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Analysis of heat-transfer performance of cross-flow fin-tube heat exchangers under dry and wet conditions
International Journal of Heat and Mass Transfer 55 (2012) 1496–1504
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Analysis of heat-transfer performance of cross-flow fin-tube heat exchangers under dry and wet conditions Cheen Su An, Do Hyung Choi ⇑ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 16 June 2011 Accepted 7 August 2011 Available online 29 November 2011 Keywords: Wave-fin heat exchanger Dehumidifying condition Fin efficiency 3D analysis
a b s t r a c t A three-dimensional analysis procedure for the detailed phenomenon in a fin-tube heat exchanger has been developed and applied to predict the heat/mass transfer characteristics of the wave-fin heat exchangers. The continuity, Navier–Stokes and energy equations together with the species equation for the air–vapor mixture are solved in a coupled manner, so that the inter-dependence between the temperature and the humidity can be properly taken into account, by using the SIMPLE-type finite volume method. Having validated the procedure, calculations have been carried out for various frontal-velocity and inlet-humidity conditions. It has been shown that the flow characteristics, such as the temperature and humidity fields, along with the local heat flux and the condensation rate, can be successfully captured. The numerical results reveal that the existing correlations considerably underestimate the fin efficiency especially for multi-row heat exchangers. For dehumidifying cases, the sensible heat-transfer rate seems insensitive to the inlet-humidity change. The rate changes mostly in the narrow band of partially wet regime between 25% and 40% of inlet relative humidity. The range of the frontal velocity that gives the best performance for various numbers of rows is also estimated. The analogy between the heat and mass transfer on the fin surface is also examined. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction It is of practical importance to determine the fin efficiency accurately in estimating the overall heat transfer of a heat exchanger. The solution generally is not known for various fin types and configurations, and numerous mathematical approximations have been suggested in the literature. Most such studies, however, are based on one-dimensional analysis together with the assumption that the fin-to-air heat-transfer coefficient is uniform. Improved methods have been suggested [1–7] by taking the variations of the heat-transfer coefficient or the air temperature into consideration by approximating them as functions of the distance from the tube. Haung and Shah [2] carried out comparative study of these methods for a simple fin shape, but no clear conclusion as to which is most accurate. Besides, all these approaches show limited success as the heat-transfer coefficient is substantially larger near the leading edge of the fin compared to the rest of the region and make it impossible to model by simple analytic functions. Estimation of heat transfer under dehumidifying condition is even more complex. Various effects of design parameters have been explored. To tackle the problems of practical interest such as optimization of fin pitch, wave depth, positions of slits or ⇑ Corresponding author. Tel.: +82 42 350 3018; fax: +82 42 350 3210. E-mail address: [email protected] (D.H. Choi). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.055
louvers, etc., however, one needs to have detailed local information which normally is beyond the reach of experimental study. CFD techniques may provide the accurate and detailed heat-transfer characteristics of heat exchangers. Tsai et al. [8], Perotin and Clodic [9] and Tao et al. [10] carried out three-dimensional conjugate heat-transfer analyses for dry condition and obtained the local heat-transfer characteristics and/or the heat-transfer enhancement for various fin types. Few in the literature, however, provide a methodology that takes the water condensation/evaporation into account. The rate of mass transfer may be estimated by the Chilton–Colburn [11] analogy assuming that the mass transfer can be decoupled from the heat transfer. Although this gives a straight-forward way of predicting the mass-transfer rate from the known heat-transfer performance, it may not be appropriate as the condensation and the temperature are interdependent as seen in the psychrometric chart. Recently, Comini et al. [12] developed an analysis procedure, in which the cooling air is treated as two-component air–vapor mixture, and successfully estimated the temperature and condensate distributions, and the heat-transfer coefficient for rectangular-finned heat exchangers. The objective of this study is to develop a fully coupled three-dimensional heat/mass-transfer analysis procedure under dry and wet conditions. The procedure is then applied to analyze the detailed heat/mass transfer characteristics of a wave fin of complex shape and the multi-row heat-exchanger performance.
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Nomenclature A Aa Afr Amin Cp D dh do ha, hs hm ia ir k L N p PT, PL qlm q00 Q Qn
heat exchange area [m2] total surface area [m2] frontal area [m2] minimum free flow area [m2] specific heat [J/kg °C] diffusion coefficient [m2/s] hydraulic diameter [m] outer tube diameter [m] air-side and sensible heat-transfer coefficients [W/m2 K] mass-transfer coefficient enthalpy of humid air [J/kg] enthalpy of saturated water vapor at mean air temperature [J/kg] thermal conductivity [W/m K] length of the fin [m] number of rows of heat exchanger pressure [Pa] transversal and longitudinal tube pitch [m], (see Fig. 1) latent heat of water [J/kg] heat flux [W/m2] overall heat-transfer rate [W] heat-transfer rate of nth row [W]
These numerical results may help evaluate the commonly invoked data reduction practices: The fin efficiency is estimated and compared with existing correlations based on various approximations. The effects of inlet relative humidity, and the heat and mass transfer analogy are also examined and/or assessed.
Qmax Rv Re T t ui
vfr
w
Greek symbols gf fin efficiency l viscosity [N s/m2] q density [kg/m3] r standard deviation Subscripts a air b fin base f fin s saturated state t tube w fin surface
Energy:
@ @ 2 T^ ^ ¼ 1 ^j TÞ ðu @ ^xj Re Pr @ ^xj @ ^xj
dh ¼
4Amin L Aa
umax
Afr ¼ v fr Amin
ð1Þ
2 @ ^ @ w ^j wÞ ¼ D ðu @ ^xj @ ^xj @ ^xj
with
Amin ¼ MinðA1 ; 2A2 Þ
ð3Þ
where A1 and A2 are the areas between two adjacent tubes shown in Fig. 1 and vfr the frontal velocity, the dimensionless governing equations can be written as Continuity:
^j Þ ^u @ðq ¼0 @ ^xj
ð4Þ
Momentum:
^ ^i @ @p 1 @2u ^i u ^j Þ ¼ ðu þ @ ^xj @ ^xi Re @ ^xj @ ^xj
ð5Þ
ð7Þ
where the carets denote dimensionless variables and
q p T Tt ^¼ ; p ; T^ ¼ ; qin qin u2max T in T t q umax dh lC p ^ D ; Pr ¼ Re ¼ in ; D¼ umax dh l k
q^ ¼
ð8Þ
Here ui is the velocity component in the xi direction, p the pressure, T the temperature, w the humidity, D the mass diffusivity, q the density, l the dynamic viscosity, k the thermal conductivity, and Cp the specific heat. The flow is incompressible, but the density variation due to the temperature or the humidity is taken into consideration as was done in O’Connell [13].
