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Nusselt number correlation for compact heat exchangers in transition regimes
Applied Thermal Engineering 151 (2019) 514–522
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Research Paper
Nusselt number correlation for compact heat exchangers in transition regimes
T
⁎
M.V.V. Morteana, , M.B.H. Mantellib a b
Joinville Technology Center, Federal University of Santa Catarina, Joinville, SC, Brazil Heat Pipe Laboratory, Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis, SC, Brazil
H I GH L IG H T S
theoretical and experimental thermal analysis is performed. • AStudy and transition flow regimes in compact heat exchanger. • Criticallaminar number is estimated to start at 1800. • A NusseltReynolds number correlation for transition regimes is proposed. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Nusselt number Diffusion bonding Compact heat exchanger Transition regimes
A compact heat exchanger is characterized by high heat transfer rates in small volumes. Basically it can be considered as a solid metal block where hot and cold fluid streams flow through mini or micro channels. The fluids usually flow in laminar regime, but the equipment can also operate in transition and in turbulent regimes. However, no model for the transition flow regimes, applied to compact heat exchangers, is available in the literature, as the few papers in this subject are usually focused in the classical correlations. The present work performed a theoretical and experimental thermal analysis of the laminar and transition flows in a diffusion bonded cross flow compact heat exchanger and proposed a new Nusselt number correlation for the transition regime. The theoretical model and the main correlations for laminar, turbulent and transition regime are presented. To validate the model, a stainless steel cut plate heat exchanger, with 450 square channels of 3 mm edge, was manufactured. The test facility was designed to operate with hot water and air at ambient temperature. The tests were performed with different mass flow rates, operating with Reynolds numbers between 360 and 2500. The theoretical and experimental results compared well, with theoretical data located within the range of the experimental uncertainty. However, for high Reynolds numbers, greater than 1650, the difference between the results tended to increase. Due to limitations of existing correlations, a Nusselt number correlation for air flow in the transition region and the liquid under constant temperature conditions, based on the asymptotic correlation method, is proposed for compact heat exchangers. The new model provides a smooth transition from the developing laminar flow to the turbulent. To validate the equation, a new test set was performed, and the proposed correlation presented a good accuracy in the transition regime.
1. Introduction Over the past years, research involving heat transfer and fluid flow in micro and mini channels has increased significantly, mainly due to industry demand for high heat transfer rates in small volumes. Compact heat exchangers were developed to fulfill this industry demand. These equipment are composed of mini or micro channels and present high surface area density, normally manufactured by the alternating stacking
⁎
of thin plates with small channels. The process of joining the plates to produce the core must be precise, efficient and cannot obstruct the channels. Conventional fusion welding techniques are not able to achieve the union requirements for these applications [1,2]. Due to the dimensional and quality requirements in microsystems, diffusion bonding process appears as a good alternative [3]. In a simple way, it can be described as a solid-state bonding technique, capable of joining large areas with a high
Corresponding author. E-mail address: [email protected] (M.V.V. Mortean).
https://doi.org/10.1016/j.applthermaleng.2019.02.017 Received 12 November 2018; Received in revised form 30 January 2019; Accepted 7 February 2019 Available online 08 February 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
Applied Thermal Engineering 151 (2019) 514–522
M.V.V. Mortean and M.B.H. Mantelli
Nomenclature A B C Dh D f K L L* ṁ n NTU Nu p P Pr q Re T u y z
Greek symbols ε
flow area [m2] dimensionless coefficients dimensionless coefficients hydraulic diameter, =4A/p [m] channel diameter [m] Darcy friction factor factor channel length [m] dimensionless thermal length mass flow rate [kg/s] superposition parameter Number of heat transfer units Nusselt number perimeter [m] pressure [kPa] Prandtl number heat transfer rate [kW] Reynolds number temperature [°C] velocity [m/s] dimensionless parameter dimensionless parameter
effectiveness
Subscript and abbreviations air b c exp h in lam out tran turb w 1 2 3 4 5 6
air bulk cold side experimental hot side inlet laminar outlet transição turbulent wall flow level 1 flow level 2 flow level 3 flow level 4 flow level 5 flow level 6
channel length and Prandtl number, respectively. L∗is the dimensionless thermal input length, defined as
mechanical strength in the interface region [3–5]. Printed Circuit Heat Exchangers [6,7], Diffusion-Bonded Plate-Fin Heat Exchangers [8] and Cut Plate Heat Exchangers [9,10] are examples of diffusion bonded compact heat exchangers. Large number of studies, models and correlations were developed for micro/mini-channels and compact heat exchangers operating in the laminar regime [11–13]. In some applications, the equipment also operates in transition and turbulent regimes, especially when they are used for the heat exchange of gases [6], as for offshore oil platforms applications. However, there are few papers and studies focused in the use of classical correlations and models to predict the heat transfer in turbulent or transition flow in micro channels. In the present work, a theoretical and experimental analysis of the thermal performance of a diffusion bonded cross flow compact heat exchangers, produced with rectangular mini channels, operating in the laminar and transition regimes, is performed. The applicability of the models and correlations for this range of operation is verified.
L∗ =
L Dh Re Pr
(2)
2. Literature review
According to Stephan and Preußer [16], Eq. (1) is valid for Prandtl numbers between 0.7 and 7 and L∗ > 0.03. A flow is considered under development when the dimensionless thermal length (L*) is lower than the dimensionless thermal input length, which, for square channels, is 0.06204 [14]. For fully developed laminar flows in square channels, the Nusselt number is constant and equal to 3.63 [18]. Mortean et al. [19] conducted a study focused on a laminar flow in a cross-flow compact heat exchanger. The author suggested a modification in Eq. (1), aiming its application on square cross-section channels, since the Stephan and Preußer [16] Nusselt number correlation was originally developed for circular channels. According to the author, the correlation that can better reproduce the thermal behavior of the square channel heat exchanger can be rewritten as:
2.1. Laminar flow
Nu = 3.63 + 0.086
Mini and micro channels applied to compact heat exchangers normally operate in the laminar regime, as presented in the researches of Kang et al. [11], Kim and No [12] and Luo et al. [13]. Mortean et al. [10] also studied the laminar flow in compact heat exchangers. The authors applied the Nusselt number correlation proposed by Lee and Garimella [14], Shah and London [15] and Stephan and Preußer [16] to predict the thermal behavior of a compact heat exchanger with rectangular channels. According to them, the correlation proposed by Stephan and Preußer [16] presented better results. Stephan and Preußer [16] proposed a Nusselt number correlation for laminar flow, considering the flow as under thermal development. The correlation was developed for circular channels, but also presented good results for different channel geometries [17] and is expressed as:
Gnielinski [20] also proposed a Nusselt number correlation for laminar flow, valid for Re < 2300 and Pr > 0.7:
Nu = 4.364 + 0.086
Nulam = {Num3 , q,1 + 0.63 + (Num, q,2 − 0.6)3 + Num3 , q,3 }1
(3)
3
(4)
where
Num, q,1 = 4.354
(5)
Num, q,2 = 1.953 3 Re Pr (D L)
(6)
Num, q,3 = 0.924 3 Pr Re (D L)
(7)
As already mentioned, a large number of studies were developed for micro/mini channels and compact exchangers operating in the laminar regime. On the other hand, there are few available studies that use classical Nusselt number correlations to predict heat transfer in turbulent or transition flow in micro/mini channels.
