the Reynolds numbers lessheat transfer engineering vol 31 no 1 2010
Trang 14Cooler-core (58 mm)
Cooler-core (78 mm) Heater-core Radiator
aDeviation between experiment and correlation = | Experiment – Correlation| /Experiment.
than 2300 This also proves that the flow might not necessarily
be laminar but could be in transition The slight difference
be-tween the results of this study and those of Olsson and Sunden
[17] could be due to the difference in the channel
configura-tions, geometry, enhancement size, and flow conditions Plus,
the experimental data in this study were taken for the integrated
heat exchanger and not a single tube, as they did The results
of this study might be interesting and useful to those who are
looking at the heat exchanger as an integrated thermal system,
but probably not to those who are focusing on single channels
or single fin arrays
In summary, it was concluded that the flow inside such
meso-channel compact heat exchangers was near turbulent even at very
low Reynolds numbers Because of this and due to the surface
enhancements inside the meso-channels, the heat transfer rate in
these heat exchangers was increased compared to conventional
smooth pipe heat exchangers Extensive analyses of enhanced
surfaces have been presented by Webb [18]
It should be noted that the experimental results presented in
this article focus mainly on the liquid single-phase flow inside
the enhanced meso-channels of the compact heat exchangers
The air pressure drop was not measured, and there was less
at-tention/accuracy on the air-side heat transfer However, overall
heat transfer correlations for the air side were approximated
us-ing the results of Wilson plot method, as presented in Table 3
Looking carefully at Figures 3 through 9 and Table 1, one can
find that the fin configuration and geometries for the compact
heat exchangers were not identical Also, the flow conditions
for testing these heat exchangers were slightly different, as
pre-sented in Figure 10 and Table 2 As a result, correlations with
different Reynolds exponent and constant coefficients on the air
side were expected and obtained for these heat exchangers
Pressure Drop Correlations
The measured pressure drop for the glycol–water mixture in
each compact heat exchanger was used to determine the
corre-sponding Fanning friction factor for each test point A
correla-tion was then fitted to the experimental data using the general
form, introduced by Shah and Wanniarachchi [14], as shown
a simpler form of the correlation provided the best fit to thedata The pressure drop correlations for the five meso-channelcompact heat exchangers are summarized in Table 4
A comparison of the correlations with the experimental dataand some other correlations is shown in Figure 13 The frictioncoefficient for the laminar flow shown in Figure 13 was foundfrom:
Cf = C
where C is 16 for the case of circular tube For the rectangularchannels, Kays and Crawford [16] evaluated and plotted theconstant C for different aspect ratio The constant C for thecooler-cores with 42, 58, and 78 mm was found to be 21.8, 22.0,and 22.4, respectively, and for the radiator 22.3
The fully turbulent flow inside the rectangular channels could
be estimated by the correlations for circular tubes using the
Figure 13 Comparison of the single-phase pressure drop correlations for the five meso-channel compact heat exchangers with other relevant correlations.
heat transfer engineering vol 31 no 1 2010
Trang 1514 A JOKAR ET AL.
hydraulic diameter Kays and Crawford [16] argued that the
ef-fects of corners on the flow pattern in the noncircular channels in
the turbulent region are negligible They presented a correlation
for the smooth circular tube in the fully turbulent region as:
Cf
It is important to point out that this correlation is used for
Reynolds numbers greater than 20,000 No specific correlation
has been reported for the transition region in the rectangular
channels The roughness of the tube walls can affect the friction
coefficient Kays and Crawford [16] presented the following
correlation for flows inside pipes in the fully rough region:
mm to form an upper level in the Cfversus Re plot, as shown in
Figure 13
Figure 13 shows that the slope of the fitted line for current
study was more similar to the turbulent region than the laminar
region This conclusion could be confirmed by looking at the
proposed correlations listed in Table 4, since the exponents of
the inverse Reynolds numbers were not unity (as in laminar
flow) Therefore, the glycol–water flows in the meso-channels
of the compact heat exchangers with the interior enhancements
were near turbulent even at low Reynolds number
The results of the work by Olsson and Sunden [17] were
also reviewed and included in Figure 13 for comparison They
studied pressure drop through single rectangular tubes that were
used in automotive radiators Their Fanning friction factors
were above the laminar flow region, and the data did not follow
the trend/slope of the laminar flow, but looked more like the
turbulent flow These results were consistent with the results of
this study
SUMMARY AND CONCLUSIONS
Single-phase heat transfer and fluid flow of five meso-channel
compact heat exchangers, with different interior channel
con-figurations and sizes, were reviewed and analyzed in this article
Fifty percent glycol–water mixture was pumped through the
en-hanced meso-channels which had either circular or rectangular
cross section On the other side, cold or hot air was pushed over
the channels, which included louvered thin plate fins
One unique feature of this study is that the measurements of
actual heat exchangers with various dimensions were detailed
under realistic operating conditions These heat exchangers were
components of the secondary fluid loop of an automotive air
conditioning system that were investigated experimentally The
refrigeration system and its two secondary fluid loops were
operated at different conditions in both air-conditioning and
heat-pump modes The experimental data included
tempera-tures, pressures, and flow rates that were collected at
steady-state conditions The previously obtained correlations for theheat transfer and pressure drop of the glycol–water mixture andair flowing through these meso-channel compact heat exchang-ers were reviewed, compared, and discussed
Investigating these correlations, one can conclude that theconventional macro-scale correlations through the circular ornoncircular channels do not perfectly match the experimentalresults obtained for the meso-channel compact heat exchangersunder study It was observed that even at low Reynolds numbers(less than 1000), the Nusselt number of the glycol–water flowwithin these meso-channels does not stay constant, as it does
in macro-channels, but increases with Reynolds number Thisdifference can probably be explained by combinations of twoeffects: (1) the surface enhancements on the channel walls, and(2) the transition from macro scale to micro-/mini-scale chan-nels On the glycol–water side of the compact heat exchangers,the enhancements inside the meso-channels promoted turbulentflow and increased heat transfer in comparison with the smoothchannels
There is still a long way to go to fully understand the fluidflow and heat transfer within the micro-/mini-scale channels,especially at the transition and turbulent flow regimes, yet theresults of this study may be used along the way for improvingthe compact heat exchanger design
NOMENCLATURE
A heat transfer area or cross-sectional area (m2)
b intercept in Wilson plot
C constant
Cp specific heat capacity (J/kg-K)
Cf Fanning friction factor
D diameter (m)
f Moody friction coefficient
F correction factor
G mass flux (kg/m2-s)
h heat transfer coefficient (W/m2-K)
h modified heat transfer coefficient (W/m2-K)
u single passage velocity (m/s)
U overall heat transfer coefficient (W/m2.K)heat transfer engineering vol 31 no 1 2010
Trang 16A JOKAR ET AL 15
W width (m)
X Wilson parameter in x direction
Y Wilson parameter in y direction
σ surface area ratio
θ louvers angle (degrees)
n Prandtl number exponent
p Reynolds number exponent
REFERENCES
[1] Kandlikar, S G., and Grande, W J., Evolution of Microchannel
Flow Passages—Thermohydraulic Performance and Fabrication
Technology, Heat Transfer Engineering, vol 24, no 1, pp 3–17,
2003
[2] Webb, R L., and Kim, N., Advances in Air-Cooled Heat
Ex-changer Technology, Journal of Enhanced Heat Transfer, vol 14,
no 1, pp 1–26, 2007
[3] Webb, R L., and Zhang, M., Heat Transfer and Friction in
Small Diameter Channels, Microscale Thermophysical ing, vol 2, no 3, pp 189–202, 1998.
Engineer-[4] Wang, B X., and Peng, X F., Experimental Investigation onLiquid Forced-Convection Heat Transfer Through Microchannels,
International Journal of Heat and Mass Transfer, vol 37, no.
suppl 1, pp 73–82, 1994
[5] Steinke, M E., and Kandlikar, S G., Single-Phase Liquid
Fric-tion Factors in Microchannels, InternaFric-tional Journal of Thermal Sciences, vol 45, no 11, pp 1073–1083, 2005.
[6] Steinke, M E., and Kandlikar, S G., Single-Phase Liquid Heat
Transfer in Plain and Enhanced Microchannels, Proc 4th ternational Conference on Nanochannels, Microchannels and Minichannels, Ireland, pp 943–951, 2006.
In-[7] Jokar, A., Eckels, S J., and Hosni, M H., Evaluation of HeatTransfer and Pressure Drop for the Heater-Core in an Automotive
Heat Pump System, Proc 2004 ASME International cal Engineering Congress and RD&D Expo, Anaheim, CA, no.
[11] Kays, W M., and London, A L., Compact Heat Exchangers, 3rd
ed., McGraw-Hill, New York, 1984
[12] Incropera, F P., and DeWitt, D P., Fundamentals of Heat and Mass Transfer, 4th Ed., John Wiley & Sons, New York,
1996
[13] AHRAE, ASHRAE Handbook of Fundamentals, Chapter 18,
American Society of Heating, Refrigerating and Air-ConditioningEngineers, Atlanta, GA, 2001
[14] Shah, R K., and Wanniarachchi, A S., Plate Heat Exchanger Design Theory, Industrial Heat Exchangers, Lecture Series No.
1991-04, J.-M Bushlin, Von Karman Institute for Fluid Dynamics,Sint-Genesius-Rode, Belgium, 1992
[15] Coleman, H W., and Steele, W G., Experimentation and tainty Analysis for Engineers, John Wiley & Sons, New York,
Uncer-1989
[16] Kays, W M., and Crawford, M E., Convective Heat and Mass Transfer, 2nd Ed., McGraw-Hill Pub., New York, 1980.
[17] Olsson, C O., and Sunden, B., Heat Transfer and Pressure
Drop Characteristics of Ten Radiator Tubes, International nal of Heat and Mass Transfer, vol 39, pp 3211–3220,
Jour-1996
[18] Webb, R L., Principles of Enhanced Heat Transfer, John Wiley
& Sons, New York, 1994
heat transfer engineering vol 31 no 1 2010
Trang 1716 A JOKAR ET AL.
Amir Jokar is an assistant professor in the School
of Engineering and Computer Science at ton State University Vancouver, WA He received his Ph.D in 2004 from Kansas State University, Manhat- tan, KS His research area is in thermal/fluid sciences with more background in micro-/mini-channel heat transfer and fluid flow, thermal systems design and simulation, condensation, and evaporation.