q¼ ð2Þ
ð6Þ
Species:
2. Equations and solution procedure The humid air can be taken as a mixture of air and water vapor. Since the mass fraction of the water vapor is very low, it has little effect on the air flow. Taking air as the carrier fluid and water vapor as a dilute species, the laminar humid air flow can be described by the equations of continuity, momentum, energy, and species transport. The effects of water film on the overall heat transfer or the thermal resistance for the condensate of the droplet shape are known to be very small and thus are neglected in the present analysis. Introducing the hydraulic diameter dh and the characteristic velocity umax as the characteristic length and velocity scales:
maximum heat-transfer rate [W]: ma(ia,in ir,in) gas constant of air [J/kg K] Reynolds number temperature [°C; K] fin thickness [m] flow velocity in xi direction [m/s] frontal velocity [m/s] humidity [kg/kg air]
RT
P
p ðm k =M k Þ k
ð9Þ
where mk and Mk are the mass fraction and the molecular weight of species k, respectively. The saturated humidity, which is identified by the saturation line in the psychrometric chart, may be described by the following formula:
ws ðT w Þ ¼
ps ðT w Þ
qRv T w
ð10Þ
The fin configuration of the heat exchanger is taken to be that of the experimental settings [14] and the boundary conditions for air and water temperature, and humidity are also matched to those of the experiment. The schematic of the heat exchanger under consideration together with its dimensions is shown in Fig. 2. It may be assumed periodic in the transverse direction and it suffices to consider only half of a unit module indicated in the figure. The
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PL
A2
Air flow
PT
A1
do
(a)
Fig. 2. Schematic of the heat exchanger and detailed fin shape: (a) heat exchanger, (b) wave-type fin.
(b) Fig. 1. Schematic of a fin-tube heat exchanger: (a) tube arrangement, (b) fin broken into sectors.
^ q00latent ¼ qqlm Drw n
ð13Þ
The humidity on the fin surface is equal to the saturation humidity, ws, but assumes the value at the inlet if the local air has not reached the saturation condition:
calculation domain is chosen to include the fin and the flow passage between the fins, and is extended to 6do upstream and 20do downstream of the heat exchanger in the streamwise direction to capture the entrance and exit regions properly. Using the second-order upwind scheme for discretization of the convective terms, the governing Eqs. (4)–(7) are solved by the SIMPLE-type finite volume method, FLUENT, with the following boundary conditions: The velocity and the temperature along with the humidity are prescribed at the inlet boundary while the gradient in the flow direction is assumed negligible at the outlet boundary. No slip condition on the solid surface and the coolant-tube temperature are specified. The symmetry or periodic condition is imposed at the respective boundaries. At the fin–air interface, the temperature and the heat flux on the fin and air sides should match, respectively, as:
To properly account for the discontinuity in the heat flux between the air and fin sides as noted in Eq. (12), the air-side and fin-side domains are solved successively by using each other’s results as boundary conditions: Eqs. (11) and (14) are applied at the air-side interface while Eq. (12) is imposed on the fin side. These interface conditions are updated after each iteration and are determined as part of the solution. The sensible heat exchanged may then be estimated by integrating the heat flux into the air over the surface:
Tjfin ¼ Tjair
where ka is the thermal conductivity of the air. The fin efficiency, defined as the ratio of actual heat-transfer rate, Qactual, to the ideal maximum rate, Qideal, can be obtained.
kf
@T @T ¼ k þ q00latent a @nfin @nair
ð11Þ
ð12Þ
where the latent heat flux q00latent is determined from the mass-transfer rate on the surface:
ww ¼ min½ws ðT w Þ; win
Q sensible ¼
Z Aa
gf ¼
Q actual Q ideal
ka
@T dA @nair
ð14Þ
ð15Þ
ð16Þ
This concept is useful in data reduction and also in estimating the actual heat transfer from the more readily obtainable ideal heat
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transfer, in which the temperature variation in the fin is ignored. Since the fin material is uniform and the fin is sufficiently thin, the air-side heat-transfer characteristic is the single most significant factor in the fin-tube heat exchangers. The exact solutions are known only for the limited fin types and configurations. It is thus, in general, obtained from a simple 1D analysis under the assumption that the heat-transfer coefficient for the fin surface is constant and the ambient temperature variation is negligible. For fin-tube heat exchangers, Schmidt’s approximation [1], in which the polygon-shaped fin area is approximated by an equivalent circular disk, is widely used. Improved results may be obtained by taking account of the varying air temperature [2], fin thickness [3], non-uniform thermal conductivity [5,6], variable heat-transfer coefficients [7], etc. Threlkeld [15], McQuiston [16], and Wu and Bong [17] proposed the fin efficiency under fully wet conditions. Wu and Bong also suggested a partially-wet 1D overall fin efficiency. Recently, Pirompugd et al. devised a reduction scheme for the multiple row fin-and-tube heat exchangers by subdividing the region into many tiny segments and analyzed the heat/mass transfer characteristics under dehumidifying conditions [18,19]. The fin efficiency approximations under both dry and wet conditions are compared with that obtained in the present study. 3. Results and discussion Taking advantage of periodicity and symmetry as pointed out in the previous section, only one half of the unit fin module is considered. Approximately 200,000 cells are used to fit the domain of each row as shown in Fig. 3. Additional 10,200 and 17,000 grid cells are distributed for the upstream and downstream regions of the fin, respectively. The calculation is performed for the inlet air temperature of 21 °C for the dry case and 27 °C with 43% humidity for the wet case, while the tube temperature is fixed at 45 °C and 6 °C for the dry and wet cases, respectively. To validate the procedure described above, the normalized overall heat-transfer rate under dry and wet conditions for heat exchangers with various row numbers is compared with the experimental data of Youn [14] in Fig. 4. The Reynolds number based on the inlet air condition ranges from 125 to 625 and 124 to 495 for dry and wet cases, respectively, for the frontal velocity range tested. The variation of the Reynolds
1499
number due to the heat transfer in the heat exchanger amounts to 6.5% for the single-row case at the lowest frontal velocity. The Prandtl number variation from inlet to outlet is not as significant and remains close to 0.72. The heat-transfer coefficient for a single module is first calculated by solving the fin–air conjugate heattransfer problem. To account for the temperature variation of the air and the coolant through the heat exchanger, the amount of heat transfer for all such modules is estimated by using the tube-bytube algorithm [20] with the fixed heat-transfer coefficient for each row. The results are in remarkably good agreement with the measurement for all conditions. The slightly less satisfactory results for the three-row cases may be attributed to the shortcomings inherent in the tube-by-tube method. The accuracy may be improved by carrying out the three-dimensional calculation for each fin segment with varying coolant and air temperatures. No attempt has been made to do this, though. The excellent agreement confirms that the procedure is capable of accurately predicting the heat-transfer performance of the heat exchanger. In subsequent calculations, it is seen that the spatial density variation of the air–vapor mixture due to the temperature change in the field is as large as ±4.2%, while those of viscosity and specific heat turn out to be smaller than ±1%. The variation of the thermal conductivity is about ±3%, but shows negligible effects on the results. This confirms that the present approach of treating the density as variable and keeping the transport properties constant is appropriate. In the leading-edge region, the hydraulic and thermal boundary layers are developing and the heat transfer is very active. The local heat transfer in the vicinity of the leading edge becomes high and the fin temperature varies rapidly. This phenomenon is clearly visible in the temperature and heat-flux distributions for the cooled case at 1 m/s frontal velocity in Fig. 5. The leading-edge area is the most effective heat-transfer region of the fin. The flow accelerates between the coolant tubes due to the narrowing flow passage; this results in the high heat flux on the fin surface there. In the certain region behind the tube, the air temperature becomes lower than the adjacent fin temperature and the negative heat transport is observed. This peculiar behavior may be explained as follows: the air temperature behind the coolant tube is directly affected by the low tube temperature while the fin surface below is kept relatively warm by the ambient air away from the spot. To see the heat-transfer behavior more clearly, the heat- and mass-flux
Fig. 3. Sample grid for the periodic computational domain.