L∗)1.33
(1 1 + 0.1Pr (Dh Re L)0.83
(1 L∗)1.33 1 + 0.1Pr (Dh Re L)0.83
(1)
where Re, Dh, L and Pr are Reynolds number, hydraulic diameter, 515
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2.2. Turbulent flow
results. Harms et al. [36] experimentally studied the turbulent flow in rectangular micro channels and used the Gnielinski [26] correlation in their theoretical model. According to the authors, the experimental data were similar to the predicted by the correlation, but again, experimental results were higher than the analytical ones. Yu-ting et al. [37] performed an experimental study of the flow in a circular duct, with Reynolds number between 4000 and 10,000, and verified that the theoretical model, using the correlation proposed by Gnielinski [26], presented a similar trend to the experimental results.
The turbulent and laminar flows in ducts can be fully developed, under hydrodynamic development, under thermal development or under hydrodynamic and thermal development. In all these cases, the entry lengths of the dimensionless thermal and hydrodynamic boundary layers are much lower than those observed in laminar flow regimes. As a result, most of the turbulent studies employ fully developed turbulent flow models in their models [21]. This consideration is also adopted in the present work. The literature reports that, in turbulent regime, the channel geometry has not large influence on the flow thermal behavior significantly, different from observed in laminar flows [22]. Therefore, Nusselt number correlations for completely developed turbulent flows in circular ducts can also be applied, as a good approximation, in rectangular channels [21]. According to Rosa et al. [23], the thermal behavior of turbulent flows inside channels can be predicted well by several correlations, including the classic Nusselt number correlations of Dittus and Boelter [24], Petukhov [25], Gnielinski [26], and, in some cases, Adams et al. [27]; this last one was developed specifically for micro channels. Hesselgreaves [5] recommends the use of the Nusselt number correlation proposed by Gnielinski [26], originally developed for circular ducts, for turbulent flows in non-circular channels. This correlation presents an average error of 20% when compared to data [28,29]. Adams et al. [30] studied the applicability of Gnielinski [26] correlation in non-circular channels, verifying that the experimental data are within the limits predicted by the correlations. This correlation proposed by Gnielinski [26] is expressed as:
Nuturb =
2 3 (f 2)(Re − 1000) Pr ⎡1 + ⎛ D ⎞ ⎤ K ⎥ 1 + 12.7(f 2)1 2 (Pr 2 3 − 1) ⎢ L ⎝ ⎠ ⎦ ⎣
2.3. Transition flow Heat exchangers can also operate in laminar to turbulent transition region, however, the heat transfer behavior in this flow regime range is still not well known [38]. Wang et al. [39] studied the transition regime in rectangular ducts. According to them, there are few studies related with this subject in the literature. The authors compared the experimental data with the theoretical thermal model given by Eq. (8), concluding that the correlation proposed by Gnielinski [26] presented good results for the fully developed turbulent regime (Re > 7500), however, it was not able to predict the heat transfer in the transition region. Abraham et al. [28] verified that the correlation of Gnielinski [26] presented good results for Reynolds number higher than 3100 and proposed another Nusselt number correlation for the transition region (2300 < Re < 3100). Lee et al. [31] studied the applicability of the Nusselt number correlations proposed by Dittus and Boelter [24], Petukhov [25] and Gnielinski [26], developed for conventional channels, and of Adams et al. [27], developed specific for microchannels, to predicting the thermal behavior of rectangular microchannels, for Reynolds number between 2000 and 3500. According to the authors, the experimental Nusselt number was higher than predicted by Adams et al. [27] and Gnielinski [26] correlations, but smaller than those predicted by Dittus and Boelter [40] and Petukhov [25]. Gnielinski [32] carried out a study focused on heat transfer in the transition region, proposing a Nusselt number correlation for the transition region, consisting of an interpolation between laminar (Eq. (4)) and turbulent (Eq. (8)) Nusselt number correlations, valid for the transition region (2300 < Re < 4000), given by:
(8)
where f and D are, respectively, Darcy friction factor and channel diameter. The term 1 + (D L)2 3 represents the influence of duct length on the heat transfer. The K factor is expressed by:
(Pr Prw )0.11 for liquid K=⎧ n ⎨ for gas ⎩ (Tb Tw )
(9)
where T is temperature, and the subscripts b and w are relative to bulk and wall, respectively. The exponent n depends on the fluid, for air n = 0.45. In cases where temperature difference between inlet and outlet are not large, Eq. (8) can be simplified and written without the K factor [18]. Eq. (8) is valid for 2300 < Re < 1 × 104 and 0.5 < Pr < 2000 [26]. However, according to Lee et al. [31] the correlation can be used, with good approximation, for Reynolds number of up to 5 × 106. According to Gnielinski [26], Darcy friction factor (f) is calculated by the correlation proposed by Filonenko [26]:
f=
1 (1.82 log Re − 1.64)−2 4
Nu = (1 − γ ) Nulam,2300 + γNuturb,4000 where
γ=
1 (1.8 log Re − 1.5)−2 4
Re−2300 4000 − 2300
(13)
where Nulam,2300 and Nuturb,4000 are the Nusselt number calculated by Eq. (4) for Re = 2300 and by Eq. (8) for Re = 4000, respectively [32]. This correlation is valid for 2300 < Re < 4000 and Pr > 0.7. The author compared their theoretical results (Eqs. (4), (8) and (12)) with experimental data, and verified a good agreement. Bertsche et al. [41] carried out an heat transfer experimental study in circular ducts for a wide range of Reynolds and Prandtl numbers (500 < Re < 23,000 and 7 < Pr < 41). Their data were compared with the theoretical model, composed by Eqs. (4), (8) and (12), showing good convergence for the three flow regimes, with 80% of the experimental data within ± 15% of the values predicted by the correlations. Therefore, as no models for the transition flow regimes that can be applied directly to the compact heat exchangers are available in the literature, the present paper aims to study the thermal performance of a diffusion bonded compact heat exchanger operating in the laminar and transition flow regimes. In the analytical model, the Nusselt number correlation proposed by Mortean et al. [19] (Eq. (3)) is implemented for developing laminar flow, while the value of 3.63 is used for fully developed laminar flow and the results from Gnielinski [32] (Eq. (12)), for
(10)
In the recent study performed by Gnielinski [32], the author suggests replacing Eq. (10) by the one proposed by Konakov [33]:
f=
(12)
(11)
According to Rosa et al. [23], the applicability of the classical Nusselt number correlations proposed by Dittus and Boelter [24], Petukhov [25] and Gnielinski [26] to micro/mini channels, still needs to be studied. Barik et al. [34] performed a numerical study of turbulent flow inside a circular duct and compared their results with experimental data of Sleicher and Rouse [35] and with the correlation of Gnielinski [26]. The numerical results agreed well with the experimental data, however the theoretical model underestimated the experimental 516
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the transition regime. A heat exchanger was manufactured and tested with two streams operation condition: hot water and air at ambient temperature, for Reynolds numbers ranging from 360 to 2500. The theoretical and experimental data were compared and a new Nusselt number correlation for the transition region, based on the asymptotic correlation method was proposed.