Washing-Steven Eckels is a professor in the Mechanical and
Nuclear Engineering Department at Kansas State University, Manhattan, KS He received his Ph.D in
1993 from Iowa State University, Ames, IA His main research areas include two-phase flow and heat trans- fer, enhanced heat transfer, thermal system modeling, and human thermal comfort He is currently director
of the Institute for Environmental Research at Kansas State University.
Mohammad Hosni is the department head of the
Mechanical and Nuclear Engineering Department at Kansas State University, Manhattan, KS He received his Ph.D in 1989 from Mississippi State University, Mississippi State, MS His area of expertise is thermal and fluid sciences, and he has extensive experience
in both experimental and computational evaluation of indoor air distribution He is a fellow of the ASME and ASHRAE.
heat transfer engineering vol 31 no 1 2010
Trang 18CopyrightC Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903263242
Exergy Efficiency of Two-Phase Flow
in a Shell and Tube Condenser
YOUSEF HASELI, IBRAHIM DINCER, and GREG F NATERER
Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada
This study deals with a comprehensive efficiency investigation of a TEMA “E” shell and tube condenser through exergy
efficiency as a potential parameter for performance assessment Exergy analysis of condensation of pure vapor in a mixture of
non-condensing gas in a TEMA “E” shell and tube condenser is presented This analysis is used to evaluate both local exergy
efficiency of the system (along the condensation path) and for the entire condenser, i.e., overall exergy efficiency The numerical
results for an industrial condenser with a steam–air mixture and cooling water as working fluids indicate significant effects
of temperature differences between the cooling water and the environment on exergy efficiency Typical predicted cooling
water and condensation temperature profiles are illustrated and compared with the corresponding local exergy efficiency
profiles, which reveal a direct (inverse) influence of the coolant (condensation) temperature on the exergy efficiency Further
results provide verification of the newly developed exergy efficiency correlation with a set of experimental data.
INTRODUCTION
Exergy is a measure of the departure of the state of a system
from that of the environment It can be defined as the maximum
obtainable work from the combined system and its environment
Unlike energy, exergy is not conserved, since it is destroyed
by irreversibilities The exergy destruction during a process is
proportionally related to the entropy generation due to these
irreversibilities Dincer [1] has examined exergy from several
perspectives and introduced the exergy analysis method as a
useful tool for developing more efficient energy-resource use
Rosen and Dincer [2] studied the effects of variations in
dead-state properties on the results of energy and exergy analyses In
their case study, a coal-fired electrical generating station was
examined to illustrate the actual influences Although energy
and exergy values are dependent on the intensive properties of
the dead state, it was shown that the main results of energy
and exergy analyses are usually not significantly sensitive to
reasonable variations in these properties
Utilization of exergy analysis has been widely adopted
re-cently in different applications For instance, Ozgener et al
[3, 4] applied the exergy method in geothermal district heating
The authors acknowledge the support provided by the Natural Sciences and
Engineering Research Council of Canada.
Address correspondence to Professor Ibrahim Dincer, Faculty of
Engi-neering and Applied Science, University of Ontario Institute of
Technol-ogy, 2000 Simcoe Street North, Oshawa, Ontario, Canada, L1H 7K E-mail:
of different pitch in the inner pipe of a double-pipe heat changer The effects of process parameters, such as the massflow rate and temperature, on the entropy generation and ex-ergy loss were theoretically and experimentally investigated byNaphon [7] for a horizontal concentric tube heat exchanger
ex-In past work of San and Jan [8] involving a wet cross-flowheat exchanger, the effectiveness, exergy recovery factor, andsecond-law efficiency of the wet heat exchanger were indi-vidually defined and numerically evaluated for various operat-ing conditions Additionally, the exergy-based thermoeconomicmethodology has been developed in [9] and [10] for optimizationpurposes
Exergy analysis of condensation of steam from a mixture ofair in a TEMA “E” shell and tube condenser has been recentlyperformed by Haseli et al [11] They proposed two correlationsfor evaluating the exergy efficiency along the exchanger andentire condenser These correlations are linear functions of a di-mensionless temperature, defined as the ratio of the difference
17
Trang 1918 Y HASELI ET AL.
Figure 1 Plan view of the condenser.
between the coolant and environment temperatures, to the
tem-perature difference between the condensation and environment
Unlike the past work, this article develops a formulation of
exergy efficiency of condensation of a pure vapor in detail,
considering the influence of the presence of a non-condensing
gas in a TEMA “E” shell and tube condenser Illustrative
ex-amples will provide new information about the effects of the
coolant, condensation, and environment temperatures on the
exergy efficiency Furthermore, the correlation of the overall
ex-ergy efficiency proposed in previous work is validated through
comparisons with a set of experimental data
FORMULATION OF EXERGY EFFICIENCY
Exergy (second law) efficiency is formulated herein for
densation of pure vapor in a TEMA “E” shell and tube
con-denser, taking into consideration the effect of non-condensing
gas leakage A plan view of this type of condenser is depicted
in Figure 1 The direction of the hot fluid, which enters from
one end of the heat exchanger and flows through the baffles to
the other end of it, is indicated with lined arrows Condensation
takes place in the shell side, due to contact between the hot fluid
and cold wall of the tubes, through which coolant is flowing
The steady-state exergy rate balance for a control volume can
where ˙Qjrepresents the time rate of heat transfer at the location
on the boundary at the instantaneous temperature of Tj; ˙Wcv
represents the time rate of energy transfer, by work other than
flow work; and ˙me accounts for the time rate of exergy transfer
accompanying mass flow and flow work, with subscripts 1 and 2
representing the inlet and outlet, respectively The specific flow
exergy, e, is evaluated using Eq (2) as follows:
e= (h − ho)− To(s− so)+V2
where h and s denote, respectively, enthalpy and entropy of the
system and ho and so are the values of the same properties, if
Figure 2 Control volume of the condenser, illustrating the inlet and outlet flows.
the system was brought to the dead state isentropically Also, To
refers to the dead state (environment) temperature
In Eq (1), the term ˙Edaccounts for the time rate of exergydestruction due to irreversibilities within the control volume.For an arbitrary control volume of a condenser shown in Figure
1, we have (see Figure 2):
With respect to the conservation of vapor mass,
˙
mv1= ˙mv2+ ˙mcond (6)Equation (5) can be rearranged as follows:
[( ˙mv2+ ˙mcond) ev1− ˙mv2ev2]+ ˙mg(eg1− eg2)− ˙mcondecond
= ˙mc(ec2− ec1)+ ˙Ed (7)or
˙
mv2(ev1− ev2)+ ˙mg(eg1− eg2)+ ˙mcond(ev1− econd)
= ˙mc(ec2− ec1)+ ˙Ed (8)heat transfer engineering vol 31 no 1 2010
Trang 20In other words, ηex is the ratio of the net increase in the flow
exergy of cold fluid (coolant) between the inlet and outlet, to
the net decrease of the flow exergy of hot fluid (binary mixture)
from the inlet to the outlet
The term e2− e1 (net change of flow exergy) is evaluated
using Eq (2) as follows:
e2− e1= h2− h1− To(s2− s1) (10)
Assuming a constant specific heat, cp, the difference of enthalpy
between two states of a process may be written as
h2− h1= cp(T2− T1) (11)Also, depending on whether the flow is incompressible or com-
pressible, the change of entropy between two states of a process
Thus, the flow exergy change in the coolant, vapor, and
non-condensable gas in Eq (9) can be formulated using Eqs (14)–
(16)
ec2− ec1= cp,c
(Tc2− Tc1)− Toln
In the bulk mixture, it is assumed that the temperature is uniform,
so that vapor and non-condensable gas temperatures are the
same in the inlet and outlet of the control volume, i.e., Tv1= Tg1
and Tv2= Tg2 Also,
ev1− econd= hv1− hcond− To(sv1− scond) (17)
The preceding method cannot be used to determine the
dif-ference between inlet flow exergy of steam and the flow exergy
of condensate As condensation occurs at Tcond ≤ Tv1, the ference between the inlet enthalpy of vapor at a temperature of
dif-Tv1 and the condensate enthalpy is determined by the sum ofheat transfer from cooling of the vapor from Tv1 to Tcond andlatent heat released at the condensation temperature It may bewritten in the form of the following expression:
hv1− hcond= cp,v(Tv1− Tcond)+ hfg|T=T cond (18)Also, entropy of the inlet vapor, sv1, may be expressed as thesum of the entropy difference due to the temperature difference
Tv1− Tcond at constant pressure Pv1 and entropy of saturatedvapor at a temperature of Tcond, sv |T=T cond Hence, the entropydifference in Eq (17) can be written as
sv1− scond= sv+ sv|T=T cond − scond|T=T cond
ev1− econd= cp,v(Tv1− Tcond)+ hfg |T=T cond
or
ev1− econd = cp,v
(Tv1− Tcond)− Toln
RESULTS AND DISCUSSION
This section deals with numerical evaluation of the exergyefficiency, ηex, for a typical counter-current TEMA “E” shelland tube exchanger of almost standard industrial design Thecondenser has an exchange area of 30 m2, with 0.438 m diameterand 2.438 m length The shell is divided into eight equal spaces
by seven baffles with a cut of 35.5% The tubes, which are of19.05 mm OD, 14 SWG, are arranged on a 25.4= mm triangularpitch with a characteristic angle of 30◦[13] Superheated steam
at 125◦C and 1 kg/s enters the condenser Condensation of steamtakes place in baffle spaces when contacting the cold wall of thetubes, through which cooling water at a mass flow rate of 62.5kg/s is flowing
It is convenient to present correlations that predict the oration/condensation enthalpy and entropy of water vapor, asrequired in Eq (20) The following correlations are used for
evap-hfgand sfg, based on the ASME International Steam Tables forheat transfer engineering vol 31 no 1 2010
Trang 2120 Y HASELI ET AL.
Figure 3 Variation of exergy efficiency along the condensation path at various
environment temperatures.