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Fig. 4. Overall heat-transfer rate for various heat exchangers: (a) dry condition, (b) wet condition.
distributions along the A1–A10 cross-section for two different frontal velocities are shown in Fig. 6. The heat flux is very high near the leading edge in the developing region. It is natural to observe that a higher frontal velocity gives a larger overall heat flux. The slightly irregular behavior in the heat-flux distribution may be attributed to the crests or troughs of the wave fin that makes the flow accelerate or decelerate. The wall-shear-stress distribution in the symmetry planes, A1–A10 , A2–A20 , and A3–A30 in Fig. 7, clearly elucidates this phenomenon. The shear stress exhibits the local maxima or minima which coincide with the crests or the troughs of the wave fin; the heat flux tends to move with the wall shear stress for obvious reasons. The procedure is capable of predicting the humidity field about the heat exchanger. The saturation surface moves gradually outward from the fin surface as the air gets cooler as it travels downstream. Fig. 8 shows the development of the saturation region in the symmetry plane for various frontal velocities, i.e., 0.5, 1.0, and 2.0 m/s. Similar to the behavior of the hydraulic boundary
layer, the fully saturated region in the first row decreases as the flow velocity increases. For subsequent rows, however, the air is sufficiently cooled to the point that the entire region becomes fully saturated. The condensation occurs as the air temperature dips below the saturation temperature. The condensation in the leading edge region depends very much on whether the saturation condition is reached there. Once past the developing region, the condensation and the heat flux move in unison as seen in Fig. 6 for the crosssection A1–A10 . The trend continues to the second and third rows. The flow reaches the saturation condition earlier when the frontal velocity is lower; the larger condensation rate follows. One of the assumptions frequently adopted in the fin-efficiency approximation is axisymmetry. This assumption usually does not hold in real situations, as the fin temperature along three different circular paths (L1, L2, L3) for the air velocity of 1 m/s in Fig. 9 exhibits. It is evident that the axisymmetry deteriorates as the circle gets farther away from the tube: along the path close to the tube, L1, the dimensionless temperature varies little, smaller than 0.03, while the variation along L3 is 0.16. The flow appears to return to symmetry about the tubes for the second and third rows as the flow properties varies little in the streamwise direction. From these results, coupled with the humidity variation in the flow direction that needs to be accounted for in the analysis, onedimensional fin efficiency approximations or simple modified models thereof that only take account of the geometric change in the fin shape, may introduce significant error when applied to the fins of complex shape. The fin efficiency under both dry and wet conditions computed in the present study is depicted in Fig. 10. The results of the existing fin-efficiency approximations are also shown in the figure for comparison. The fin-efficiency curve is constructed by varying the frontal velocity. Under dry conditions shown in Fig. 10(a), the fin efficiency becomes higher as the number of rows increases. It is because that the fin–air temperature difference becomes smaller towards the downstream end and so does the temperature variation. This tends to make the actual heat transfer close to that of the ideal heat-transfer rate and pushes the fin efficiency up close to unity. For fins with large flow depth, the fin efficiency approximation underestimates the value up to 10–15%. The flow depth needs to be small for the fin efficiency approximation to remain valid. Among various approximations, the Han and Lefkowitz method underestimates the results most. The heat-transfer coefficient is very high near the leading edge of the fin and the fin efficiency is likely to be underestimated if it is based on the average heat-transfer coefficient. The sector method and the methods of Schmidt, and Huang and Shah agree quite well with one another but the values are appreciably smaller than the present results. The fin temperature variation is much less in the second and the third row than in the first row, i.e., the fin efficiency depends less on the distance from the tube there. The close agreement of Huang and Shah, which takes account of the variation of ambient air temperature in the transverse direction, with other methods seems to imply that the effects of transverse mixing is insignificant. For the fin efficiency under wet condition shown in Fig. 10(b), the earlier studies [15–17] are again seen to give significantly lower values. The coolant temperature in those studies is slightly different from that of the present, 7 °C vs. 6 °C. The discrepancy is attributed to the assumptions made in devising the approximation formulae. If the fully wet condition is met, i.e., the condensation occurs from the fin leading edge, it is reported that the fin efficiency varies only slightly with the relative humidity at the inlet [17–19]. It is seen in the figure that, when the inlet humidity changes from 60% to 100%, the fin efficiency by Wu and Bong [17] results in the difference of up to 6% while that in the present study 2.8%. For multi-row heat exchangers, the difference becomes
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Fig. 5. Temperature and local heat-transfer rate for
smaller as hs increases, i.e., with increasing frontal velocity, while the opposite is true for the single-row heat exchanger. The insensitive behavior of fin efficiency to the inlet relative humidity is due to the fact that the flow is in near fully wet condition. To see how the fin efficiency varies from the dry to wet condition, calculations have been carried out with varying inlet relative humidity and the results are plotted in Fig. 11. This is for the single-row case with three different inlet velocities. The fin efficiency remains fairly constant when the condition is either dry or fully wet. It varies mostly in the narrow region of partially wet regime between 25% and 40% of inlet relative humidity. This band of partially wet regime becomes narrower as the frontal velocity decreases. Similar behavior, but with narrower partially wet regime, is also observed for two- and three-row heat exchangers. For heat exchangers with multiple rows, the total heat-transfer rate increases with the number of rows. Unfortunately, that also accompanies higher pressure loss. From the perspective of wallshear stress depicted in Fig. 7, the shear stress becomes periodic immediately after the leading-edge region of the first row. The pressure drop for the heat exchanger, therefore, is expected to increase almost linearly with the number of rows. The performance of each row for various conditions is summarized in Table 1. In fact, the heat-transfer rate for the second or third row is much smaller than that of the first row. The lower the frontal velocity, the less effective the heat-transfer rate of the second and third rows. To assess the performance of the multi-row heat exchangers, the heat-transfer rate is plotted against the power consumed in Fig. 12. The driving power per unit face area is given by the product
^ (b) heat-transfer rate vfr = 1.0 m/s: (a) temperature T,
q00 . q00av
of the pressure drop through the heat exchanger DP and the frontal velocity vfr:
ðdriving powerÞ ¼ DP v fr ½W=m2 ðunit face areaÞ
ð17Þ
Naturally, for both dry and wet conditions, a single-row heat exchanger is superior to the multi-row heat exchangers when the power is low. As the power increases, however, the performance of the two- or three-row heat exchangers surpasses that of the single-row heat exchanger at point A indicated in the figure. Eventually, the performance of the three-row heat exchanger comes out to be on top at relatively high power beyond point B. To find the optimal range of operation for a given number of rows, the variation in the tube temperature must also be considered using, for example, the tube-by-tube method. This is not pursued in the present study to keep the analysis simple. As one of the many design considerations, it may be useful to point out that, at the crossover point, B in Fig. 12(b), for twoand three-row heat exchangers under the wet condition, the frontal velocities are 1.32 and 1.14 m/s, respectively. Similarly, for single- and two-row exchangers, the frontal velocities at the crossover point are 0.63 and 0.50 m/s. These numerical results may be vital in designing the heat exchangers. The effect of inlet humidity on the sensible heat transfer is presented in Fig. 13. The temperature change between the inlet and the outlet relative to that of the dry condition is plotted against the frontal velocity in the figure for two different inlet-humidity
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(a)
(a)
(b) (b)
Fig. 6. Heat-flux and condensation-rate distributions along A1–A10 section for two different frontal velocities: (a) vfr = 0.5 m/s, (b) vfr = 2.0 m/s.