3. Heat exchanger and experimental facility 3.1. Heat exchanger manufacture The cut-plate heat exchanger was produced with a new manufacturing method, presented for the first time by Mortean et al. [9]. This technology employs the water jet cutting process to produce minichannels on flat plates and diffusion bonding process to join the heat exchanger core. It should be noted that Mortean et al. [9] first produced a diffusion bonded compact heat exchanger prototype with C-12000 copper as base metal, but later, aiming industrial applications, stainless steel AISI 316L heat exchanger was produced and the data regarding its performance are used in the present work. Their manufacturing process consists of alternately stacking comb shape machined plates, produced with water jet cutting process, and non-machined flat plates. The sandwich between flat non-machined plates and comb shape plates forms one layer of the heat exchanger core (Fig. 1). The flat plates are responsible for separating the flows, and the comb shape machined plates are responsible for creating the channels sidewalls. After completing the layer stacking process, the assembly is inserted into a hot press furnace and diffusion bonded, producing the heat exchanger core. Diffusion bonding is a solid-state joining technique, where the assembly is submitted to uniaxial pressure and high temperature, in a vacuum atmosphere. Due to high temperatures and interface contact, the union of the two solid materials occurs through atomic diffusion. The main parameters involved in the process are: temperature, pressure and time [42]. Besides using basically the same stacking and diffusion bonding process, the present technology differs from the usual printed circuit heat exchangers in the sense that the geometry of the channels can be easily controlled by the design of the grooves of the middle plate and of the thickness of the sheets. The channels of the present heat exchanger was design to have square cross section of 3 mm of edge and rectangular cross section solid metal side walls (fins), 2 mm wide and 3 mm high. To separate the layers, 0.5 mm thickness flat plates were used (see Fig. 1). Boarder plates of 3 mm of thickness closes the core. The core was produced with 15 layers and 450 square channels for each flow, in a cross flow configuration. The complete heat exchanger was 167 mm wide, 167 mm long and 111 mm high (Fig. 1). The entire assembly was bonded at 1050 °C for 60 min under a constant pressure of 18 MPa. Finishing the manufacturing process, headers were welded on the sides of the heat exchanger. The complete heat exchanger is shown in Fig. 2.
Fig. 2. Compact heat exchanger with headers.
Fig. 3. Core sample.
To evaluate the joint quality of the heat exchanger, a metallography analysis of the bonded interface was performed in an optical microscope. The studied region is shown in Fig. 3. The metallography analysis was carried out on the top and bottom part of the channel. In total, four channels were investigated. Fig. 4 presents the result. As it can be seen in Fig. 4, the sample presented a small deformation, resulting in some failures in the bonding interface, as evidenced by horizontal dark lines in images I, J, L, M and N. The main effect of these failures in the bonding interface is to reduce the maximum working pressure that the heat exchanger can bear. In the current work, the core was tested at low pressures, between 1.5 and 3 bar; therefore the failures did not influence the results. To produce a core without failures and able to work with high pressures, it is recommended the conduction
Fig. 1. Stainless steel heat exchanger: (a) stacking process, (b) complete core and (c) core geometric details. 517
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Fig. 4. Metallography result.
minimum to the maximum (c1 to c6). This same procedure was performed for all flow levels, resulting in 30 tests. In the tests, the air flow ranged from laminar to transition region: 364 < Re < 2496. Table 3 presents the inlet and outlet temperatures, as well as the Reynolds numbers for both flows, the experimental heat transfer rate (qexp) and the experimental effectiveness (ε exp), this last defined as the rate between the experimental heat transfer rate and the maximum heat transfer rate that is thermodynamically possible. As it can be seen in Table 3, the heat transfer rate varied from 0.44 to 2.23 kW. It is worth noting that the high temperature of the outlet air is close to the water inlet temperature, evidencing the high effectiveness of the heat exchanger, ranging between 0.77 and 0.96. The analytical model, used to perform the thermal design of the heat exchanger, was the same presented by Mortean et al. [10]. The authors use the core geometric parameters and ε-NTU method to predict the thermal behavior of the compact heat exchanger. This method depends on the Nusselt number to estimate the convection heat transfer coefficient. For more details of the design procedure, see Mortean et al. [10]. In the current work, the Nusselt number model proposed by Mortean et al. [19] (see Eq. (3)) was used for the laminar flow under thermal development. For fully developed laminar flow, the Nu number was taken as 3.63. For the transition regime, the correlation proposed by Gnielinski [32] (see Eq. (12)) was used. The fan used did not allowed to reach turbulent flow regimes and so Eq. (8) proposed by Gnielinski [26] was not applied. Fig. 6 shows the heat transfer rate (q) as a function of the cold stream Reynolds number (Rec), for the test set data h1_c1 to h1_c6 (see Table 1). In this case, the hot side mass flow rate (water) was kept constant at the lowest level (h1), while the cold side mass flow rate
of further analysis of the diffusion bonding process for compact heat exchanger applications to establish the optimal bonding parameters of stainless steel AISI 316L. 3.2. Experimental tests An experimental apparatus was developed with the objective of evaluating the thermal performance of the cut-plate heat exchanger and verifying the application of Nusselt number correlations for these heat transfer flows. This setup was designed to work with air at ambient temperature and water at high temperature. On the hot side, the water is stored in a vessel and heated by resistors, up to the desired working temperature. On the cold side, a centrifugal fan is responsible for the air flow at ambient temperature. The fan power allows the heat exchanger to be tested from the laminar flow regime to the beginning of the transition flow. Fig. 5 shows the experimental apparatus. It is basically composed of tubes, a centrifugal fan, a vessel, a pump and a bank of electrical resistances. Type T thermocouples and RTD temperature sensors monitor the temperatures. Liquid turbine flow meter FTB-1316 and an anemometer measure the liquid and air mass flow rates. Besides, among other equipment, the setup counts with two manometers, a data acquisition system National Instrument NI cDAQ -9178 and a computer. The tests are performed varying the mass flow rates: five levels for the hot side stream (water) and six levels for the cold side stream (air), as shown in Table 1. During the tests, the inlet air temperature was maintained at constant levels: ambient temperature for the cold side and 70 °C for the hot side (water). The absolute pressure ranged from 195 kPa to 230 kPa, for the hot side, and from 103.5 kPa to 105.5 kPa, for the cold side. The following parameters were measured: inlet and outlet temperatures of the hot and cold flows (Th,in, Tc,in, Th,out, and Tc,out), inlet pressure, mass flow rates on the hot side and cold air velocity (Ph,in, Pc,in, ṁ h and uair , respectively). Based on the air velocity, measured with the anemometer, the mass flow rate was calculated. The data uncertainty is presented in Tables 1 and 2. 4. Results and discussion As already noted, in order to study the influence of the Reynolds number on the heat transfer, the tests were performed by varying the mass flow of one stream while the other one was kept constant. Due to fan power limitations, it was not possible to achieve turbulent flow inside the channels. Initially, the hot side was kept constant at h1 (0.348 kg/s) and the cold flow was varied from the lowest (c1 = 0.0095 kg/s) to the highest value (c6 = 0.0645 kg/s). Then, the hot flow was set to the second level, h2 (0.607 kg/s) and the cold flow was again varied from the
Fig. 5. Experimental facility. 518
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3.0
Table 1 Mass flow rate. Cold side (kg/s) - air
Theoretical model Experimental data
2.5
Hot side (kg/s) - water Uncertainty
ṁ h
Uncertainty
c1 = 0.0095 c2 = 0.0211 c3 = 0.0314 c4 = 0.0422 c5 = 0.0539 c6 = 0.0645
0.0015 0.0022 0.0055 0.0062 0.0069 0.0075
h1 = 0.348 h2 = 0.607 h3 = 0.850 h4 = 1.122 h5 = 1.349 –
0.04 0.04 0.04 0.04 0.04 –
2.0
q (kW)
ṁ c
1.5 1.0 0.5 0.0 0
Table 2 Uncertainty parameters. Parameter Uncertainty
Th, in (°C) 0.18
500
1000
1500
2000
2500
3000
Re_c Th, out (°C) 0.18
Tc, in (°C) 0.08
Tc, out (°C) 0.18
Fig. 6. Comparison between theoretical and experimental heat transfer rate for the test set h1_c1 to h1_c6 (Reh = 640).