Industrial Use [14] They correlate the variables by linear
func-tions of temperature, at which the condensation/evaporation of
water occurs, with less than 0.3% error in the range of 10–70◦C
hfg= 2501.6 − 2.3981T h : [kJ/kg] , T : [◦C] (21)
sfg= 9.0093 − 0.0324T s : [kJ/kg.K] , T : [◦C] (22)
Equation (9) can be utilized to evaluate either exergy
effi-ciency along the condensation path (local exergy effieffi-ciency) or
for the entire condenser (overall exergy efficiency) In the first
case, one needs to obtain the variation of temperatures (including
the shell side, tube side, and condensate) along the
condensa-tion path For this purpose, the numerical model of Haseli and
Roudaki [13], which uses film theory with heat and mass
trans-fer equations, is utilized to determine the relevant parameters
By accounting for a diversity of parameters, such as the effects
of heat and mass transfer rates, variations of physical
proper-ties with temperature and the effects of exchanger geometry on
performance, the model can accurately predict the temperature
profiles and rate of condensation of steam in the presence of
air along a shell and one-path tube condenser The previous
exergy methodology is merged with the past model of Haseli
and Roudaki [13] to obtain the variation of exergy efficiency
along the condenser In order to determine the overall exergy
efficiency of the condenser, one must use the outlet values of
temperatures and mass flow rates, as well as the appropriate
average condensation temperature resulting from the previously
mentioned model
Figure 3 illustrates the variation of exergy efficiency, ηex,
along the condensation path at various environment
tempera-tures for a typical process condition In this figure, a higher
environment temperature at a constant inlet cooling-water
tem-perature results in a lower exergy efficiency along the condenser
In other words, as the temperature difference between the
in-let cooling water and environment, i.e., Tin= Tc,in− To,
de-Figure 4 Similarity of exergy efficiency curves along the condenser.
creases, then ηex is reduced It can also be observed from thefigure that ηex decreases from the entrance of the steam–airmixture to the location after the midpoint in the condenser—aregion around the fifth baffle space Then it starts to increasefrom this point to the outlet of the heat exchanger It seemsthat there is a relation between the condensation heat trans-fer and the exergy efficiency Past studies [13, 15] have shownthat from the region where exergy efficiency is a minimum,the condensation and heat transfer rates diminish significantly.Beyond this region, the relative role of sensible heat transferincreases
Figure 4 depicts the set of curves that correspond to the same
T It is seen that when T= 0, the exergy graphs decreaseconsistently along the exchanger An important result from Fig-
ure 4 is that T has a significant influence on the exergy
ef-ficiency, rather than the inlet cooling-water temperature itself
As T increases, ηexbecomes higher This result agrees withthe definition of exergy When a system carries more exergy, itdeviates more from the environment Based on these results, itcan be understood why ηexdecreases from the entrance of thesteam–air mixture to around the fifth baffle space, consideringthe configuration of the condenser in this study (a counterflowtype) As cooling water flows in the opposite direction of thesteam–air mixture, its temperature increases from the last bafflespace to the first one An example is shown in Figure 5 At a
given environment temperature, the local T = (Tc− To) willincrease in the same direction Except for the region where thecurves of Figure 3 have a positive slope, it seems that the lo-
cal T plays a dominant role in the variation of ηex However,
as reported in our previous work [11], in addition to T, the
condensation temperature may influence the exergy efficiency.Additional explanation about this trend will be given hereafter
In our recent paper [16], the predictive model exhibits error inthe last baffle spaces (as an example, in Figure 4, with bafflenumbers 6 to 8) This may explain the divergence of curves afterheat transfer engineering vol 31 no 1 2010
Trang 22Y HASELI ET AL 21
Figure 5 Variation of cooling water temperature along the condensation path.
baffle number 6 in Figure 4 Reference [16] presents the detailed
derivation of an explicit expression for the exergy efficiency
Figure 6 shows the effects of air mass flow rate on ηex It
reveals that higher exergy efficiency may result from higher air
leakage Irreversibility of the process, as characterized by the
entropy generation rate in the condenser, diminishes with air
leakage since this reduces the rate of condensation heat transfer
[17] Based on the relation between the local T and exergy
efficiency, one expects that the profile of cooling water
temper-ature corresponding to the higher air mass flow rate in Figure 6
is higher The predicted cooling water temperature and
conden-sation temperature profiles along the condenser are depicted in
Figure 7 for the same air mass flow rates shown in Figure 6
As seen in Figure 7, since the temperature profiles of cooling
water are approximately the same in the first few baffle spaces,
Figure 6 Effect of air mass flow rate on exergy efficiency along the
conden-sation path and predicted outlet steam mass flow rates.
Figure 7 Predicted cooling-water and condensation temperatures along the condensation path at three different air mass flow rates.
the local T is the same (environment temperature is 10◦C inFigure 6) in this region for all cases On the other hand, a signif-icant difference between the condensation temperature profilesoccurs, as a higher air mass flow rate results in a lower conden-sation temperature in the first baffle spaces Thus, comparingcurves of Figures 6 and 7, it may be implied that the conden-sation temperature may have an inverse impact on the exergyefficiency, while other process parameters remain nearly con-stant A comparison of the temperature and ηexprofiles in thelast baffle space in Figures 6 and 7, respectively, shows that thecondensation temperature may inversely influence the exergyefficiency, since the cooling-water temperature has the same
profile for all cases in this region, so T is constant The
pre-dicted outlet steam mass flow rates are also given in Figure 6.Air is a non-condensable component that provides resistance toheat and mass transfer processes Increasing the air mass flowrate leads to a lower heat transfer rate and condensation rate.Further discussion may be found elsewhere (such as [15]) Inorder to explain the trend of curves in Figure 3, as long as allsteam has not condensed, ηexdecreases along the condensationpath When condensation of steam finishes (usually by the lastbaffle space), ηexwill increase
Dependence of the overall exergy efficiency of the condenser,
ηex,overall, on the environment temperature at different inlet ing water temperatures is illustrated in Figure 8 At a fixed ex-ternal environment temperature, a higher inlet cooling watertemperature leads to a higher overall exergy efficiency Sincethe cooling water carries more exergy at the higher inlet tem-perature (and therefore a higher mean temperature of coolingwater), the irreversibility within the system will diminish Inaddition, in order to establish a specific ηex,overallat different en-vironment conditions, it is required to change the inlet coolingwater temperature A similarity between the illustrated curvescan be observed in Figure 8 A common trend that agrees withprevious results is that ηex,overallincreases when the differenceheat transfer engineering vol 31 no 1 2010
Trang 23cool-22 Y HASELI ET AL.
Figure 8 Dependence of the overall condenser exergy efficiency on
environ-ment temperatures at various inlet cooling-water temperatures.
between the inlet cooling-water temperature and environment
temperature increases
In a previous study [11], the following correlation was
devel-oped to predict the overall exergy efficiency of a heat exchanger,
where condensation of steam takes place in the presence of air:
correlation described in Eq (23) are compared with past
ex-perimental data of Webb et al [18], which were taken from
Figure 9 Illustration of the experimental configuration of Webb et al [18].
Table 1 Inlet and outlet measured performance parameters
Cooling water temperature ( ◦C) 47.63 51.57Cooling water mass flow rate (kg/s) 43.45
Note Source: Ref [18].
a similar industrial-scale exchanger Webb and his coworkersmeasured an extensive set of experimental data of condensation
of steam and steam–air mixtures at atmospheric and reducedpressures in TEMA “E” shell and “J” shell condensers Theexperiments consist of a condenser, an after condenser, cool-ing water circuit, vacuum pump, and plant instrumentation Theexperimental setup is illustrated in Figure 9 Steam was gener-ated by a boiler at a rate of up to 1 MW and the system wasusually operated under partial vacuum, maintained by a liquidring pump Cooling water was circulated in a pump-around withcontrolled addition to give the desired temperature Calibratedthermocouples were used to measure the coolant and vapor inletand outlet temperatures The condensate flow rate was measured
by direct collection in each half of the condenser and the coolingwater flow rate by an orifice plate A typical reported set of mea-sured performance parameters of a TEMA “E” shell condenser
is given in Table 1
Comparisons of Eq (23) with experimental data (based onTable 1) are illustrated in Figure 10 at various ambient tem-peratures (represented by the numbers beside the points) in therange of 10–45◦C From this figure, it is apparent that Eq (23)overpredicts the second law efficiency of the exchanger within
an offset of –15% This overprediction is even lower at coolerambient temperatures Close agreement between this correlationand experimental data over a range of temperatures in Figure 10
Figure 10 Comparison of proposed correlation, Eq (23), with measured data
of Webb et al [18].
heat transfer engineering vol 31 no 1 2010
Trang 24Y HASELI ET AL 23
provides useful validation of the predictive model and resulting
correlation in Eq (23)
CONCLUSIONS
The exergy efficiency has been formulated for condensation
of a binary mixture with one non-condensable component in
a TEMA “E” shell and tube condenser The exergy efficiency
model may be expressed as a function of the inlet and outlet
temperatures and mass flow rates of both streams across the
boundary of a control volume, as well as the condensation
tem-perature Numerical results are obtained through a combination
of the exergy formulation with a recent calculation method for
an industrial-scale countercurrent condenser, where
condensa-tion of steam occurs in the presence of air with cooling water
as a coolant The results show that the temperature difference
between the cooling water and environment has a considerable
influence on the exergy efficiency, whereas the condensation
temperature has an inverse effect on the exergy efficiency In
addition, when the inlet cooling water temperature is greater
than the environment temperature, the exergy efficiency
de-creases along the heat exchanger, as long as condensation of
steam occurs However, when condensation of steam almost
finishes, it tends to increase The correlation of overall exergy
efficiency of the condenser proposed in previous work was also
verified through a comparison of the model results with a set of
hfg condensation latent heat, kJ/kg
ho enthalpy at reference environment (dead state),
T difference between a given cooling water
tempera-ture and environment temperatempera-ture,◦C
T in difference between cooling water temperature at
the inlet of control volume and environment perature,◦C
tem-θ dimensionless temperature, Eq (23)
[2] Rosen, M A., and Dincer, I., Effect of Varying Dead-State
Prop-erties on Energy and Exergy Analyses of Thermal Systems, ternational Journal of Thermal Science, vol 43, pp 121–133,
In-2004
[3] Ozgener, L., Hepbasli, A., and Dincer, I., Energy and ExergyAnalysis of Geothermal District Heating Systems: An Applica-
tion, Building and Environment, vol 40, pp 1309–1322, 2005.
[4] Ozgener, L., Hepbasli, A., and Dincer, I., Effect of ReferenceState on the Performance of Energy and Exergy Evaluation of
Geothermal District Heating Systems: Balcova Example, Building and Environment, vol 41, pp 699–709, 2006.
[5] Fiaschi, D., and Manfrida, G., Exergy Analysis of the Semi-Closed
Gas Turbine Combined Cycle (SCGT/CC), Energy Conversion and Management, vol 39, pp 1643–1652, 1998.