conditions. This compares the sensible heat-transfer drop from that of the dry condition for various numbers of rows. The drop is most pronounced, above 20%, for the single row heat exchanger at the highest frontal velocity. This may be elucidated as follows: the air is most humid in the first row and thus the latent heat transfer becomes largest. This results in the smallest sensible heat transfer there. Increasing frontal velocity makes the moisture flux in the air stream larger and also reduces the sensible heat transfer. Naturally the drop becomes less significant as the frontal velocity decreases and/or the number of rows increases. This observation is in line with the experimental study of Wang et al. [21] but the present analytical procedure provides much finer details as to how the performance is affected. Let us now examine the analogy between the heat- and masstransfer on the fin surface. The following Pearson product–moment correlation coefficient Cor(hs, hm), which lies between 0 and 1 with 1 indicating the perfect dependence, is useful in quantifying the degree of linear dependence of the two parameters.
Corðhs ; hm Þ ¼
E½ðhs Eðhs ÞÞðhm Eðhm ÞÞ
rH rM
Fig. 7. Shear-stress distribution on the fin surface along the symmetry planes for two different frontal velocities: (a) vfr = 0.5 m/s, (b) vfr = 2.0 m/s.
ð18Þ
where E denotes the area average of the argument over the fin surface and rH, rM the standard deviation of hs, hm, respectively:
h n
rH ¼ E ðhs Eðhs ÞÞ2
oi1=2
ð19Þ
Fig. 8. Saturation lines in the A1–A10 section of the first row for various frontal velocities.
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Fig. 9. Fin surface temperature along various circular paths. Fig. 11. Effects of inlet relative humidity on fin efficiency for various inlet velocities.
(a) Table 1 Heat-transfer rates for the second and third rows relative to the first row.
vfr [m/s] 0.50 1.00 2.00
Dry
Wet
Q2/Q1
Q3/Q1
Q2/Q1
Q3/Q1
0.1444 0.3462 0.5752
0.02133 0.1104 0.2517
0.1785 0.3803 0.6125
0.03186 0.1364 0.2901
row. In the leading-edge region, the condensation rate is relatively low compared to the heat flux as seen in Fig. 6. This is presumably due to the dissimilarity of the developing profiles of temperature and humidity and/or the fact that the flow is not fully saturated in the leading-edge region. This tendency is well supported by the significantly small value of the correlation for the first row in the table. If the upstream 25% of the first row is treated separately, the correlation for this region is about 0.502, for the higher frontal velocity case, whereas that for the remaining 75% is 0.968, higher than that of the second row. One may deduce from this result that for fins with louvers or slits, as they have much larger overall developing area compared to the wave fins, the heat and mass transfer analogy would be less accurate.
(b)
4. Conclusions
Fig. 10. Fin efficiency against the heat-transfer rate: (a) dry condition, (b) wet condition.
h n
rM ¼ E ðhm Eðhm ÞÞ2
oi1=2
ð20Þ
The results are presented in Table 2. It is shown that the two are more strongly related in the second and third rows than in the first
A numerical procedure that predicts the overall heat-transfer rate of the heat exchanger has been developed. The three-dimensional continuity, momentum, energy, and species transport equations coupled with the condensation model at the interface are solved by using the finite volume method of SIMPLE type to obtain the heat-transfer rate of multi-row cross-flow fin-tube heat exchangers under both dry and wet conditions. The details of the heat-transfer characteristics including the local condensation rate are seen to be accurately captured. The optimal range of frontal velocity is also identified for one-, two-, and three-row heat exchangers. The fin efficiency concept is found to underestimate the heattransfer rate noticeably for all cases. This is attributed to the simplified assumptions adopted in approximating the fin efficiency. Although the results may be improved by making modifications
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(a)
vfr [m/s]
Row
Correlation factor
0.5
First Second Third First Second Third
0.8321 0.9384 0.9138 0.7513 0.9536 0.9451
2.0
to the formulae, the improvement is limited and leaves much to be desired in accurately estimating the fin efficiency, especially, for multi-row heat exchangers. For dehumidifying cases, the overall sensible heat-transfer rate seems insensitive to the inlet humidity change if the fully wet condition is met. The fin efficiency varies mostly in the narrow region of partially wet regime between 25% and 40% of inlet relative humidity. A quantitative assessment of the heat- and mass-transfer analogy by using the product–moment correlation shows that the analogy holds almost perfectly for most of the fin surface except the small developing entrance region. The analogy may still be useful and accurate when the inlet air is in near saturation state.
(b)
References
Fig. 12. Heat-transfer rate vs. power consumption for various numbers of rows: (a) dry condition, (b) wet condition.
1
(ΔTin-out )wet / (ΔTin-out )dry
0.95
0.9
0.85
0.8
RH = 60 % 0.75
0.7
RH = 100 % 0.5
1
N=1 N=2 N=3 1.5
2
vfr [m/s] Fig. 13. Sensible heat transfer relative to the dry case for two inlet-humidity conditions and varying row numbers.