presented similar behaviors, within the range of experimental uncertainty. The mean difference between the theoretical and experimental data, taking into account all tests, was approximately 12%. Also, it can be observed that for low Reynolds numbers (the first four test points: 350 < Re < 1650), within the laminar region, the theoretical and experimental results presented a very similar trend, with an average difference of approximately 8%. However, for higher Reynolds numbers, the difference tends to increase, reaching approximately 19.8% for Re > 1650 (the last 2 test points). This difference between theoretical and experimental data can be imputed to the fact that the data is located within the beginning of the transition region, i.e., from laminar to turbulent flow.
varied from the lowest (c1) to highest value (c6). Fig. 7 shows the results for the last set of tests: h5_c1 to h5_c6, where hot side mass flow rate was kept constant at level h5, while the cold side mass flow rate varied from c1 to c6. In the first three experimental points (Rec ≈ 320, 820 and 1220), the cold side flow was at the fully developed laminar regime, sinceL∗ > 0.06204 (see Eq. (2)). In the next two experimental points (Rec ≈ 1660 and 2110), the flow was in the limit of the laminar condition and the flow is considered under development since L∗ ⩽ 0.06204 . The last experimental point (Rec ≈ 2490), the flow was in the transition region. As already explained, in the theoretical model, the appropriate Nusselt number correlation was used according to each of these flow conditions. From Figs. 6 and 7, one can see that, for the range of Reynolds numbers studied, the theoretical model curves and experimental data
Table 3 Parameters collected during the tests. Test
Reh
Rec
Th,in (°C)
Th,out (°C)
Tc,in (°C)
Tc,out (°C)
qexp (kW)
εexp
c1_h1 c2_h1 c3_h1 c4_h1 c5_h1 c6_h1
638 644 638 636 644 635
364 821 1216 1658 2083 2493
70.12 70.22 70.42 70.35 71.78 70.54
69.93 69.84 69.73 69.48 70.65 69.25
22.27 22.13 21.17 21.85 24.32 26.62
68.29 62.78 60.16 59.43 60.21 59.58
0.44 0.87 1.22 1.61 1.94 2.14
0.96 0.84 0.79 0.77 0.75 0.75
c1_h2 c2_h2 c3_h2 c4_h2 c5_h2 c6_h2
1104 1131 1115 1112 1113 1116
365 824 1213 1662 2112 2487
70.24 70.37 70.53 70.39 70.40 70.45
70.13 70.17 70.16 69.92 69.82 69.74
22.14 22.26 20.95 22.14 24.47 26.62
68.48 63.33 60.68 60.13 60.15 60.35
0.44 0.88 1.24 1.63 1.95 2.18
0.96 0.85 0.80 0.78 0.77 0.76
c1_h3 c2_h3 c3_h3 c4_h3 c5_h3 c6_h3
1561 1560 1562 1560 1557 1564
363 818 1246 1674 2112 2487
70.07 70.14 70.41 70.18 70.24 70.38
69.99 69.99 70.17 69.82 69.81 69.87
21.87 22.25 20.81 22.26 24.34 26.58
68.41 63.34 60.81 60.24 60.37 60.60
0.44 0.87 1.28 1.65 1.97 2.20
0.96 0.85 0.80 0.79 0.78 0.77
c1_h4 c2_h4 c3_h4 c4_h4 c5_h4 c6_h4
2040 2055 2068 2064 2059 2061
364 815 1229 1603 2071 2492
69.98 70.07 70.22 70.16 70.17 70.20
69.91 69.95 70.03 69.91 69.87 69.84
21.36 22.33 20.74 22.33 24.23 26.56
68.23 63.41 60.80 60.45 60.42 60.73
0.45 0.87 1.26 1.58 1.94 2.22
0.96 0.86 0.80 0.79 0.78 0.78
c1_h5 c2_h5 c3_h5 c4_h5 c5_h5 c6_h5
2473 2467 2480 2471 2469 2472
355 797 1227 1598 2093 2496
69.92 69.91 70.15 70.18 70.12 70.22
69.87 69.80 70.00 69.98 69.88 69.91
20.71 21.90 20.48 22.24 24.36 26.58
68.13 63.29 60.75 60.53 60.56 60.86
0.44 0.86 1.27 1.58 1.96 2.23
0.96 0.86 0.81 0.79 0.79 0.78
519
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3.0
16
2.5
Theoretical model
14
Experimental data
12 10 Nu
q (kW)
2.0
Mortean et al. [19] - Eq. (3) Gnielinski [26] - Eq. (8) Proposed - Eq. (18) Gnielinski [32] - Eq. (12)
1.5
8 6
1.0
4
0.5
2
0.0
0 1000
0
500
1000
1500
2000
2500
3000
Re_c
1500
2000
2500 Re
3000
3500
4000
Fig. 8. Nusselt number as a function of Reynolds number.
Fig. 7. Comparison between theoretical and experimental heat transfer rate for the test set h5_c1 to h5_c6 (Reh = 2472).