[6] Akpinar, E K., Evaluation of Heat Transfer and Exergy Loss in aConcentric Double Pipe Exchanger Equipped with Helical Wires,
Energy Conversion and Management, vol 47, pp 3473–3486,
2006
heat transfer engineering vol 31 no 1 2010
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[7] Naphon, P., Second Law Analysis on the Heat Transfer of the
Horizontal Concentric Tube Heat Exchanger, International
Com-munications in Heat and Mass Transfer, vol 33, pp 1029–1041,
2006
[8] San, J Y., and Jan, C L., Second-Law Analysis of a Wet Cross
flow Heat Exchanger, Energy, vol 25, pp 939–955, 2000.
[9] Selbas¸, R., Kızılkan, O., and S¸encan, A., Thermoeconomic
Opti-mization of Subcooled and Superheated Vapor Compression
Re-frigeration Cycle, Energy, vol 31, pp 1772–1792, 2006.
[10] Accadia, M D., Fichera, A., Sasso, M., and Vidiri, M.,
Deter-mining the Optimal Configuration of a Heat Exchanger (With
a Two-Phase Refrigerant) Using Exergoeconomics, Applied
En-ergy, vol 71, pp 191–203, 2002.
[11] Haseli, Y., Dincer, I., and Naterer, G F., Exergy Analysis of
Con-densation of a Binary Mixture With One Non-Condensable
Com-ponent in a Shell and Tube Condenser, Journal of Heat Transfer–
Transactions of the ASME, vol 130, no 8, pp
084504-1–084504-5, 2008
[12] Moran, M J., and Shapiro, H N., Fundamental of Engineering
Thermodynamics, 5th ed., Wiley, Hoboken, NJ, 2004.
[13] Haseli, Y., and Roudaki, S J M., A Calculation Method for
Anal-ysis Condensation of a Pure Vapor in the Presence of a
Non-Condensable Gas on a Shell and Tube Condenser, ASME Heat
Transfer/Fluids Engineering Summer Conf., Charlotte, North
Car-olina, vol 3, pp 155–163, 2004
[14] ASME International Steam Tables for Industrial Use, American
Society of Mechanical Engineering, CRTD, New York, vol 58,
2000
[15] Haseli, Y., and Roudaki, S J M., Simultaneous Modeling of Heat
and Mass Transfer of Steam–Air Mixture on a Shell and Tube
Condenser Based on Film Theory, ASME Summer Heat Transfer
Conf., Las Vegas, Nevada, vol 2, pp 251–259, 2003.
[16] Haseli, Y Dincer, I., and Naterer, G F., Thermal Effectiveness
Correlation for a Shell and Tube Condenser with Noncondensing
Gas, AIAA Journal of Thermophysics and Heat Transfer, vol 22,
no 3, pp 501–507, 2008
[17] Haseli, Y., Dincer, I., and Naterer, G F., Entropy Generation of
Vapor Condensation in the Presence of a Non-Condensable Gas
in a Shell and Tube Condenser, International Journal of Heat and
Mass Transfer, vol 51 no 7–8, pp 1596–1602, 2008.
[18] Webb, D R., Dell, A J., Williams, J., and Stevenson, R W.,
An Experimental Comparison of the Performance of TEMA ‘E’
and ‘J’ Shell Condensers, Chemical Engineering Research and
Design, vol 75, pp 646–651, 1997.
Yousef Haseli received his master’s degree in
me-chanical engineering from the University of Ontario, Institute of Technology, Ontario, Canada (received the Governor General’s Gold Medal) He received his B.Sc degree in mechanical engineering, ther- mal power plant option (first class honors), from the Power and Water University of Technology, Tehran, Iran His research interests are mathematical model- ing of condensation of steam–air mixture, second law analysis of energy systems, thermodynamic model- ing of integrated gas turbine SOFC power plants, and transport phenomena in fluidized beds He has published more than 20 articles in journals and conference proceedings.
Ibrahim Dincer is a full professor in the Faculty
of Engineering and Applied Science at University of Ontario, Institute of Technology, Ontario, Canada Renowned for his pioneering works, he has authored and co-authored several books and book chapters, over 450 refereed journal and conference papers, and numerous technical reports He has chaired many na- tional and international conferences, symposia, work- shops, and technical meetings He has delivered over
100 plenary, keynote, and invited lectures He is an active member of various international scientific organizations and societies, and serves as editor-in-chief, associate editor, regional editor, and editorial board member on various prestigious international journals He is a recipient
of several research, teaching, and service awards, including the Premier’s search excellence award in 2004 He has made innovative contributions to the understanding and development of sustainable energy technologies and their implementation.
re-Greg Naterer is a Tier 1 Canada Research Chair in
Advanced Energy Systems and a professor of chanical engineering at the University of Ontario In- stitute of Technology He is an associate dean in the Faculty of Engineering and Applied Science He re- ceived his Ph.D in mechanical engineering from the University of Waterloo in 1995 His research interests involve design of energy systems, hydrogen technolo- gies, and heat transfer, with over 210 journal and con- ference publications in these fields He has authored
me-a book entitled Heme-at Trme-ansfer in Single me-and Multiphme-ase Systems (CRC Press, 2003), as well as another book in 2008 entitled Entropy Based Design of Fluids Engineering Systems He is a fellow of the Canadian Society for Mechanical
Engineering and an associate fellow of the American Institute of Aeronautics and Astronautics.
heat transfer engineering vol 31 no 1 2010
Trang 26CopyrightC Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903263291
Optimum Fins Spacing and Thickness
of a Finned Heat Exchanger Plate
MARCELO MOREIRA GANZAROLLI and CARLOS A C ALTEMANI
Department of Energy, State University of Campinas, Campinas, Brazil
The thermal design of a counterflow heat exchanger using air as the working fluid was performed with two distinct goals:
minimum inlet temperature difference and minimum number of entropy generation units The heat exchanger was constituted
by a double-finned conductive plate closed by adiabatic walls at the fin tips on both sides The hot and cold air flows were
considered in the turbulent regime, driven by a constant pressure head The thermal load was constant, and an optimization
was performed to obtain the optimum fin spacing and thickness, according to the two design criteria A computer program
was employed to evaluate the optimum conditions based on correlations from the literature The results obtained from both
design criteria were compared to each other A scale analysis was performed considering the first design goal and the
corresponding dimensionless parameters were compared with the results from the correlations.
INTRODUCTION
Large machine tools have cabinets containing the electronics
used for their operation and control The electric power
dissi-pated in the cabinets (q) must be removed and environmental air
is the most convenient cooling fluid, but in some shop floors it
may be contaminated by oil vapor resulting from the
manufac-turing processes This may cause an undesirable oil deposition
on the electronic equipment in the cabinets To prevent this
problem and to keep air as the cooling fluid, the cabinets may
be sealed, so that the air inside will not be contaminated by oil
The heat load (q) may then be removed by recirculating the air
inside the cabinet through a heat exchanger located at a
cabi-net wall, as indicated in Figure 1 Environmental air, forced in
counterflow in a single pass, becomes the cooling fluid, without
any contact with the electronic equipment A heat exchanger
constituted by a double-finned aluminum plate was selected for
this analysis, assuming that the fin tips were closed by adiabatic
walls The finned plate has length L and thickness W The
rect-angular channels were defined by two adjacent fins (of thickness
t ), with spacing s and height H The main purpose of this work
was to select the fin thickness (t) and spacing (s) to minimize the
recirculating hot air inlet temperature T h,i, indicated in Figure
1, considering a constant cold inlet temperature T c,i The heat
Address correspondence to Marcelo Moreira Ganzarolli, Department of
Energy, State University of Campinas (UNICAMP–FEM–DE), Caixa Postal
6122, CEP 13.083-970, Campinas–SP, Brazil E-mail: ganza@fem.unicamp.br
load and the heat exchanger plate length were constant tionally, the entropy generation rate, due to the irreversible heatexchanger operation, was evaluated by a dimensionless number
Addi-of entropy generation units (N s), as described by Bejan [1] The
configurations for minimum N s were compared with those for
minimum T h,i.The optimization of stacks of heated parallel plates and offinned heat sinks with straight rectangular fins employed inelectronics cooling was performed in previous works Bejanand Sciubba [2] obtained an optimal spacing to maximize theheat transfer from a stack of parallel boards cooled by lam-inar forced convection The pressure head was kept constantacross the stack and the calculated results were based on corre-lations for the Nusselt number and for the pressure losses Mereu
et al [3] performed a numerical simulation to optimize the ing between heat generating boards cooled by laminar forcedconvection Bejan and Morega [4] considered the cooling of astack of plates by turbulent forced convection For both regimes,
spac-a scspac-ale spac-anspac-alysis bspac-ased on correlspac-ations for the spac-asymptotic its of the board-to-board spacing was presented This analysisdetermined, in an order-of-magnitude sense, the adequate di-mensionless groups for the optimal spacing and for the maxi-mum power dissipation Bar-Cohen and Jelinek [5] presented aprocedure to optimize arrays of longitudinal rectangular fins forthe least material They illustrated their work with several exam-ples of the optimization procedure and quantified the theoreticalthermal performance of some particular natural and forced con-vection air-cooled fin arrays An optimization of plate fin heatsinks for electronic applications using the entropy generation
lim-25
Trang 2726 M M GANZAROLLI AND C A C ALTEMANI
q
Th,o
Th,i
Figure 1 Heat exchanger schematic view.
minimization was presented by Culham and Musychka [6]
Relevant design parameters, including geometric parameters,
heat dissipation, material properties, and flow conditions, were
included to characterize a heat sink for minimum entropy
gen-eration Their procedure was based on the use of correlations
for heat transfer and fluid friction Ganzarolli and Altemani [7]
presented an optimization of the fin spacing (s) of a
double-finned counterflow heat exchanger with fixed fin thickness (2
mm) The design was based on correlations and the purpose
was either minimum hot fluid inlet temperature (T h,i) or
mini-mum dimensionless entropy generation (N s) It was verified that
the designs based on both criteria were quite similar
In the present work, a new optimization was performed to
determine simultaneously the fin thickness (t) and their
spac-ing (s) to obtain either the minimum T h,i or the minimum N s
This investigation was performed because as the fins become
thicker, the fin efficiency improves, but their number on the heat
exchanger plate is reduced Since these two effects are opposed,
there must be an optimum fin thickness and an optimum fin
spac-ing A scale analysis based on correlations for the asymptotic
behavior of the heat exchanger operation evaluated the optimum
design for minimum T h,i A design procedure based on
correla-tions for turbulent heat transfer and fluid flow, together with the
ε-NTU method of heat exchanger analysis [8], was applied to
minimize either T h,i or the dimensionless entropy generation N s
SCALE ANALYSIS FOR MINIMUM T
For a counterflow heat exchanger with equal heat capacity
rates, the maximum temperature difference T = (T h,i – T c,i)
between the hot and cold inlet fluids can be expressed by
1
The existence of minimum T for a fixed total heat transfer
rate q or, conversely, maximum heat transfer rate q when T is
fixed can be verified examining the sum in parentheses in Eq
(1) Imposing a constant pressure head P to the heat exchanger
channels, two asymptotic limits can be identified
(i) As the spacing s decreases, the total heat transfer area A
increases and the outlet temperature of one stream in a terflow heat exchanger approaches the inlet temperature of
coun-the ocoun-ther, so that coun-the temperature difference T → q/ ˙mc p.(ii) In the opposite limit, due to a mass flow rate and heat capac-ity increase, the logarithmic mean temperature difference
approaches T and consequently T → q/UA.