[1] T.E. Schmidt, Heat transfer calculations for extended surfaces, Refrig. Eng. 49 (1949) 351–357. [2] L.J. Huang, R.K. Shah, Assessment of calculation methods for efficiency of straight fins of rectangular profile, Int. J. Heat Fluid Flow 13 (1992) 282–293. [3] W. Lau, C.W. Tan, Errors in the one-dimensional heat-transfer analysis in straight and annular fins, J. Heat Transfer 95 (1973) 549–551. [4] J.B. Aparecido, R.M. Cotta, Improved one-dimensional fin solutions, Heat Trans. Eng. 11 (1990) 49–59. [5] H.M. Hung, F.C. Appl, Heat transfer of thin fins with temperature dependent thermal properties and internal heat generation, J. Heat Transfer 89 (1967) 155–162. [6] H. Barrow, J. Mistry, D. Clayton, Numerical and exact mathematical analyses of two-dimensional rectangular composite fins, Heat Transfer 2 (1986) 367–372. [7] L.S. Han, S.G. Lefkowitz, Constant cross-section fin efficiencies for nonuniform surface heat-transfer coefficients, ASME Paper 60-WA-41, 1960. [8] S.F. Tasi, T.W.H. Sheu, S.M. Lee, Heat transfer in a conjugate heat exchanger with a wavy fin surface, Int. J. Heat Mass Transfer 42 (1999) 1735–1745. [9] T. Perrotin, D. Clodic, Thermal-hydraulic CFD study in louvered fin-and-flattube heat exchangers, Int. J. Refrig. 27 (2004) 422–432. [10] Y.B. Tao, Y.L. He, J. Huang, Z.G. Wu, W.Q. Tao, Numerical study of local heat transfer coefficient and fin efficiency of wavy fin-and-tube heat exchangers, Int. J. Thermal Sci. 46 (2007) 768–778. [11] T.H. Chilton, A.P. Colburn, Mass transfer (absorption) coefficients-prediction from data on heat transfer and fluid function, Ind. Eng. Chem. 26 (1934) 1183– 1187. [12] G. Comini, C. Nonino, S. Savino, Numerical evaluation of fin performance under dehumidifying conditions, J. Heat Transfer 129 (2007) 1395–1402. [13] J.P. O’Connell, J.M. Prausnitz, B.E. Poling, Properties of Gases and Liquids, fifth ed., McGraw-Hill, 2001. [14] B. Youn, Private Communication, Samsung Electronics Co., 2004. [15] J.L. Threlkeld, Thermal Environmental Engineering, Prentice-Hall, Inc., New York, 1970. [16] F.C. McQuiston, Fin efficiency with combined heat and mass transfer, ASHRAE Trans. 81 (1975) 350–355. [17] G. Wu, T.-Y. Bong, Overall efficiency of a straight fin with combined heat and mass transfer, ASHRAE Trans. 100 (1994) 367–374. [18] W. Pirompugd, S. Wongwises, C.-C. Wang, Simultaneous heat and mass transfer characteristics for wavy fin-and-tube heat exchangers under dehumidifying conditions, Int. J. Heat Mass Transfer 49 (2009) 132–143. [19] W. Pirompugd, C.C. Wang, S. Wongwises, A fully wet and fully dry tiny circular fin method for heat and mass transfer characteristics for plain fin-and-tube heat exchangers under dehumidifying conditions, J. Heat Transfer 129 (2006) 1256–1267. [20] P.A. Domenski, Computer modeling and prediction of performance of an air source heat pump with a capillary tube, Ph.D. Dissertation, The Catholic University of America, Washington D.C., 1989, 62–93. [21] C.-C. Wang, Y.-T. Lin, C.-J. Lee, Heat and momentum transfer for compact louvered fin-and-tube heat exchangers in wet conditions, Int. J. Heat Mass Transfer 43 (2000) 3443–3452.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Analysis of heat-transfer performance of cross-flow fin-tube heat exchangers under dry and wet conditions Cheen Su An, Do Hyung Choi ⇑ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 16 June 2011 Accepted 7 August 2011 Available online 29 November 2011 Keywords: Wave-fin heat exchanger Dehumidifying condition Fin efficiency 3D analysis
a b s t r a c t A three-dimensional analysis procedure for the detailed phenomenon in a fin-tube heat exchanger has been developed and applied to predict the heat/mass transfer characteristics of the wave-fin heat exchangers. The continuity, Navier–Stokes and energy equations together with the species equation for the air–vapor mixture are solved in a coupled manner, so that the inter-dependence between the temperature and the humidity can be properly taken into account, by using the SIMPLE-type finite volume method. Having validated the procedure, calculations have been carried out for various frontal-velocity and inlet-humidity conditions. It has been shown that the flow characteristics, such as the temperature and humidity fields, along with the local heat flux and the condensation rate, can be successfully captured. The numerical results reveal that the existing correlations considerably underestimate the fin efficiency especially for multi-row heat exchangers. For dehumidifying cases, the sensible heat-transfer rate seems insensitive to the inlet-humidity change. The rate changes mostly in the narrow band of partially wet regime between 25% and 40% of inlet relative humidity. The range of the frontal velocity that gives the best performance for various numbers of rows is also estimated. The analogy between the heat and mass transfer on the fin surface is also examined. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction It is of practical importance to determine the fin efficiency accurately in estimating the overall heat transfer of a heat exchanger. The solution generally is not known for various fin types and configurations, and numerous mathematical approximations have been suggested in the literature. Most such studies, however, are based on one-dimensional analysis together with the assumption that the fin-to-air heat-transfer coefficient is uniform. Improved methods have been suggested [1–7] by taking the variations of the heat-transfer coefficient or the air temperature into consideration by approximating them as functions of the distance from the tube. Haung and Shah [2] carried out comparative study of these methods for a simple fin shape, but no clear conclusion as to which is most accurate. Besides, all these approaches show limited success as the heat-transfer coefficient is substantially larger near the leading edge of the fin compared to the rest of the region and make it impossible to model by simple analytic functions. Estimation of heat transfer under dehumidifying condition is even more complex. Various effects of design parameters have been explored. To tackle the problems of practical interest such as optimization of fin pitch, wave depth, positions of slits or ⇑ Corresponding author. Tel.: +82 42 350 3018; fax: +82 42 350 3210. E-mail address: [email protected] (D.H. Choi). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.055
louvers, etc., however, one needs to have detailed local information which normally is beyond the reach of experimental study. CFD techniques may provide the accurate and detailed heat-transfer characteristics of heat exchangers. Tsai et al. [8], Perotin and Clodic [9] and Tao et al. [10] carried out three-dimensional conjugate heat-transfer analyses for dry condition and obtained the local heat-transfer characteristics and/or the heat-transfer enhancement for various fin types. Few in the literature, however, provide a methodology that takes the water condensation/evaporation into account. The rate of mass transfer may be estimated by the Chilton–Colburn [11] analogy assuming that the mass transfer can be decoupled from the heat transfer. Although this gives a straight-forward way of predicting the mass-transfer rate from the known heat-transfer performance, it may not be appropriate as the condensation and the temperature are interdependent as seen in the psychrometric chart. Recently, Comini et al. [12] developed an analysis procedure, in which the cooling air is treated as two-component air–vapor mixture, and successfully estimated the temperature and condensate distributions, and the heat-transfer coefficient for rectangular-finned heat exchangers. The objective of this study is to develop a fully coupled three-dimensional heat/mass-transfer analysis procedure under dry and wet conditions. The procedure is then applied to analyze the detailed heat/mass transfer characteristics of a wave fin of complex shape and the multi-row heat-exchanger performance.
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Nomenclature A Aa Afr Amin Cp D dh do ha, hs hm ia ir k L N p PT, PL qlm q00 Q Qn
heat exchange area [m2] total surface area [m2] frontal area [m2] minimum free flow area [m2] specific heat [J/kg °C] diffusion coefficient [m2/s] hydraulic diameter [m] outer tube diameter [m] air-side and sensible heat-transfer coefficients [W/m2 K] mass-transfer coefficient enthalpy of humid air [J/kg] enthalpy of saturated water vapor at mean air temperature [J/kg] thermal conductivity [W/m K] length of the fin [m] number of rows of heat exchanger pressure [Pa] transversal and longitudinal tube pitch [m], (see Fig. 1) latent heat of water [J/kg] heat flux [W/m2] overall heat-transfer rate [W] heat-transfer rate of nth row [W]
These numerical results may help evaluate the commonly invoked data reduction practices: The fin efficiency is estimated and compared with existing correlations based on various approximations. The effects of inlet relative humidity, and the heat and mass transfer analogy are also examined and/or assessed.