Nulam for Re → 0 Nutran = ⎧ ⎨ ⎩ Nuturb for Re → ∞
(16)
4.1. Transition region analysis
Nutran = [(Nulam )n + (Nuturb )n]1
According to the literature, the transition region between the laminar and turbulent flows is characterized when a critical Reynolds number (Recrit) [21] value is exceeded. For circular channels and fully developed flows, the critical Reynolds number is approximately 2300 [21]. However, some researches concerning mini and micro channels presented smaller transition Reynolds numbers. Wang and Peng [43] investigated water and methanol flow in a rectangular micro channel and reported that the turbulent flow started at Reynolds number close to 1500. Harms et al. [29] also verified that the transition region occurred for Reynolds number equal to 1500. Lee et al. [31] verified a change in the laminar flow behavior for Reynolds number between 1500 and 2000. The experimental and numerical results of Sahar et al. [44] showed that the transition from laminar to turbulent occurred for 1600 < Re < 2000. Natrajan and Christensen [45] attributed the change of regime flows for Reynolds numbers of 1800. Mylavarapu [46] studied the flow in mini channels and also reported the change from laminar to turbulent regime for Reynolds number equal to 1800. One can see that in most studies the transition from laminar to turbulent regime occurred for Reynolds numbers lower than 2300 and near to 1800. In the current work, this change behavior occurred for Reynolds number between 1650 and 2050. So, based on the literature reports, in the present work, the critical Reynolds number is defined as 1800. However, Eq. (12) [32] and the existing correlations for the transition region, as proposed by Abraham et al. [28], are not valid for Reynolds numbers smaller than 2300. Therefore, it is not possible to use these correlations in the comparison with experimental data in the transition region, considered to start at 1800. In order to better predict the heat exchanger thermal behavior in the transition region, a Nusselt number correlation is proposed (Nutrans ) , based on the asymptotic correlation method proposed by Churchill and Usagi [47]. The model presents an interpolation method that mathematically associates two distinct equation behaviors, for the same dependent dimensionless parameter y, which are functions of the independent dimensionless parameter z. For the case of the following functions:
y = Bz p for z → 0 y=⎧ 0 q ⎨ ⎩ y∞ = Cz for z → ∞
n
(17)
where Nulam and Nuturb are the Nusselt numbers for the developing laminar flow, Eq. (3), and turbulent flow, Eq. (8), respectively. This method proposes a correlation that presents a smooth transition between the laminar and turbulent flow regimes, able to capture better the observed experimental results. To estimate the parameter n, the difference between Eqs. (3) and (8), for Re = 1800 and Re = 4000, respectively, and those estimated by Eq. (17), should be the lowest possible value. Based on the mentioned criteria, the value of n, which minimizes the root mean square error comparing the theoretical data by Eq. (17) with the experimental results of Table 3, was estimated to be 6. Therefore, the following model is proposed:
Nutran = [(Nulam )6 + (Nuturb )6]1
6
(18)
Fig. 8 shows the Nusselt number based on Eqs. (3), (8), (12) and (18), i.e., using the developing laminar flow correlation developed by Mortean et al. [19], the turbulent flow correlation of Gnielinski [26], the transition region correlation based on the work of Gnielinski [32] and finally the model proposed on the present work (Eq. (18)), respectively. As expected, the proposed solution provides a smooth transition from the developing laminar flow to the turbulent flow. Comparing the experimental data from the transition region (presented in Table 3) with the theoretical model using Eq. (18), the mean difference between the results is approximately 13.9%, lower than the 19.8% previous calculated with Eq. (12) [32]. In order to check the efficiency of the proposed model, additional tests were carried out, focusing on the transition region. During the tests, the water mass flow rate was kept constant at 0.853 kg/s and air flow was varied from 0.037 kg/s (Rec = 1440) to 0.069 kg/s (Rec = 2690). Fig. 9 shows the comparison between the results, using Eq. (12) [32], the proposed correlation (Eq. (18)), and the experimental data. As can it be seen in Fig. 9, the theoretical model compare quite well with the experimental data. The mean difference between experimental data and theoretical model, for transition region, using Eq. (12) [32] was approximately 21.6%; however, with the proposed model (Eq. (18)), the difference reduced to 16.3%. Besides, the model was able to reproduce all the data trends observed. Therefore, based on the present results, it is suggested the use of the proposed correlation to the calculation of the Nusselt number (and so the convective heat transfer coefficients) for flows in the laminar to turbulent transition regions, in compact heat exchangers operating in similar conditions of the device tested in the current work.
(14)
where B and C are dimensionless coefficients, one has, for q > p :
y = [(y0 )n + (y∞ )n]1
n
(15)
where n is a superposition parameter determined by comparison with experimental data. For the current case, the model takes the form: 520
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3.0
https://doi.org/10.4028/www.scientific.net/KEM.297-300.2795. [2] J.J. Brandner, L. Bohn, T. Henning, U. Schygulla, K. Schubert, Microstructure heat exchanger applications in laboratory and industry, Heat Transf. Eng. 28 (2007) 761–771, https://doi.org/10.1080/01457630701328528. [3] Y. Li, F. Xuan, S. Li, S. Tu, Quality evaluation of diffusion bonded joints by electrical resistance measuring and microscopic fatigue testing, Chin. J. Mech. Eng. 24 (2011) 1. [4] N. Bouquet, E. Rigal, S.C. Grenoble, F. Bernard, O.H. Dijon, Interface formation during HIP-bonding of austenitic stainless steel, in: Brazing High Temp. Brazing Diffus. Bond. LÖT 2013, DVS Media, Aachen, 2013, pp. 118–122. [5] J.E. Hesselgreaves, Compact Heat Exchangers: Selection, Design, and Operation, Pergamon, Amsterdam ; New York, 2001. [6] S.K. Mylavarapu, X. Sun, R.E. Glosup, R.N. Christensen, M.W. Patterson, Thermal hydraulic performance testing of printed circuit heat exchangers in a high-temperature helium test facility, Appl. Therm. Eng. 65 (2014) 605–614, https://doi. org/10.1016/j.applthermaleng.2014.01.025. [7] S.K. Mylavarapu, X. Sun, R.N. Christensen, R.R. Unocic, R.E. Glosup, M.W. Patterson, Fabrication and design aspects of high-temperature compact diffusion bonded heat exchangers, Nucl. Eng. Des. 249 (2012) 49–56, https://doi.org/ 10.1016/j.nucengdes.2011.08.043. [8] Q. Li, G. Flamant, X. Yuan, P. Neveu, L. Luo, Compact heat exchangers: A review and future applications for a new generation of high temperature solar receivers, Renew. Sustain. Energy Rev. 15 (2011) 4855–4875, https://doi.org/10.1016/j.rser. 2011.07.066. [9] M.V.V. Mortean, L.H.R. Cisterna, K.V. Paiva, M.B.H. Mantelli, Development of diffusion welded compact heat exchanger technology, Appl. Therm. Eng. 93 (2016) 995–1005, https://doi.org/10.1016/j.applthermaleng.2015.09.021. [10] M.V.V. Mortean, K.V. Paiva, M.B.H. Mantelli, Diffusion bonded cross-flow compact heat exchangers: theoretical predictions and experiments, Int. J. Therm. Sci. 110 (2016) 285–298, https://doi.org/10.1016/j.ijthermalsci.2016.07.010. [11] S.-W. Kang, Y.-T. Chen, G.-S. Chang, The manufacture and test of (110) orientated silicon based micro heat exchanger, Tamkang J. Sci. Eng. 5 (2002) 129–136. [12] I.H. Kim, H.C. No, Thermal hydraulic performance analysis of a printed circuit heat exchanger using a helium–water test loop and numerical simulations, Appl. Therm. Eng. 31 (2011) 4064–4073, https://doi.org/10.1016/j.applthermaleng.2011.08. 012. [13] L. Luo, Y. Fan, W. Zhang, X. Yuan, N. Midoux, Integration of constructal distributors to a mini crossflow heat exchanger and their assembly configuration optimization, Chem. Eng. Sci. 62 (2007) 3605–3619, https://doi.org/10.1016/j.ces.2007.02.058. [14] P.