Since in Eq (1) the two terms in parentheses vary in opposite
directions with the spacing s, it is expected that T will be
minimum at an intermediate spacing, with an order of magnitude
obtained from the condition U A ∼ = ˙mc p.The order of magnitude of these two terms was obtained as-suming fully developed turbulent flow regime Turbulent forcedconvection was also assumed by Bejan and Morega [4] to reportthe optimal spacing of a stack of parallel plates cooled by forcedconvection The present analysis applies the method introduced
by Bejan and Sciubba [2] to a counterflow heat exchanger withequal heat capacity rates The analysis was performed per unit
width in the direction normal to the fin profile L × H.
Scale for m.c˙ p A longitudinal force balance for the flow
in the rectangular channel (cross section s × H) leads to an expression for the channel flow mean velocity V :
m = ρ s H V N f Substituting V from Eq (2) and defining
β = s/H, an expression for the flow thermal capacity is
Scale for UA Some simplifications were adopted to
deter-mine the order of magnitude of the product UA First, the total
heat transfer area per unit plate width was calculated consideringonly the fins area Also, the thermal resistance across the plate
thickness W was assumed negligible With these assumptions,
an order of magnitude of UA was
¯h2
(s + t)ηf (4)
where ¯h is the average convection coefficient in the rectangular
channel and ηf is the fin efficiency This coefficient can be
evaluated evoking the Colburn analogy (f /2 = St Pr 2/3) betweenmomentum and heat transfer Substituting the expression for
V from Eq (2) in the Stanton number (St = ¯h/ρ c p V) andheat transfer engineering vol 31 no 1 2010
Trang 28M M GANZAROLLI AND C A C ALTEMANI 27
introducing the heat transfer coefficient ¯h in Eq (4), we obtain
The order of magnitude of the optimal spacing s that minimizes
T was found from the condition U A ∼ = ˙mc p Hence, from
This expression, when replaced in Eqs (3) and (5), gave rise to
a dimensionless form of Eq (1):
For each fin thickness t there is an optimum spacing s for
minimum T The last two factors of Eq (7) are functions
of the fin thickness, since the fin efficiency ηf depends on this
parameter Introducing the fin efficiency for straight rectangular
fins with adiabatic tip, the product of these two factors was
defined as the function F (t/s, b):
F (t/s, b) = (1 + t/s)
tanh
The order of magnitude of the parameter b can also be obtained
from the scale analysis The heat transfer coefficient ¯h in the
definition of b was expressed with the help of the Colburn
analogy and the friction factor was eliminated using Eq (6)
The result was expressed by
(9)
The function F defined by Eq (8) exhibits a minimum for each
value of the parameter b, as shown in Figure 2 This indicates
an optimal ratio (t/s) opt which minimizes T The dashed line
in Figure 2 indicates the locus of this optimum, which, in the
range of the parameter b from 2 to 20, can be expressed by the
following correlation:
(t/s) opt = 0.33 b −0.9 (10)
Using the Blasius equation (f = 0.079 Re −1/4) for the friction
factor f , and Eq (2), the ratio (s/L) optpresented in Eq (6) was
1.201.401.60
spac-The scales derived in Eqs (7), (9), and (11) were used to
define new dimensionless variables T *, s*, and b* as follows:
the minimum temperature difference T
FLUID FLOW AND HEAT TRANSFER
The heat exchanger operating conditions were obtained by
an iterative process based on correlations for turbulent fluid flowand heat transfer, together with the effectiveness method Thechannels flow rate was obtained matching the available constantpressure head with the channel flow losses
The same pressure head was imposed to both sides of theheat exchanger The air properties for the hot and cold streamswere evaluated at their averaged mixed mean temperatures at theentrance and exit of the heat exchanger Due to this procedure,
the hot (recirculating) fluid heat capacity rate (C h) was always
slightly smaller than that of the cold (environmental) fluid (C c),mainly due to the effect of temperature on the air density.heat transfer engineering vol 31 no 1 2010
Trang 2928 M M GANZAROLLI AND C A C ALTEMANI
The fins’ height (H ) and their spacing (s) defined the
rect-angular channels geometry on both sides of the heat exchanger
plate A model for flow losses in the rectangular channels was
employed to obtain the hot and cold flow rates For a fixed
ther-mal load (q) per unit plate width, the heat transfer problem was
to determine the inlet and exit flow temperatures The heat
ex-changer global thermal conductance (UA) per unit plate width
was assumed one-dimensional
On each side, the overall surface efficiency ηo was obtained
from the finned and the total heat transfer areas and from the fin
efficiency, considering adiabatic fin tips
The global thermal conductance (UA) and the minimum heat
capacity rate (C h) were used to evaluate the number of
trans-fer units (NTU = UA/C h) The effectiveness (ε) resulted from
appropriate ε–NTU relations (Kays and London [9]) for the
counterflow heat exchanger The hot fluid inlet temperature was
obtained invoking the effectiveness definition
T h,i = T c,i+ q
ε C h
(16)
An energy balance, assuming no heat losses from the heat
ex-changer, furnished the outlet temperatures of both fluid streams
Pressure Drop Correlations
The interfin channels head loss was associated to the flow in
rectangular ducts with aspect ratio β= (s/H) ≤ 1 During the
search for the optimum s, if the ratio (s/H ) > 1, then the aspect
ratio was changed to (H /s) ≤ 1 The friction losses for the
rectangular channels were evaluated from the apparent Fanning
friction factor f app presented by Phillips [10] It includes the
pressure drop due to flow momentum change and the increased
wall shear in the entrance region, so that the contraction losses
are already included For turbulent flow,
duct hydraulic diameter (D h), and φ is a correction function,
presented by Jones [11], based on the duct aspect ratio β:
φ=2
3 +11
The pressure loss at the duct exit was obtained from values of
the expansion coefficient K efor parallel plates (β= 0) and for
square ducts (β= 1), presented in Kays and London [9] This
coefficient depends on the contraction area ratio [σ= s/(s +
t)] and the Reynolds number For turbulent flows, it changes
Table 1 Coefficients for Eq (20)
only about 10% when the Reynolds number varies from 2,000
to infinity, so that this weak dependence was disregarded and a
mean value of K e was adopted for the whole Re range Thus,for any aspect ratio (0≤ β ≤ 1), a correlation was obtained bylinear interpolation from quadratic fittings obtained by Reis andAltemani [12] for the parallel plates and the square duct:
Convective Heat Transfer Correlations
For turbulent flow in the rectangular channels, the Nusseltnumber was obtained from two modifications of the Gnielinskiequation (Kays and Crawford [8]) First, the Reynolds numberwas corrected by Eq (19), and second, the apparent frictionfactor, defined by Eq (17), was used
N u∞= (f app / 2)(φRe − 1000)Pr
1+ 12.7(f app /2)0.5 (Pr 0.67− 1) (22)The effect of the simultaneous heat and flow development wastaken into account using the correction indicated by Phillips[10] for rectangular ducts:
The average convection coefficients ¯h for the hot and cold
streams were evaluated separately, due to the temperature ence on the air properties, and the resulting values were replaced
influ-in Eq (15) to obtainflu-in the global thermal conductance (UA).
Analogous procedures based on similar correlations for sure drop and heat transfer were employed before by Knight
pres-et al [13] and by Tomazpres-eti and Altemani [14], with good sults when compared to experimental data In the present work,
re-a single test wre-as performed through numericre-al computre-ationre-alfluid dynamics (CFD) simulation, in order to compare the re-sults with those from the correlations The case for minimum
T h,i, considering a constant pressure head equal to 20 Pa andfin height equal to 40 mm, was simulated with the commercialheat transfer engineering vol 31 no 1 2010
Trang 30M M GANZAROLLI AND C A C ALTEMANI 29
software Phoenics [15] A three-dimensional numerical
simula-tion indicated the same flow rate and a heat transfer coefficient
14% larger than that of the correlations
ENTROPY GENERATION
An ideal gas behavior was assumed for both air streams in
the heat exchanger Thus, the total entropy generation rate due
to heat transfer and viscous flow was expressed by
The inlet pressures (P c,i and P h,i) were assumed equal to 105Pa
and the pressure drops along the channels were comparatively
very small ( P < 50 Pa), so that for each stream
The dimensionless number of entropy generation units (Bejan
[1]) was obtained dividing Eq (24) by the smallest heat capacity
rate (hot fluid)
1−T h
T h,i
(26a)and
P c
P c,i
(26b)
According to Bejan [1], N s,T includes the T effect due to
heat capacity imbalance (C h < C c) and that due to the heat
exchanger finite area for heat transfer
The previously evaluated inlet and outlet air mixed mean
tem-peratures and pressure drops were used to evaluate N sin order
to compare the heat exchanger thermal design with minimum
N s to that with minimum T h,i
RESULTS AND DISCUSSION
The rectangular fins distribution on both sides of the heatexchanger plate indicated in Figure 1 was uniform and sym-metric The results were obtained for a unit width of the heatexchanger plate, considering the following set of geometric andphysical parameters, in order to illustrate the described proce-dure The heat exchanger plate was a double-finned aluminum
alloy (k p = 175 W/m-K) plate with a length (L) of 1 m and a
thickness (W) of 4 mm The cold environmental air at the heat
exchanger inlet was assumed at T c,i = 30◦C and the thermal
load was constant, q = 800 W Four fin heights H (2, 3, 4, and 6 cm) were considered, assuming a range of fin thickness (t) from
0.3 mm to 3 mm, with 0.1-mm increments The fin spacing wasthen varied, always checking whether the resulting flow was inthe turbulent regime The results were obtained assuming thesame pressure head on both sides of the heat exchanger plate,varying in the range from 10 Pa to 40 Pa The operating pointswere obtained iteratively, matching the available head with theflow head losses
The inlet temperatures difference T = (T h,i − T c,i) depends
on the fin spacing, as indicated in Figure 3 for a 6-cm high finarray subject to a constant pressure head of 40 Pa For any finthickness, a reduction of the fin spacing is always accompanied
by smaller flow rates due to narrower ducts It also causes,however, an increase of the total heat transfer area, resulting in
a trade-off effect with minimum T for a particular spacing.