Qmax Rv Re T t ui
vfr
w
Greek symbols gf fin efficiency l viscosity [N s/m2] q density [kg/m3] r standard deviation Subscripts a air b fin base f fin s saturated state t tube w fin surface
Energy:
@ @ 2 T^ ^ ¼ 1 ^j TÞ ðu @ ^xj Re Pr @ ^xj @ ^xj
dh ¼
4Amin L Aa
umax
Afr ¼ v fr Amin
ð1Þ
2 @ ^ @ w ^j wÞ ¼ D ðu @ ^xj @ ^xj @ ^xj
with
Amin ¼ MinðA1 ; 2A2 Þ
ð3Þ
where A1 and A2 are the areas between two adjacent tubes shown in Fig. 1 and vfr the frontal velocity, the dimensionless governing equations can be written as Continuity:
^j Þ ^u @ðq ¼0 @ ^xj
ð4Þ
Momentum:
^ ^i @ @p 1 @2u ^i u ^j Þ ¼ ðu þ @ ^xj @ ^xi Re @ ^xj @ ^xj
ð5Þ
ð7Þ
where the carets denote dimensionless variables and
q p T Tt ^¼ ; p ; T^ ¼ ; qin qin u2max T in T t q umax dh lC p ^ D ; Pr ¼ Re ¼ in ; D¼ umax dh l k
q^ ¼
ð8Þ
Here ui is the velocity component in the xi direction, p the pressure, T the temperature, w the humidity, D the mass diffusivity, q the density, l the dynamic viscosity, k the thermal conductivity, and Cp the specific heat. The flow is incompressible, but the density variation due to the temperature or the humidity is taken into consideration as was done in O’Connell [13].
q¼ ð2Þ
ð6Þ
Species:
2. Equations and solution procedure The humid air can be taken as a mixture of air and water vapor. Since the mass fraction of the water vapor is very low, it has little effect on the air flow. Taking air as the carrier fluid and water vapor as a dilute species, the laminar humid air flow can be described by the equations of continuity, momentum, energy, and species transport. The effects of water film on the overall heat transfer or the thermal resistance for the condensate of the droplet shape are known to be very small and thus are neglected in the present analysis. Introducing the hydraulic diameter dh and the characteristic velocity umax as the characteristic length and velocity scales:
maximum heat-transfer rate [W]: ma(ia,in ir,in) gas constant of air [J/kg K] Reynolds number temperature [°C; K] fin thickness [m] flow velocity in xi direction [m/s] frontal velocity [m/s] humidity [kg/kg air]
RT
P
p ðm k =M k Þ k
ð9Þ
where mk and Mk are the mass fraction and the molecular weight of species k, respectively. The saturated humidity, which is identified by the saturation line in the psychrometric chart, may be described by the following formula:
ws ðT w Þ ¼
ps ðT w Þ
qRv T w
ð10Þ
The fin configuration of the heat exchanger is taken to be that of the experimental settings [14] and the boundary conditions for air and water temperature, and humidity are also matched to those of the experiment. The schematic of the heat exchanger under consideration together with its dimensions is shown in Fig. 2. It may be assumed periodic in the transverse direction and it suffices to consider only half of a unit module indicated in the figure. The
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PL
A2
Air flow
PT
A1
do
(a)
Fig. 2. Schematic of the heat exchanger and detailed fin shape: (a) heat exchanger, (b) wave-type fin.
(b) Fig. 1. Schematic of a fin-tube heat exchanger: (a) tube arrangement, (b) fin broken into sectors.
^ q00latent ¼ qqlm Drw n
ð13Þ
The humidity on the fin surface is equal to the saturation humidity, ws, but assumes the value at the inlet if the local air has not reached the saturation condition:
calculation domain is chosen to include the fin and the flow passage between the fins, and is extended to 6do upstream and 20do downstream of the heat exchanger in the streamwise direction to capture the entrance and exit regions properly. Using the second-order upwind scheme for discretization of the convective terms, the governing Eqs. (4)–(7) are solved by the SIMPLE-type finite volume method, FLUENT, with the following boundary conditions: The velocity and the temperature along with the humidity are prescribed at the inlet boundary while the gradient in the flow direction is assumed negligible at the outlet boundary. No slip condition on the solid surface and the coolant-tube temperature are specified. The symmetry or periodic condition is imposed at the respective boundaries. At the fin–air interface, the temperature and the heat flux on the fin and air sides should match, respectively, as:
To properly account for the discontinuity in the heat flux between the air and fin sides as noted in Eq. (12), the air-side and fin-side domains are solved successively by using each other’s results as boundary conditions: Eqs. (11) and (14) are applied at the air-side interface while Eq. (12) is imposed on the fin side. These interface conditions are updated after each iteration and are determined as part of the solution. The sensible heat exchanged may then be estimated by integrating the heat flux into the air over the surface:
Tjfin ¼ Tjair
where ka is the thermal conductivity of the air. The fin efficiency, defined as the ratio of actual heat-transfer rate, Qactual, to the ideal maximum rate, Qideal, can be obtained.
kf
@T @T ¼ k þ q00latent a @nfin @nair
ð11Þ
ð12Þ
where the latent heat flux q00latent is determined from the mass-transfer rate on the surface:
ww ¼ min½ws ðT w Þ; win
Q sensible ¼
Z Aa
gf ¼
Q actual Q ideal
ka
@T dA @nair
ð14Þ
ð15Þ
ð16Þ
This concept is useful in data reduction and also in estimating the actual heat transfer from the more readily obtainable ideal heat
C.S. An, D.H. Choi / International Journal of Heat and Mass Transfer 55 (2012) 1496–1504
transfer, in which the temperature variation in the fin is ignored. Since the fin material is uniform and the fin is sufficiently thin, the air-side heat-transfer characteristic is the single most significant factor in the fin-tube heat exchangers. The exact solutions are known only for the limited fin types and configurations. It is thus, in general, obtained from a simple 1D analysis under the assumption that the heat-transfer coefficient for the fin surface is constant and the ambient temperature variation is negligible. For fin-tube heat exchangers, Schmidt’s approximation [1], in which the polygon-shaped fin area is approximated by an equivalent circular disk, is widely used. Improved results may be obtained by taking account of the varying air temperature [2], fin thickness [3], non-uniform thermal conductivity [5,6], variable heat-transfer coefficients [7], etc. Threlkeld [15], McQuiston [16], and Wu and Bong [17] proposed the fin efficiency under fully wet conditions. Wu and Bong also suggested a partially-wet 1D overall fin efficiency. Recently, Pirompugd et al. devised a reduction scheme for the multiple row fin-and-tube heat exchangers by subdividing the region into many tiny segments and analyzed the heat/mass transfer characteristics under dehumidifying conditions [18,19]. The fin efficiency approximations under both dry and wet conditions are compared with that obtained in the present study. 3. Results and discussion Taking advantage of periodicity and symmetry as pointed out in the previous section, only one half of the unit fin module is considered. Approximately 200,000 cells are used to fit the domain of each row as shown in Fig. 3. Additional 10,200 and 17,000 grid cells are distributed for the upstream and downstream regions of the fin, respectively. The calculation is performed for the inlet air temperature of 21 °C for the dry case and 27 °C with 43% humidity for the wet case, while the tube temperature is fixed at 45 °C and 6 °C for the dry and wet cases, respectively. To validate the procedure described above, the normalized overall heat-transfer rate under dry and wet conditions for heat exchangers with various row numbers is compared with the experimental data of Youn [14] in Fig. 4. The Reynolds number based on the inlet air condition ranges from 125 to 625 and 124 to 495 for dry and wet cases, respectively, for the frontal velocity range tested. The variation of the Reynolds
1499
number due to the heat transfer in the heat exchanger amounts to 6.5% for the single-row case at the lowest frontal velocity. The Prandtl number variation from inlet to outlet is not as significant and remains close to 0.72. The heat-transfer coefficient for a single module is first calculated by solving the fin–air conjugate heattransfer problem. To account for the temperature variation of the air and the coolant through the heat exchanger, the amount of heat transfer for all such modules is estimated by using the tube-bytube algorithm [20] with the fixed heat-transfer coefficient for each row. The results are in remarkably good agreement with the measurement for all conditions. The slightly less satisfactory results for the three-row cases may be attributed to the shortcomings inherent in the tube-by-tube method. The accuracy may be improved by carrying out the three-dimensional calculation for each fin segment with varying coolant and air temperatures. No attempt has been made to do this, though. The excellent agreement confirms that the procedure is capable of accurately predicting the heat-transfer performance of the heat exchanger. In subsequent calculations, it is seen that the spatial density variation of the air–vapor mixture due to the temperature change in the field is as large as ±4.2%, while those of viscosity and specific heat turn out to be smaller than ±1%. The variation of the thermal conductivity is about ±3%, but shows negligible effects on the results. This confirms that the present approach of treating the density as variable and keeping the transport properties constant is appropriate. In the leading-edge region, the hydraulic and thermal boundary layers are developing and the heat transfer is very active. The local heat transfer in the vicinity of the leading edge becomes high and the fin temperature varies rapidly. This phenomenon is clearly visible in the temperature and heat-flux distributions for the cooled case at 1 m/s frontal velocity in Fig. 5. The leading-edge area is the most effective heat-transfer region of the fin. The flow accelerates between the coolant tubes due to the narrowing flow passage; this results in the high heat flux on the fin surface there. In the certain region behind the tube, the air temperature becomes lower than the adjacent fin temperature and the negative heat transport is observed. This peculiar behavior may be explained as follows: the air temperature behind the coolant tube is directly affected by the low tube temperature while the fin surface below is kept relatively warm by the ambient air away from the spot. To see the heat-transfer behavior more clearly, the heat- and mass-flux
Fig. 3. Sample grid for the periodic computational domain.