-S. Lee, S.V. Garimella, Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios, Int. J. Heat Mass Transf. 49 (2006) 3060–3067, https://doi.org/10.1016/j.ijheatmasstransfer.2006.02.011. [15] R.K. Shah, A.L. London, Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data, Academic Press, New York, 1978. [16] K. Stephan, P. Preußer, Wärmeübergang und maximale Wärmestromdichte beim Behältersieden binärer und ternärer Flüssigkeitsgemische: Wärmeübergang und maximale Wärmestromdichte beim Behältersieden binärer und ternärer Flüssigkeitsgemische, Chem. Ing. Tech. 51 (1979), https://doi.org/10.1002/cite. 330510112 37 37. [17] P.-S. Lee, Heat Transport in Silicon Microchannel Arrays, Purdue University, 2007. [18] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed, Wiley, New York, 1996. [19] M.V.V. Mortean, L.H.R. Cisterna, K.V. Paiva, M.B.H. Mantelli, Thermal and hydrodynamic analysis of a cross-flow compact heat exchanger, Appl. Therm. Eng. 150 (2019) 750–761, https://doi.org/10.1016/j.applthermaleng.2019.01.038. [20] V. Gnielinski, G1 Heat Transfer in Pipe Flow, in: VDI e. V. (Ed.), VDI Heat Atlas, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010, pp. 691–700. http://link. springer.com/10.1007/978-3-540-77877-6_34 (accessed May 16, 2016). [21] S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-phase Convective Heat Transfer, Wiley, New York, 1987. [22] R.K. Shah, D.P. Sekulić, Fundamentals of Heat Exchanger Design, John Wiley & Sons, Hoboken, NJ, 2003. [23] P. Rosa, T.G. Karayiannis, M.W. Collins, Single-phase heat transfer in microchannels: the importance of scaling effects, Appl. Therm. Eng. 29 (2009) 3447–3468, https://doi.org/10.1016/j.applthermaleng.2009.05.015. [24] F.W. Dittus, L.M.K. Boelter, Heat Transfer in Automobile Radiators of the Tubular Type, vol. 2, Univ. Calif. Publ. Eng., 1930, pp. 443–461. [25] B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, in: Adv. Heat Transf., Elsevier, 1970, pp. 503–564. [26] V. Gnielinski, New equations for heat and mass transfer in turbulent pipe and channel flow, Int. Chem. Eng. 16 (1976) 359–368. [27] T.M. Adams, S.I. Abdel-Khalik, S.M. Jeter, Z.H. Qureshi, An experimental investigation of single-phase forced convection in microchannels, Int. J. Heat Mass Transf. 41 (1998) 851–857, https://doi.org/10.1016/S0017-9310(97)00180-4. [28] J.P. Abraham, E.M. Sparrow, J.C.K. Tong, Heat transfer in all pipe flow regimes: laminar, transitional/intermittent, and turbulent, Int. J. Heat Mass Transf. 52 (2009) 557–563, https://doi.org/10.1016/j.ijheatmasstransfer.2008.07.009. [29] A.J. Ghajar, L.-M. Tam, Heat transfer measurements and correlations in the transition region for a circular tube with three different inlet configurations, Exp. Therm. Fluid Sci. 8 (1994) 79–90. [30] T.M. Adams, M.F. Dowling, S.I. Abdel-Khalik, S.M. Jeter, Applicability of traditional turbulent single-phase forced convection correlations to non-circular microchannels, Int. J. Heat Mass Transf. 42 (1999) 4411–4415. [31] P.-S. Lee, S.V. Garimella, D. Liu, Investigation of heat transfer in rectangular microchannels, Int. J. Heat Mass Transf. 48 (2005) 1688–1704, https://doi.org/10. 1016/j.ijheatmasstransfer.2004.11.019.
2.5 q (kW)
2.0
1.5 1.0
Gnielinski [32] - Eq. (12) Experimental data Proposed - Eq. (18)
0.5
0.0 1300
1550
1800
2050
2300
2550
2800
Re_c Fig. 9. Comparison between proposed model and experimental data of additional tests.
5. Conclusion In this work, a theoretical and experimental study of the thermal performance of cut plate heat exchanger was performed, for the equipment operating in the laminar and transition regimes, aiming to verify the applicability of the existing models and correlations for this range of operation. The theoretical model and the main correlations for laminar and transition regimes were presented. The compact heat exchanger, produced in 316L stainless steel and with 450 square channels of 3 mm edge, was used in the experimental tests. An experimental facility was developed to operate with air at ambient temperature and hot water at 70 °C. Initially 30 tests were performed, for laminar and transition flows. To measure the heat exchanger thermal performance, the analytical model presented by Mortean et al. [10] was applied. The Nusselt number correlation proposed by Mortean et al. [19] (Eq. (3)), for developing laminar flow, and the correlation of Gnielinski [32] (Eq. (12)), for transition regime, were used. The theoretical and experimental results showed similar behaviors, with the theoretical data laying within the experimental uncertainty range. However, the difference between the results tended to increase as the Reynolds number increased, presenting an average difference of 19.8% for Reynolds numbers higher than 1650, showing limitations of the of the existing correlations. Therefore, a new Nusselt number correlation for air flows in the transition regime, and water under close to constant liquid temperature conditions, using 1800 as the critical Reynolds number, was proposed for compact heat exchanger. The mean difference between the theoretical model, using the proposed correlation, and experimental data, from the transition region, reduced to approximately 13.9%. Also, the new correlation was able to follow all the trends observed in the data. Aiming to confirm the applicability of the new correlation, an additional set of test was performed, in which the Reynolds number was varied from Rec = 1440 to Rec = 2690, therefore in the transition region. Comparing the new experiment data with the theoretical model, it was verified that the new correlation presented better results than those tested before, as the mean difference between experimental data and theoretical model, using the proposed correlation, was 16.3%. Finally, the present authors propose the use of this model to calculate the Nusselt number for the laminar to turbulent transition region, for compact heat exchangers, especially those with square cross section mini channels. Due to the limitations of the experiments, further study still needs to be performed in order to reduce the error and increase the accuracy of the proposed equation. References [1] Z.L. An, W.-L. Luan, F.Z. Xuan, S.T. Tu, High temperature performance of 316L-SS joint produced by diffusion bonding, Key Eng. Mater. 297–300 (2005) 2795–2799,
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2012.08.010. [40] W.M. Rohsenow, J.P. Hartnett, Y.I. Cho (Eds.), Handbook of Heat Transfer, 3rd ed., McGraw-Hill, New York, 1998. [41] D. Bertsche, P. Knipper, T. Wetzel, Experimental investigation on heat transfer in laminar, transitional and turbulent circular pipe flow, Int. J. Heat Mass Transf. 95 (2016) 1008–1018, https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.009. [42] D.J. Stephenson, ed., Diffusion Bonding 2, Springer Netherlands, Dordrecht, 1991. http://doi.org/10.1007/978-94-011-3674-7. [43] B.X. Wang, X.F. Peng, Experimental investigation on liquid forced-convection heat transfer through microchannels, Int. J. Heat Mass Transf. 37 (1994) 73–82, https:// doi.org/10.1016/0017-9310(94)90011-6. [44] A.M. Sahar, M.R. Özdemir, E.M. Fayyadh, J. Wissink, M.M. Mahmoud, T.G. Karayiannis, Single phase flow pressure drop and heat transfer in rectangular metallic microchannels, Appl. Therm. Eng. 93 (2016) 1324–1336, https://doi.org/ 10.1016/j.applthermaleng.2015.08.087. [45] V.K. Natrajan, K.T. Christensen, Non-intrusive measurements of transitional and turbulent convective heat transfer in a rectangular microchannel, J. Micromech. Microeng.. 21 (2011) 085001, https://doi.org/10.1088/0960-1317/21/8/085001. [46] S.K. Mylavarapu, Design, Fabrication, Performance Testing, and Modeling of Diffusion Bonded Compact in a High-temperature Helium Test Facility, The Ohio State University, (2011). [47] S.W. Churchill, R. Usagi, A general expression for the correlation of rates of transfer and other phenomena, AIChE J. 18 (1972) 1121–1128, https://doi.org/10.1002/ aic.690180606.