Similarly, for any fin height, the optimum fin spacing is not muchinfluenced by the fin thickness—too thin fins have low thermalefficiency, but too thick fins reduce the total heat transfer area.The optimum thickness reflects a compromise between thesetwo effects Figure 4 also indicates that for the fin height tested,
s [mm]
8101214
Figure 3 Effect of fin spacing on T
heat transfer engineering vol 31 no 1 2010
Trang 3130 M M GANZAROLLI AND C A C ALTEMANI
Figure 4 Effect of fin thickness on T
the optimum thickness for the smallest T is attained very
smoothly within a narrow fin thickness and spacing range This
corroborates the results presented previously in Figure 2 The
optimum fin thickness increases slightly with the fin height,
while the corresponding optimum fin spacing presents a slight
decrease with the fin height, as predicted by Eq (11) Figure 4
also presents the design based on minimum N s; these results are
detailed with the discussion of Figure 7
The results obtained from the heat transfer and fluid flow
correlations are now compared with the predictions of the scale
analysis For all the numerical tests performed, the
dimension-less parameters T∗and s∗associated to the optimum conditions
for minimum T collapsed in the shadowed region indicated in
Figure 5, which can be specified by
0.13 ≤ (s∗
opt≤ 0.18 and 4.3 ≤ (T∗
min≤ 5.1 (27)
These results confirmed that the scale analysis and the
cor-relations indicated the same order of magnitude for minimum
temperature difference and optimum spacing The optimum
con-ditions also correspond to a value of b∗, defined by Eq (14),
around 104 These values are useful to estimate the optimum fin
spacing, thickness, and height, as well as the minimum T.
Consideration is now given to the heat exchanger design
based on minimum N s The contributions of N s,T and N s,P
to N shave a distinct behavior Considering a constant fin height,
the results presented in Figure 6 indicate that for each fin
thick-ness there is an optimum fin spacing for minimum N s,T This
can be justified considering an approximation to Eq (26a) The
heat capacity ratio (C r = C h /C c) was always close to unity
and the temperature changes for both fluid streams from inlet to
outlet (T c and T h) were similar and small compared to their
0.1 0.12 0.14 0.16 0.18 0.2
S*
44.44.85.25.66
Figure 5 Region of best thermal design.
inlet absolute inlet temperatures (T c,i and T h,i), so that
T c,i − 1
T h,i
(28)For constant pressure head and thermal load, the fluid flow rate
decreases for narrower fin spacing, causing an increase of T c
At the same time, T h,iinitially decreases to a minimum, as
indi-cated in Figure 3 Thus, Eq (28) predicts that minimum N s,T
will be attained for a fin spacing larger than that for minimum
s [mm]
0.81.21.622.4
Figure 6 Effect of fin spacing on N s,T.
heat transfer engineering vol 31 no 1 2010
Trang 32M M GANZAROLLI AND C A C ALTEMANI 31
Figure 7 Effect of fin thickness on N s.
T h,i Decreasing even more the fin spacing, Eq (28) predicts that
both T c and T h,iwill contribute to a pronounced increase of
N s,T The results also indicated that the contributions of N s,P
to N s were almost constant, and minimum N swas attained with
the same fin spacing as for minimum N s,T
The effects of fin height and thickness on N s are illustrated
in Figure 7 for a constant pressure head For any fin height,
minimum N s is attained very smoothly with the fin thickness
As the fins become taller, N sdecreases due to larger flow rates
and smaller fluid temperature changes from inlet to outlet, and
minimum values are attained for thicker fins due to their higher
efficiency Compared to the data presented in Figure 4 for
min-imum T h,i , the solution for minimum N scorresponds to thicker
fins with larger spacing These results indicate a trend for larger
flow rates with smaller fluid temperature changes and more
ef-ficient fins, compatible with the minimum N sdesign goal The
N s -based design presents a slightly higher T h,i (smaller than
0.5◦C), but due to the larger fin spacing, the heat exchanger
plate requires less material
CONCLUSIONS
The present analysis was performed to improve the
produc-tivity in the early stages of a heat exchanger design cycle The
analysis was based on turbulent fluid flow and heat transfer
cor-relations from the literature The results obtained are as good as
the correlations employed in the model, but they are very useful
in the early phases of a thermal design The presented procedure
constitutes a fast way to select competing designs and suggested
improvements, as well as to analyze eventual changes of
oper-ational conditions From these studies, the thermal design must
proceed with a more detailed analysis employing a CFD method
on the previously selected alternatives for the heat exchanger.The thermal design of a counterflow heat exchanger was
performed considering two distinct design goals: minimum T and minimum N s The design variables were the fin thicknessand spacing of a double-finned heat exchanger plate For anyheat exchanger plate geometry and pressure head information,the fluid flow rates across the heat exchanger and the overallthermal conductance were obtained from turbulent flow andheat transfer correlations For fixed thermal load, the standardε–NTU method was used to obtain the desired temperatures
at the heat exchanger inlet and outlet The simulations werevery fast in a microcomputer and a parametric change of the finthickness and spacing was used to select the best configurationfor a specific design goal
The obtained results indicated that the desired minimum,
either T or N s, was attained very smoothly with the fin ness and more sharply with the fin spacing Thus, the adequatethermal design should carefully select the optimum fin spacing.The results from the correlations served also to infer values forthree dimensionless parameters obtained from a scale analysis
thick-performed for the condition of minimum T From these, a
first approximation to optimum fin spacing, fin thickness, and
minimum T can be obtained The design based on minimum
N s indicated optimum values for thicker fins and larger
spac-ing than the design for minimum T When compared to each other, the design based on minimum N s presented a penaltydue to a slightly higher fluids inlet temperatures difference (lessthan 0.5◦C) It had, however, the advantage of smaller heat ex-changer plate mass, due to larger fin spacing, reducing materialsand manufacturing costs
NOMENCLATURE
A heat transfer area per unit plate width, m2
b inverse of the dimensionless fin height, Eq (8)
C heat capacity rate, W/K
C r heat capacity ratio (Ch/C c)
c p fluid specific heat at constant pressure, J/kg-K
D h hydraulic diameter, m
F dimensionless function, Eq (8)
f Fanning friction factor
H fin height, m
¯h average convection coefficient, W/m2-K
K e expansion coefficient, Eq (20)
k fluid thermal conductivity, W/m-K
k p plate thermal conductivity, W/m-K
L heat exchanger plate length, m
˙
m mass flow rate per unit plate width, kg/s
N f number of fins per unit plate width
N s number of entropy generation units, Eq (26)
N u average Nusselt number, Eq.(23)
Nu∞ fully developed Nusselt number
NTU number of transfer unitsheat transfer engineering vol 31 no 1 2010
Trang 3332 M M GANZAROLLI AND C A C ALTEMANI
P pressure, Pa
P pressure difference, Pa
Pr fluid Prandtl number
q heat transfer rate per unit plate width, W
R gas (air) constant, J/kg-K
U overall heat transfer coefficient, W/m2-K
V channel flow mean velocity, m/s
W plate thickness, m
Greek Symbols
α fluid thermal diffusivity, m2/s
β rectangular duct aspect ratio, β < 1
ε heat exchanger effectiveness
φ correction function, Eq (19)
ηf fin efficiency
ηo overall finned surface efficiency
µ fluid viscosity, Pa-s
opt optimum value
p heat exchanger plate
Superscript
* dimensionless variable
REFERENCES
[1] Bejan, A., The Concept of Irreversibility in Heat Exchanger
De-sign: Counterflow Heat Exchangers for Gas-to-Gas Applications,
ASME Journal of Heat Transfer, vol 99, no 8, pp 374–380, 1977.
[2] Bejan, A., and Sciubba, E., The Optimal Spacing of Parallel Plates
Cooled by Forced Convection, International Journal of Heat
and-Mass Transfer, vol 35, no 12, pp 3259–3264, 1992.
[3] Mereu, S., Sciubba, E., and Bejan, A., The Optimal Cooling of aStack of Heat Generating Boards With Fixed Pressure Drop, Flow
Rate or Pumping Power, International Journal of Heat and Mass Transfer, vol 36, no 15, pp 3677–3686, 1993.
[4] Bejan, A., and Morega, A M., The Optimal Spacing of a Stack
of Plates Cooled by Turbulent Forced Convection, International Journal of Heat and Mass Transfer, vol 37, no 6, pp 1045–1048,
1994
[5] Bar-Cohen, A., and Jelinek, M., Optimum Arrays of
Longitudi-nal, Rectangular Fins in Convective Heat Transfer, Heat Transfer Engineering, vol 6, no 3, pp 68–78, 1985.
[6] Culham, J R., and Muzychka, Y S., Optimization of Plate Fin
Heat Sinks Using Entropy Generation Minimization, IEEE actions on Components and Packaging Technologies, vol 4, no.
Trans-2, pp 159–165, 2001
[7] Ganzarolli, M M., and Altemani, C A C., Thermal Optimization
of a Double Finned Heat Exchanger Plate, Proceedings of 16th ECOS, Copenhagen, Denmark, vol 2, pp 1161–1168, 2003 [8] Kays, W M., and Crawford, M E., Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, Int Ed., Singapore, 1993 [9] Kays, W M., and London, A L., Compact Heat Exchangers, 2nd
ed., McGraw-Hill, New York, 1964
[10] Phillips, R J., Microchannel Heat Sinks, in Advances in Thermal Modeling of Electronic Components and Systems, eds A Bar-
Cohen and A D Kraus, vol 2, pp 109–184, 1990
[11] Jones, O C., Jr., An Improvement in the Calculation of Turbulent
Friction in Rectangular Ducts, Journal of Fluids Engineering, vol.
98, pp 173–181, 1976
[12] Reis, E., and Altemani, C A C., Design of Heat Sinks and Planar
Spreaders with Airflow Bypass, ASME Advances in Electronic Packaging, ASME, New York, vol 1, pp 477–484, 1999.