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Fig. 4. Overall heat-transfer rate for various heat exchangers: (a) dry condition, (b) wet condition.
distributions along the A1–A10 cross-section for two different frontal velocities are shown in Fig. 6. The heat flux is very high near the leading edge in the developing region. It is natural to observe that a higher frontal velocity gives a larger overall heat flux. The slightly irregular behavior in the heat-flux distribution may be attributed to the crests or troughs of the wave fin that makes the flow accelerate or decelerate. The wall-shear-stress distribution in the symmetry planes, A1–A10 , A2–A20 , and A3–A30 in Fig. 7, clearly elucidates this phenomenon. The shear stress exhibits the local maxima or minima which coincide with the crests or the troughs of the wave fin; the heat flux tends to move with the wall shear stress for obvious reasons. The procedure is capable of predicting the humidity field about the heat exchanger. The saturation surface moves gradually outward from the fin surface as the air gets cooler as it travels downstream. Fig. 8 shows the development of the saturation region in the symmetry plane for various frontal velocities, i.e., 0.5, 1.0, and 2.0 m/s. Similar to the behavior of the hydraulic boundary
layer, the fully saturated region in the first row decreases as the flow velocity increases. For subsequent rows, however, the air is sufficiently cooled to the point that the entire region becomes fully saturated. The condensation occurs as the air temperature dips below the saturation temperature. The condensation in the leading edge region depends very much on whether the saturation condition is reached there. Once past the developing region, the condensation and the heat flux move in unison as seen in Fig. 6 for the crosssection A1–A10 . The trend continues to the second and third rows. The flow reaches the saturation condition earlier when the frontal velocity is lower; the larger condensation rate follows. One of the assumptions frequently adopted in the fin-efficiency approximation is axisymmetry. This assumption usually does not hold in real situations, as the fin temperature along three different circular paths (L1, L2, L3) for the air velocity of 1 m/s in Fig. 9 exhibits. It is evident that the axisymmetry deteriorates as the circle gets farther away from the tube: along the path close to the tube, L1, the dimensionless temperature varies little, smaller than 0.03, while the variation along L3 is 0.16. The flow appears to return to symmetry about the tubes for the second and third rows as the flow properties varies little in the streamwise direction. From these results, coupled with the humidity variation in the flow direction that needs to be accounted for in the analysis, onedimensional fin efficiency approximations or simple modified models thereof that only take account of the geometric change in the fin shape, may introduce significant error when applied to the fins of complex shape. The fin efficiency under both dry and wet conditions computed in the present study is depicted in Fig. 10. The results of the existing fin-efficiency approximations are also shown in the figure for comparison. The fin-efficiency curve is constructed by varying the frontal velocity. Under dry conditions shown in Fig. 10(a), the fin efficiency becomes higher as the number of rows increases. It is because that the fin–air temperature difference becomes smaller towards the downstream end and so does the temperature variation. This tends to make the actual heat transfer close to that of the ideal heat-transfer rate and pushes the fin efficiency up close to unity. For fins with large flow depth, the fin efficiency approximation underestimates the value up to 10–15%. The flow depth needs to be small for the fin efficiency approximation to remain valid. Among various approximations, the Han and Lefkowitz method underestimates the results most. The heat-transfer coefficient is very high near the leading edge of the fin and the fin efficiency is likely to be underestimated if it is based on the average heat-transfer coefficient. The sector method and the methods of Schmidt, and Huang and Shah agree quite well with one another but the values are appreciably smaller than the present results. The fin temperature variation is much less in the second and the third row than in the first row, i.e., the fin efficiency depends less on the distance from the tube there. The close agreement of Huang and Shah, which takes account of the variation of ambient air temperature in the transverse direction, with other methods seems to imply that the effects of transverse mixing is insignificant. For the fin efficiency under wet condition shown in Fig. 10(b), the earlier studies [15–17] are again seen to give significantly lower values. The coolant temperature in those studies is slightly different from that of the present, 7 °C vs. 6 °C. The discrepancy is attributed to the assumptions made in devising the approximation formulae. If the fully wet condition is met, i.e., the condensation occurs from the fin leading edge, it is reported that the fin efficiency varies only slightly with the relative humidity at the inlet [17–19]. It is seen in the figure that, when the inlet humidity changes from 60% to 100%, the fin efficiency by Wu and Bong [17] results in the difference of up to 6% while that in the present study 2.8%. For multi-row heat exchangers, the difference becomes
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Fig. 5. Temperature and local heat-transfer rate for
smaller as hs increases, i.e., with increasing frontal velocity, while the opposite is true for the single-row heat exchanger. The insensitive behavior of fin efficiency to the inlet relative humidity is due to the fact that the flow is in near fully wet condition. To see how the fin efficiency varies from the dry to wet condition, calculations have been carried out with varying inlet relative humidity and the results are plotted in Fig. 11. This is for the single-row case with three different inlet velocities. The fin efficiency remains fairly constant when the condition is either dry or fully wet. It varies mostly in the narrow region of partially wet regime between 25% and 40% of inlet relative humidity. This band of partially wet regime becomes narrower as the frontal velocity decreases. Similar behavior, but with narrower partially wet regime, is also observed for two- and three-row heat exchangers. For heat exchangers with multiple rows, the total heat-transfer rate increases with the number of rows. Unfortunately, that also accompanies higher pressure loss. From the perspective of wallshear stress depicted in Fig. 7, the shear stress becomes periodic immediately after the leading-edge region of the first row. The pressure drop for the heat exchanger, therefore, is expected to increase almost linearly with the number of rows. The performance of each row for various conditions is summarized in Table 1. In fact, the heat-transfer rate for the second or third row is much smaller than that of the first row. The lower the frontal velocity, the less effective the heat-transfer rate of the second and third rows. To assess the performance of the multi-row heat exchangers, the heat-transfer rate is plotted against the power consumed in Fig. 12. The driving power per unit face area is given by the product