[32] V. Gnielinski, On heat transfer in tubes, Int. J. Heat Mass Transf. 63 (2013) 134–140, https://doi.org/10.1016/j.ijheatmasstransfer.2013.04.015. [33] P.K. Konakov, Eine neue Formel für den Reibungskoeffizienten glatter Rohre (Orig. Russ), Berichte Akad, Wiss. UdSSR. 51 (1946) 503–506. [34] A.K. Barik, A. Mukherjee, P. Patro, Heat transfer enhancement from a small rectangular channel with different surface protrusions by a turbulent cross flow jet, Int. J. Therm. Sci. 98 (2015) 32–41, https://doi.org/10.1016/j.ijthermalsci.2015.07. 003. [35] C.A. Sleicher, M.W. Rouse, A convenient correlation for heat transfer to constant and variable property fluids in turbulent pipe flow, Int. J. Heat Mass Transf. 18 (1975) 677–683, https://doi.org/10.1016/0017-9310(75)90279-3. [36] T.M. Harms, M.J. Kazmierczak, F.M. Gerner, Developing convective heat transfer in deep rectangular microchannels, Int. J. Heat Fluid Flow. 20 (1999) 149–157, https://doi.org/10.1016/S0142-727X(98)10055-3. [37] W. Yu-ting, L. Bin, M. Chong-fang, G. Hang, Convective heat transfer in the laminar–turbulent transition region with molten salt in a circular tube, Exp. Therm. Fluid Sci. 33 (2009) 1128–1132, https://doi.org/10.1016/j.expthermflusci.2009. 07.001. [38] J.P. Abraham, E.M. Sparrow, W.J. Minkowycz, Internal-flow Nusselt numbers for the low-Reynolds-number end of the laminar-to-turbulent transition regime, Int. J. Heat Mass Transf. 54 (2011) 584–588, https://doi.org/10.1016/j. ijheatmasstransfer.2010.09.012. [39] C. Wang, P. Gao, S. Tan, Z. Wang, Forced convection heat transfer and flow characteristics in laminar to turbulent transition region in rectangular channel, Exp. Therm. Fluid Sci. 44 (2013) 490–497, https://doi.org/10.1016/j.expthermflusci.
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Research Paper
Nusselt number correlation for compact heat exchangers in transition regimes
T
⁎
M.V.V. Morteana, , M.B.H. Mantellib a b
Joinville Technology Center, Federal University of Santa Catarina, Joinville, SC, Brazil Heat Pipe Laboratory, Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis, SC, Brazil
H I GH L IG H T S
theoretical and experimental thermal analysis is performed. • AStudy and transition flow regimes in compact heat exchanger. • Criticallaminar number is estimated to start at 1800. • A NusseltReynolds number correlation for transition regimes is proposed. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Nusselt number Diffusion bonding Compact heat exchanger Transition regimes
A compact heat exchanger is characterized by high heat transfer rates in small volumes. Basically it can be considered as a solid metal block where hot and cold fluid streams flow through mini or micro channels. The fluids usually flow in laminar regime, but the equipment can also operate in transition and in turbulent regimes. However, no model for the transition flow regimes, applied to compact heat exchangers, is available in the literature, as the few papers in this subject are usually focused in the classical correlations. The present work performed a theoretical and experimental thermal analysis of the laminar and transition flows in a diffusion bonded cross flow compact heat exchanger and proposed a new Nusselt number correlation for the transition regime. The theoretical model and the main correlations for laminar, turbulent and transition regime are presented. To validate the model, a stainless steel cut plate heat exchanger, with 450 square channels of 3 mm edge, was manufactured. The test facility was designed to operate with hot water and air at ambient temperature. The tests were performed with different mass flow rates, operating with Reynolds numbers between 360 and 2500. The theoretical and experimental results compared well, with theoretical data located within the range of the experimental uncertainty. However, for high Reynolds numbers, greater than 1650, the difference between the results tended to increase. Due to limitations of existing correlations, a Nusselt number correlation for air flow in the transition region and the liquid under constant temperature conditions, based on the asymptotic correlation method, is proposed for compact heat exchangers. The new model provides a smooth transition from the developing laminar flow to the turbulent. To validate the equation, a new test set was performed, and the proposed correlation presented a good accuracy in the transition regime.
1. Introduction Over the past years, research involving heat transfer and fluid flow in micro and mini channels has increased significantly, mainly due to industry demand for high heat transfer rates in small volumes. Compact heat exchangers were developed to fulfill this industry demand. These equipment are composed of mini or micro channels and present high surface area density, normally manufactured by the alternating stacking
⁎
of thin plates with small channels. The process of joining the plates to produce the core must be precise, efficient and cannot obstruct the channels. Conventional fusion welding techniques are not able to achieve the union requirements for these applications [1,2]. Due to the dimensional and quality requirements in microsystems, diffusion bonding process appears as a good alternative [3]. In a simple way, it can be described as a solid-state bonding technique, capable of joining large areas with a high
Corresponding author. E-mail address: [email protected] (M.V.V. Mortean).
https://doi.org/10.1016/j.applthermaleng.2019.02.017 Received 12 November 2018; Received in revised form 30 January 2019; Accepted 7 February 2019 Available online 08 February 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
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M.V.V. Mortean and M.B.H. Mantelli
Nomenclature A B C Dh D f K L L* ṁ n NTU Nu p P Pr q Re T u y z
Greek symbols ε
flow area [m2] dimensionless coefficients dimensionless coefficients hydraulic diameter, =4A/p [m] channel diameter [m] Darcy friction factor factor channel length [m] dimensionless thermal length mass flow rate [kg/s] superposition parameter Number of heat transfer units Nusselt number perimeter [m] pressure [kPa] Prandtl number heat transfer rate [kW] Reynolds number temperature [°C] velocity [m/s] dimensionless parameter dimensionless parameter
effectiveness
Subscript and abbreviations air b c exp h in lam out tran turb w 1 2 3 4 5 6
air bulk cold side experimental hot side inlet laminar outlet transição turbulent wall flow level 1 flow level 2 flow level 3 flow level 4 flow level 5 flow level 6
channel length and Prandtl number, respectively. L∗is the dimensionless thermal input length, defined as
mechanical strength in the interface region [3–5]. Printed Circuit Heat Exchangers [6,7], Diffusion-Bonded Plate-Fin Heat Exchangers [8] and Cut Plate Heat Exchangers [9,10] are examples of diffusion bonded compact heat exchangers. Large number of studies, models and correlations were developed for micro/mini-channels and compact heat exchangers operating in the laminar regime [11–13]. In some applications, the equipment also operates in transition and turbulent regimes, especially when they are used for the heat exchange of gases [6], as for offshore oil platforms applications. However, there are few papers and studies focused in the use of classical correlations and models to predict the heat transfer in turbulent or transition flow in micro channels. In the present work, a theoretical and experimental analysis of the thermal performance of a diffusion bonded cross flow compact heat exchangers, produced with rectangular mini channels, operating in the laminar and transition regimes, is performed. The applicability of the models and correlations for this range of operation is verified.