[13] Knight, R W., Hall, D J., Goodling, J S., and Jaeger, R C.,
Heat Sink Optimization With Application to Microchannels, IEEE Trans CHMT, vol 15, no 5, pp 832–842, 1992.
[14] Tomazeti, C A., and Altemani, C A C., A Compact Model for
Heat Sinks with Experimental Verification, Proceedings of the 8th International Workshop on Thermal Investigation of ICs and Systems, Madrid, Spain, pp 266–271, 2002.
[15] Phoenics CFD from CHAM Ltd., UK; http://www.cham.co.uk
Marcelo Moreira Ganzarolli is an associate
profes-sor of mechanical engineering (Energy Department)
at State University of Campinas (UNICAMP), Brazil.
He teaches thermodynamics, heat transfer, and putational fluid dynamics His main areas of research are numerical heat transfer, natural and forced con- vection, and cooling of electronic equipment He is a member of the Brazilian Society of Mechanical Sci- ences and Engineering (ABCM).
com-Carlos A C Altemani is an associate professor
of mechanical engineering (Energy Department) at State University of Campinas (UNICAMP), Brazil.
He received his Ph.D in 1980 from the University
of Minnesota, Minneapolis, Minnesota His main search interests are in convective heat transfer, in- cluding the thermal cooling of electronics, heat ex- changers, and the numerical simulation of fluid flow and heat transfer He is a senior member of the Brazil- ian Society of Mechanical Sciences and Engineering (ABCM).
re-heat transfer engineering vol 31 no 1 2010
Trang 34CopyrightC Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903263325
Numerical Simulation of
Buoyancy-Induced Turbulent Flow
Between Two Concentric Isothermal Spheres
ALI AKBAR DEHGHAN and MASIH KHOSHAB
School of Mechanical Engineering, Yazd University, Yazd, Iran
Buoyancy-induced turbulent flow and natural convection heat transfer between two differentially heated concentric isothermal
spheres is studied numerically The low-Reynolds-number k–ω model is used for turbulence modeling The two-dimensional
governing equations are discretized using control volume method and solved by employing the alternating direction implicit
scheme Results are presented in the form of streamline and temperature patterns, and local and average Nusselt numbers,
over the heated and cooled boundaries for a wide range of Rayleigh numbers (10 2 –10 10 ), extending the previous studies
to the turbulent flow regime and for the radius ratio of 2 The results of the flow pattern and average Nusselt numbers
were compared with the previously published experimental and numerical investigations and very good agreements were
observed For low values of Rayleigh numbers, regions with conduction-dominated flow pattern accompanied with low
values of Nusselt numbers were observed, while for higher Rayleigh numbers, the flow pattern was changed to the convection
dominated boundary layer type flow, resulting in an increase in the rate of heat transfer and flow velocities adjacent to
both inner and outer boundaries The average Nusselt numbers were correlated against Rayleigh number and a 1/4 power
dependence of Ra in both laminar and turbulent regimes is obtained.
INTRODUCTION
Natural convection heat transfer in the annulus between two
concentric spheres has received much attention due to both
the-oretical interests and experimental applications This geometry
has been employed in many engineering design problems such
as nuclear reactor design, thermal energy storage (TES) systems,
and solar energy collectors and storage tanks, to name a few
Predicting the transient and steady state behavior of fluid flow
and heat transfer rates is a necessary task to these engineering
design problems
The first investigation of this geometry was reported by
Bishop et al [1, 2], in which heat transfer in the space between
two isothermal concentric spheres was investigated
experimen-tally for various diameter ratios and air as the working fluid
In their investigations, they observed two steady patterns, i.e.,
the crescent eddy pattern and the kidney-shaped pattern, and
Address correspondence to Professor Ali Akbar Dehghan, Department of
Mechanical Engineering, Yazd University, Yazd, Iran E-mail: adehghan@
yazduni.ac.ir
one unsteady pattern Natural convective heat transfer for otherfluids (water and silicone oils) in spherical shells for mediumvalues of diameter ratio was the subject of experimental study
of Scanlan et al [3] Yin et al [4] experimentally visualized urally induced flow patterns between concentric spheres Theycategorized the type of the flow in terms of Grashof numberand the inverse of the relative gap width for each fluid Forsmall Rayleigh numbers, Mack and Hardee [5] obtained an-alytical solutions by expanding the dependent variables in apower series with respect to the Rayleigh number Fujii et al [6]obtained a numerical solution of transient laminar natural con-vection between concentric spheres at a Prandtl number of 0.7and Rayleigh number of 100 Later, they extended their study
nat-to larger values of Prandtl number, ranging from 0.7 nat-to 100[7] Transient natural convection between concentric isothermalspheres was also the subject of the numerical investigation ofChu and Lee [8] They employed stream function and vorticityformulation of the governing equation in their study for Rayleighnumbers ranging from 102to 5.0× 105and for different values
of radius ratios Chiu and Chen [9] obtained a numerical solutionfor transient free convection between concentric and vertically
33
Trang 3534 A A DEHGHAN AND M KHOSHAB
eccentric spheres and for Rayleigh numbers ranging from 103
to 5.0× 105 Natural convection in concentric spherical
enclo-sures was the subject of analytical study of Teertstra et al [10]
Their analytical model was based on the linear superposition
of conduction and convection solutions where the convective
component was determined based on the combination of two
limiting cases, i.e., laminar boundary layer convection and
tran-sition flow convection They validated their model by comparing
their results with the previously published experimental and
nu-merical data for the same problem
Based on an in-depth literature review, it appears that
turbu-lent flow characteristics in spherical coordinate have not been
fully understood Therefore, the purpose of this study is to
nu-merically investigate the buoyancy-induced flow between the
annulus of two concentric differentially heated spheres for a
wider range of Rayleigh numbers, which include both laminar
and turbulent flow regions (Ra= 102–1010) and hence to
ex-tend the previously published investigations, which were mainly
done in the range of laminar flow, to the turbulence region
GOVERNING EQUATIONS
The configuration under study is shown in Figure 1 Two
concentric spheres with radius of riand roare considered The
inner sphere having a radius of riis assumed to be the hot surface
with a uniform temperature of Ti, while the outer one with a
radius of rois the cold surface with a uniform temperature of To
Spherical coordinates (r, θ ) are used for the present computation
with the origin at the center of spheres The angular coordinate
is measured in the clockwise direction with θ = 0 at the top
and θ = π at the bottom of spheres
The gap between the spheres is assumed to be filled with a
viscous and incompressible Newtonian fluid The
thermophys-ical properties of fluid are assumed constant except for
den-sity variation with temperature in the buoyancy term—i.e., the
Figure 1 Physical model and coordinate system.
Boussinesq approximation is utilized The viscous dissipationterm in the energy equation is neglected due to its small in-fluence in natural convection problems with low fluid velocity.Radiation heat exchange between the two surfaces is also ne-glected The two-dimensional (2D) nondimensional governingequations in spherical coordinate in the annular enclosure arewritten as follows
Pr + ν∗t
Prt
(∇2T)(4)where
The low-Reynolds-number (LRN) k–ω model (WX) introduced
by Wilcox [11, 12] was used for turbulence modeling in thiswork To describe the evolution of the turbulence field and definethe turbulent scales, an equation for turbulence energy is oftenused when using Reynolds-averaging two-equation models Forsteady flows, the transport equation for turbulent kinetic energy,
an extra equation is required to construct the eddy viscosity, µt.heat transfer engineering vol 31 no 1 2010
Trang 36A A DEHGHAN AND M KHOSHAB 35
Table 1 Terms in turbulence model (LRN k–ω Wilcox) used in the present
The right-hand side terms of Eq (7) have interpretations similar
to those in Eq (6) The eddy viscosity in the k–ω model is
defined as
µt = cµfµ
ρk
The production term in the k-equation, P k, is modeled by
as-suming that the turbulent Reynolds stresses are proportional to
the strain rate tensor, S ij This gives:
The terms appearing in Eqs (6) and (7) are defined in Table 1
The source term in the k-equation, S k, includes the buoyancy
term, which represents the exchange between potential energy
and turbulent kinetic energy Several sophisticated alternatives
have been used to model the source term in the k-equation, e.g.,
the GGDH model by Ince and Launder [14] and the AFM
ap-proach by Hanjalic and Vasic [15] For its simplicity, the SGDH
model remains most widely used in engineering applications,
which represents the buoyancy term, G k, by the following
The closure constants for the k–ω model are summarized in
Table 2 The damping functions, f k , fµ, and f1, for the present
k–ω turbulence model are given in Table 3 [11, 12] The
gen-eralized form of the nondimensional governing equations with
the appropriate coefficients and source terms in the spherical
coordinate, which are extracted from Anderson et al [16], are
listed in Appendix A
From the nondimensional form of the governing equations,
it is seen that the governing parameters for the present study
are the Rayleigh number (Ra), the Prandtl number (Pr), and the
radius ratio (R*) The local and average Nusselt numbers are
Table 2 Turbulence model constants [13]
The boundary values for velocity components and k were set
to zero, while the scaled temperatures were set to 1 and zero
at the hot and cold boundaries, respectively The wall conditionfor ω was specified according to the following relation:
ω= 6ν
c2y2
P
Equation (13) was employed for assigning ω at the first grid
point close to the wall surface (y+ <2.5) [13] All boundaryconditions employed in this study are also listed in Appendix A
SOLUTION PROCEDURE
The computations have been performed on the basis of theBoussinesq assumption The air properties are evaluated at thereference temperature: i.e., Tcold is assumed to be referencetemperature (To= Tcold= Tref), and the density and the dynamicviscosity are assumed to be constant
The governing differential equations are solved by employing
a finite-volume method and SIMPELER algorithm described byPatankar [17] The convective terms, in the momentum and en-ergy equations, were discretized using a power-law differencingscheme, while for the turbulence equations, a hybrid upwind–central differencing scheme is employed Peng and Davidson[13] showed that the low numerical accuracy in the turbulence–convection discretization has not contaminated the prediction
of the transition solution Derivatives at the boundaries wereapproximated by three-point forward or backward differencingformulas The Alternating Direction Implicit method is usedfor the solution of the discrete equations The pseudo-transientapproach with appropriate under-relaxation parameters for theheat transfer engineering vol 31 no 1 2010
Trang 3736 A A DEHGHAN AND M KHOSHAB
Figure 2 Variation of local Nusselt number over the cold boundary for various
mesh sizes and for Ra = 10 9 , Pr= 0.7 and R* = 2.0.