^ (b) heat-transfer rate vfr = 1.0 m/s: (a) temperature T,
q00 . q00av
of the pressure drop through the heat exchanger DP and the frontal velocity vfr:
ðdriving powerÞ ¼ DP v fr ½W=m2 ðunit face areaÞ
ð17Þ
Naturally, for both dry and wet conditions, a single-row heat exchanger is superior to the multi-row heat exchangers when the power is low. As the power increases, however, the performance of the two- or three-row heat exchangers surpasses that of the single-row heat exchanger at point A indicated in the figure. Eventually, the performance of the three-row heat exchanger comes out to be on top at relatively high power beyond point B. To find the optimal range of operation for a given number of rows, the variation in the tube temperature must also be considered using, for example, the tube-by-tube method. This is not pursued in the present study to keep the analysis simple. As one of the many design considerations, it may be useful to point out that, at the crossover point, B in Fig. 12(b), for twoand three-row heat exchangers under the wet condition, the frontal velocities are 1.32 and 1.14 m/s, respectively. Similarly, for single- and two-row exchangers, the frontal velocities at the crossover point are 0.63 and 0.50 m/s. These numerical results may be vital in designing the heat exchangers. The effect of inlet humidity on the sensible heat transfer is presented in Fig. 13. The temperature change between the inlet and the outlet relative to that of the dry condition is plotted against the frontal velocity in the figure for two different inlet-humidity
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(a)
(a)
(b) (b)
Fig. 6. Heat-flux and condensation-rate distributions along A1–A10 section for two different frontal velocities: (a) vfr = 0.5 m/s, (b) vfr = 2.0 m/s.
conditions. This compares the sensible heat-transfer drop from that of the dry condition for various numbers of rows. The drop is most pronounced, above 20%, for the single row heat exchanger at the highest frontal velocity. This may be elucidated as follows: the air is most humid in the first row and thus the latent heat transfer becomes largest. This results in the smallest sensible heat transfer there. Increasing frontal velocity makes the moisture flux in the air stream larger and also reduces the sensible heat transfer. Naturally the drop becomes less significant as the frontal velocity decreases and/or the number of rows increases. This observation is in line with the experimental study of Wang et al. [21] but the present analytical procedure provides much finer details as to how the performance is affected. Let us now examine the analogy between the heat- and masstransfer on the fin surface. The following Pearson product–moment correlation coefficient Cor(hs, hm), which lies between 0 and 1 with 1 indicating the perfect dependence, is useful in quantifying the degree of linear dependence of the two parameters.
Corðhs ; hm Þ ¼
E½ðhs Eðhs ÞÞðhm Eðhm ÞÞ
rH rM
Fig. 7. Shear-stress distribution on the fin surface along the symmetry planes for two different frontal velocities: (a) vfr = 0.5 m/s, (b) vfr = 2.0 m/s.
ð18Þ
where E denotes the area average of the argument over the fin surface and rH, rM the standard deviation of hs, hm, respectively:
h n
rH ¼ E ðhs Eðhs ÞÞ2
oi1=2
ð19Þ
Fig. 8. Saturation lines in the A1–A10 section of the first row for various frontal velocities.
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Fig. 9. Fin surface temperature along various circular paths. Fig. 11. Effects of inlet relative humidity on fin efficiency for various inlet velocities.
(a) Table 1 Heat-transfer rates for the second and third rows relative to the first row.
vfr [m/s] 0.50 1.00 2.00
Dry
Wet
Q2/Q1
Q3/Q1
Q2/Q1
Q3/Q1
0.1444 0.3462 0.5752
0.02133 0.1104 0.2517
0.1785 0.3803 0.6125
0.03186 0.1364 0.2901
row. In the leading-edge region, the condensation rate is relatively low compared to the heat flux as seen in Fig. 6. This is presumably due to the dissimilarity of the developing profiles of temperature and humidity and/or the fact that the flow is not fully saturated in the leading-edge region. This tendency is well supported by the significantly small value of the correlation for the first row in the table. If the upstream 25% of the first row is treated separately, the correlation for this region is about 0.502, for the higher frontal velocity case, whereas that for the remaining 75% is 0.968, higher than that of the second row. One may deduce from this result that for fins with louvers or slits, as they have much larger overall developing area compared to the wave fins, the heat and mass transfer analogy would be less accurate.
(b)
4. Conclusions
Fig. 10. Fin efficiency against the heat-transfer rate: (a) dry condition, (b) wet condition.
h n
rM ¼ E ðhm Eðhm ÞÞ2
oi1=2
ð20Þ
The results are presented in Table 2. It is shown that the two are more strongly related in the second and third rows than in the first
A numerical procedure that predicts the overall heat-transfer rate of the heat exchanger has been developed. The three-dimensional continuity, momentum, energy, and species transport equations coupled with the condensation model at the interface are solved by using the finite volume method of SIMPLE type to obtain the heat-transfer rate of multi-row cross-flow fin-tube heat exchangers under both dry and wet conditions. The details of the heat-transfer characteristics including the local condensation rate are seen to be accurately captured. The optimal range of frontal velocity is also identified for one-, two-, and three-row heat exchangers. The fin efficiency concept is found to underestimate the heattransfer rate noticeably for all cases. This is attributed to the simplified assumptions adopted in approximating the fin efficiency. Although the results may be improved by making modifications
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(a)
vfr [m/s]
Row
Correlation factor
0.5
First Second Third First Second Third
0.8321 0.9384 0.9138 0.7513 0.9536 0.9451
2.0
to the formulae, the improvement is limited and leaves much to be desired in accurately estimating the fin efficiency, especially, for multi-row heat exchangers. For dehumidifying cases, the overall sensible heat-transfer rate seems insensitive to the inlet humidity change if the fully wet condition is met. The fin efficiency varies mostly in the narrow region of partially wet regime between 25% and 40% of inlet relative humidity. A quantitative assessment of the heat- and mass-transfer analogy by using the product–moment correlation shows that the analogy holds almost perfectly for most of the fin surface except the small developing entrance region. The analogy may still be useful and accurate when the inlet air is in near saturation state.
(b)
References
Fig. 12. Heat-transfer rate vs. power consumption for various numbers of rows: (a) dry condition, (b) wet condition.
1
(ΔTin-out )wet / (ΔTin-out )dry
0.95
0.9
0.85
0.8
RH = 60 % 0.75
0.7
RH = 100 % 0.5
1
N=1 N=2 N=3 1.5
2
vfr [m/s] Fig. 13. Sensible heat transfer relative to the dry case for two inlet-humidity conditions and varying row numbers.
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