field parameters is employed for obtaining the steady-state
re-sults
Nonuniform grids in radial direction and uniform grids in the
angular direction were used A nonuniform grid is generated by
using Eq (14) In order to obtain a grid-independent solution
for every Rayleigh number, the numerical experimentation is
done for four different mesh size configurations The effect of
grid size on the local Nusselt number distribution over the
in-ner and outer walls is investigated In most studies, comparison
between the maximum values of Nusselt number is chosen in
order to verify the grid-independence of the solution, while it
is believed that comparison between local values is more
ap-propriate and would produce more accurate results For
exam-ple, at Ra = 1 × 109, by comparison of the results obtained
for different grid spacing, it was concluded that increasing the
mesh number beyond 250θ× 50r had no noticeable effect on
the results and hence the grid size of 250θ× 50r is selected
throughout this study The mesh-dependent results in the form
of the variation of local Nusselt number on the cold boundary
are shown in Figure 2 However, the compactness of the grids
along the boundaries is appropriately increased for higher
val-ues of Rayleigh numbers in order to capture the possible higher
gradients of the field variables This can be accomplished by
increasing α1 in Eq (14) Moreover, at each Rayleigh number
considered in this study, y+values near the solid boundaries are
calculated to ensure that at least three grid points nearest to the
wall have y+below 2.5 This ensures that the grid compactness is
tanh(α1/2)
, 3.5≤ α1≤ 9(14)
The solution was considered convergent when the global relative
errors of the field variables over the all control volume cells were
less than prescribed criteria A convergence criterion of 10−7is
Figure 3 Comparison of streamline for R*= 2.17, Pr = 0.7, and Ra = 7.392 × 10 5 : (a) present results; (b) experimental results [4].
selected for all field variables except the temperature field, forwhich the value of 10−8is considered
RESULTS AND DISCUSSION
Results in the form of flow and thermal field variables andNusselt numbers are presented for various values of the Rayleighnumbers and for a radius ratio of 2 Results were also obtainedfor different values of ro/ri It was seen that at fixed Rayleighnumbers, the average Nusselt number increases with increasingradius ratio The presented results are categorized into laminarand turbulent flow In order to investigate the accuracy of thenumerical procedure developed in this study, typical results werecompared with the results available in the open literature
Laminar Flow Between Two Concentric Isothermal Spheres
In this section the flow field patterns are presented for thelaminar flow region Figure 3 shows the comparison betweenthe streamline pattern obtained in this study and experimentalresults of Yin et al [4] It is seen that the agreement betweenthe two flow patterns is very good
The numerical data of Astill et al [18] and analytical results
of Teertstra et al [10] for concentric spheres are used to describethe characteristics of the average heat transfer rate as a function
of Rayleigh number These authors introduced a critical valuefor the Rayleigh number, where the dominant mode of heattransfer changes from conduction to convection Three distinctregions are identified by Astill et al [18] The critical Rayleighnumber depends on the geometry and fluid properties (radiusratio and Prandtl number)
The predicted Nu, ψmaxand its corresponding angular tion, θ∗, for various values of Ra are presented in Table 4 Theheat transfer engineering vol 31 no 1 2010
Trang 38posi-A posi-A DEHGHAN AND M KHOSHAB 37
Table 4 Comparison of the calculated average Nusselt number, N u,
maximum value of stream function ψmax, and angular position of vortex center
θ ∗at steady state, for various values of R∗, Pr= 0.7 and for Ra = 10 2 to 5.0 ×
Chiu and Chen [9] 1.1021 3.2360 81 ◦
Chu and Lee [8] 1.9730 17.2800 67.5 ◦ Chiu and Chen [9] 1.9110 17.9400 67.5 ◦
Chiu and Chen [9] 3.3555 35.9240 54 ◦
Chu and Lee [8] 5.3780 53.4200 49.5 ◦
a For these R∗values and Ra numbers, different velocity scaling is employed
and hence direct comparison between the nondimensional ψmaxvalues is not
possible.
same values from previously published data are also added for
comparison It may be seen that the heat transfer and flow field
data obtained in this study are in close agreement with the other
investigations However, for large values of Ra, small
discrep-ancies were observed in predicting the angular position of the
vortex center This is believed to be due to the different
formula-tion employed in the present study compared to the study of Chu
and Lee [8] and Chiu and Chen [9] and also due to uncertainty
in predicting the exact location of ψmaxfor large values of Ra
due to the slight oscillatory behavior of the flow Figure 4 shows
the predicted thermal and flow fields in the form of isotherms
and streamlines contours for different values of Rayleigh
num-ber in the laminar flow region As the flow is symmetric with
respect to the vertical axis, the isotherms are plotted on the
left and the streamlines are drawn on the right hand of each
figure As the fluid adjacent to the inner wall is heated, the
lower density fluid moves upward due to the buoyancy effect,
while the relatively colder and denser fluid will eventually flow
downward along the cold surface of the outer sphere Thermal
boundary layers start to develop at the lower edge of inner hot
wall and the upper edge of outer cold one The thickness ofboth boundary layers decreases when the Rayleigh number isincreased The thermal boundary layer thickens when the flowmoves upward along the inner wall, which results in a decrease
of the local heat transfer rate along this wall The opposite fect is seen adjacent to the outer cold wall when the warm fluidstarts to descend along it For higher values of the Rayleigh, theboundary layers are confined in the vicinity of the hot and coldsurfaces, while the core region of the annuli is occupied with alow-velocity fluid It is also observed that for higher values ofthe Rayleigh number, the bottom portion of the annular cavity isoccupied with almost stagnant cold fluid, which can be seen inFigure 4c–e
ef-Turbulent Flow Between Two Concentric Isothermal Spheres
As the published results for the turbulent flow in spherical ometries are scarce, we validated our solution procedure in theturbulent regime through comparison with numerical and exper-imental solutions for natural convection in rectangular tall cav-ity The same procedure is followed by Barhaghi [19] in the study
ge-of “DNS and LES ge-of Turbulent Natural Convection BoundaryLayer” in cylindrical coordinate Various cavity configurationshave been used in previous experimental and numerical works.Henkes and Hoogendoorn [20] proposed, in a Eurotherm Work-shop, the use of a square enclosure as the benchmark test case Intheir comparison study using the workshop results, they showedthat the results for cavities with aspect ratios of 1 and 5 (at nearlythe same Rayleigh number) are very close only if they are scaled
by the cavity height Hanjalic and Vasic [15] found that the diction was more erroneous for the turbulent flow in a tall cavity
pre-with an LRN k–ε model than for the flow in a square or
lower-aspect ratio cavity with the same LRN model [13] In the presentstudy, for a direct comparison with the available experimentaldata, the flow in a rectangular enclosure with the height to widthratio of 5 was computed The Rayleigh number is selected to be
5× 1010, Thot = 77.2◦C, and T
cold= 31.4◦C as the experimental
data, and the benchmark solution was available for the tions just described The horizontal walls are assumed to beadiabatic
condi-Figure 5 shows the predicted results of natural convection
in a rectangular cavity with the conditions just described Thetemperature, velocity, and turbulent kinetic energy profiles areshown in Figure 5 at the mid-height of the enclosure The ex-perimental results of Cheesewright et al [21] and the numericalprediction of Peng and Davidson [13] are also presented for
comparison It is seen that the LRN k–ω model used in this
study is in close agreement with the experimental data, ensuringthe validation of our turbulence modeling and solution proce-dure
Figure 6 shows the predicted thermal and flow fields in theconcentric annuli under investigation and for various values ofRayleigh number that fall in the turbulent flow region Again,the isotherms are plotted on the left while the streamlines areheat transfer engineering vol 31 no 1 2010
Trang 3938 A A DEHGHAN AND M KHOSHAB
Figure 4 Streamlines (right) and isotherm contours (left) for Pr= 0.7, R* = 2.0 and (a) Ra = 1.0 × 103 ; (b) Ra = 1.0 × 10 4 ; (c) Ra = 1.0 × 10 5 ; (d) Ra = 1.0
× 10 6 ; (e) Ra = 5.0 × 10 6
drawn on the right half section of each plot In order to facilitate
comparison between each figure, 20 streamlines and isotherms
are plotted in each figure The general shape of the flow and
thermal patterns are the same for laminar flow as shown in
Figure 4 Again, the thermal and velocity boundary layers are
formed adjacent to the both hot and cold walls of annular cavity
However, for high values of Ra, the flow fields in the
vicin-ity of both walls are more strengthened and the thicknesses
of both thermal and velocity boundary layers are significantlylower It is also noticed that the angular position of the mainvortex, θ∗, moves upward and approaches to the upper edge
of the cold wall It is also seen in Figure 6 that a stable mal stratification is formed in the core region of the annuli forhigh values of Rayleigh number, while the bottom portion isheat transfer engineering vol 31 no 1 2010
Trang 40ther-A ther-A DEHGHAN AND M KHOSHAB 39
Figure 5 Comparison between profiles of (a) temperature, (b) velocity, and (c) turbulence kinetic energy at y/H = 1/2 and (d) local Nusselt number on the hot
and cold surfaces , Numerical results with PDH model + damping function [13] ando , experimental results [21].
occupied with almost stagnant cold fluid The change in the
shape of the main vortex for increasing Rayleigh number is also
interesting
Figure 7 represents the fluid temperature distribution across
the annular cavity at an angular position of 60 and 90 degrees
for three different values of Rayleigh number The slope of the
dimensionless temperature distribution in the vicinity of both
walls increases with an increase in Rayleigh number, giving
higher values of the local heat transfer coefficient For high
values of Ra, temperature inversion is noticeable in the core
region, especially at the 60 degrees angle This is due to the
existence of a relatively strong shear driven vortex, which causes
part of the descending fluid adjacent to the cold surface to beredirected toward the inner core region before completely losingits energy
Figure 8 represents fluid tangential velocity distributionacross the spherical cavity at angular positions of 60 and 90degrees for three different values of Rayleigh number The ve-locity magnitudes presented in Figure 8 are not comparable, due
to the reference velocity selected in this study In order to makethe velocity magnitudes be comparable, either a dimensionalprofile should be plotted or the velocities should be scaled to acommon velocity reference The second alternative is selectedand all the values are normalized for the velocity reference in
Figure 6 Streamlines (right) and isotherm contours (left) for Pr= 0.7, R* = 2.0 and (a) Ra = 1.0 × 108 ; (b) Ra = 1.0 × 10 9 , and (c) Ra = 1.0 × 10 10
heat transfer engineering vol 31 no 1 2010