L∗ =
L Dh Re Pr
(2)
2. Literature review
According to Stephan and Preußer [16], Eq. (1) is valid for Prandtl numbers between 0.7 and 7 and L∗ > 0.03. A flow is considered under development when the dimensionless thermal length (L*) is lower than the dimensionless thermal input length, which, for square channels, is 0.06204 [14]. For fully developed laminar flows in square channels, the Nusselt number is constant and equal to 3.63 [18]. Mortean et al. [19] conducted a study focused on a laminar flow in a cross-flow compact heat exchanger. The author suggested a modification in Eq. (1), aiming its application on square cross-section channels, since the Stephan and Preußer [16] Nusselt number correlation was originally developed for circular channels. According to the author, the correlation that can better reproduce the thermal behavior of the square channel heat exchanger can be rewritten as:
2.1. Laminar flow
Nu = 3.63 + 0.086
Mini and micro channels applied to compact heat exchangers normally operate in the laminar regime, as presented in the researches of Kang et al. [11], Kim and No [12] and Luo et al. [13]. Mortean et al. [10] also studied the laminar flow in compact heat exchangers. The authors applied the Nusselt number correlation proposed by Lee and Garimella [14], Shah and London [15] and Stephan and Preußer [16] to predict the thermal behavior of a compact heat exchanger with rectangular channels. According to them, the correlation proposed by Stephan and Preußer [16] presented better results. Stephan and Preußer [16] proposed a Nusselt number correlation for laminar flow, considering the flow as under thermal development. The correlation was developed for circular channels, but also presented good results for different channel geometries [17] and is expressed as:
Gnielinski [20] also proposed a Nusselt number correlation for laminar flow, valid for Re < 2300 and Pr > 0.7:
Nu = 4.364 + 0.086
Nulam = {Num3 , q,1 + 0.63 + (Num, q,2 − 0.6)3 + Num3 , q,3 }1
(3)
3
(4)
where
Num, q,1 = 4.354
(5)
Num, q,2 = 1.953 3 Re Pr (D L)
(6)
Num, q,3 = 0.924 3 Pr Re (D L)
(7)
As already mentioned, a large number of studies were developed for micro/mini channels and compact exchangers operating in the laminar regime. On the other hand, there are few available studies that use classical Nusselt number correlations to predict heat transfer in turbulent or transition flow in micro/mini channels.
L∗)1.33
(1 1 + 0.1Pr (Dh Re L)0.83
(1 L∗)1.33 1 + 0.1Pr (Dh Re L)0.83
(1)
where Re, Dh, L and Pr are Reynolds number, hydraulic diameter, 515
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2.2. Turbulent flow
results. Harms et al. [36] experimentally studied the turbulent flow in rectangular micro channels and used the Gnielinski [26] correlation in their theoretical model. According to the authors, the experimental data were similar to the predicted by the correlation, but again, experimental results were higher than the analytical ones. Yu-ting et al. [37] performed an experimental study of the flow in a circular duct, with Reynolds number between 4000 and 10,000, and verified that the theoretical model, using the correlation proposed by Gnielinski [26], presented a similar trend to the experimental results.
The turbulent and laminar flows in ducts can be fully developed, under hydrodynamic development, under thermal development or under hydrodynamic and thermal development. In all these cases, the entry lengths of the dimensionless thermal and hydrodynamic boundary layers are much lower than those observed in laminar flow regimes. As a result, most of the turbulent studies employ fully developed turbulent flow models in their models [21]. This consideration is also adopted in the present work. The literature reports that, in turbulent regime, the channel geometry has not large influence on the flow thermal behavior significantly, different from observed in laminar flows [22]. Therefore, Nusselt number correlations for completely developed turbulent flows in circular ducts can also be applied, as a good approximation, in rectangular channels [21]. According to Rosa et al. [23], the thermal behavior of turbulent flows inside channels can be predicted well by several correlations, including the classic Nusselt number correlations of Dittus and Boelter [24], Petukhov [25], Gnielinski [26], and, in some cases, Adams et al. [27]; this last one was developed specifically for micro channels. Hesselgreaves [5] recommends the use of the Nusselt number correlation proposed by Gnielinski [26], originally developed for circular ducts, for turbulent flows in non-circular channels. This correlation presents an average error of 20% when compared to data [28,29]. Adams et al. [30] studied the applicability of Gnielinski [26] correlation in non-circular channels, verifying that the experimental data are within the limits predicted by the correlations. This correlation proposed by Gnielinski [26] is expressed as:
Nuturb =
2 3 (f 2)(Re − 1000) Pr ⎡1 + ⎛ D ⎞ ⎤ K ⎥ 1 + 12.7(f 2)1 2 (Pr 2 3 − 1) ⎢ L ⎝ ⎠ ⎦ ⎣
2.3. Transition flow Heat exchangers can also operate in laminar to turbulent transition region, however, the heat transfer behavior in this flow regime range is still not well known [38]. Wang et al. [39] studied the transition regime in rectangular ducts. According to them, there are few studies related with this subject in the literature. The authors compared the experimental data with the theoretical thermal model given by Eq. (8), concluding that the correlation proposed by Gnielinski [26] presented good results for the fully developed turbulent regime (Re > 7500), however, it was not able to predict the heat transfer in the transition region. Abraham et al. [28] verified that the correlation of Gnielinski [26] presented good results for Reynolds number higher than 3100 and proposed another Nusselt number correlation for the transition region (2300 < Re < 3100). Lee et al. [31] studied the applicability of the Nusselt number correlations proposed by Dittus and Boelter [24], Petukhov [25] and Gnielinski [26], developed for conventional channels, and of Adams et al. [27], developed specific for microchannels, to predicting the thermal behavior of rectangular microchannels, for Reynolds number between 2000 and 3500. According to the authors, the experimental Nusselt number was higher than predicted by Adams et al. [27] and Gnielinski [26] correlations, but smaller than those predicted by Dittus and Boelter [40] and Petukhov [25]. Gnielinski [32] carried out a study focused on heat transfer in the transition region, proposing a Nusselt number correlation for the transition region, consisting of an interpolation between laminar (Eq. (4)) and turbulent (Eq. (8)) Nusselt number correlations, valid for the transition region (2300 < Re < 4000), given by:
(8)
where f and D are, respectively, Darcy friction factor and channel diameter. The term 1 + (D L)2 3 represents the influence of duct length on the heat transfer. The K factor is expressed by:
(Pr Prw )0.11 for liquid K=⎧ n ⎨ for gas ⎩ (Tb Tw )
(9)
where T is temperature, and the subscripts b and w are relative to bulk and wall, respectively. The exponent n depends on the fluid, for air n = 0.45. In cases where temperature difference between inlet and outlet are not large, Eq. (8) can be simplified and written without the K factor [18]. Eq. (8) is valid for 2300 < Re < 1 × 104 and 0.5 < Pr < 2000 [26]. However, according to Lee et al. [31] the correlation can be used, with good approximation, for Reynolds number of up to 5 × 106. According to Gnielinski [26], Darcy friction factor (f) is calculated by the correlation proposed by Filonenko [26]:
f=
1 (1.82 log Re − 1.64)−2 4
Nu = (1 − γ ) Nulam,2300 + γNuturb,4000 where
γ=
1 (1.8 log Re − 1.5)−2 4
Re−2300 4000 − 2300
(13)
where Nulam,2300 and Nuturb,4000 are the Nusselt number calculated by Eq. (4) for Re = 2300 and by Eq. (8) for Re = 4000, respectively [32]. This correlation is valid for 2300 < Re < 4000 and Pr > 0.7. The author compared their theoretical results (Eqs. (4), (8) and (12)) with experimental data, and verified a good agreement. Bertsche et al. [41] carried out an heat transfer experimental study in circular ducts for a wide range of Reynolds and Prandtl numbers (500 < Re < 23,000 and 7 < Pr < 41). Their data were compared with the theoretical model, composed by Eqs. (4), (8) and (12), showing good convergence for the three flow regimes, with 80% of the experimental data within ± 15% of the values predicted by the correlations. Therefore, as no models for the transition flow regimes that can be applied directly to the compact heat exchangers are available in the literature, the present paper aims to study the thermal performance of a diffusion bonded compact heat exchanger operating in the laminar and transition flow regimes. In the analytical model, the Nusselt number correlation proposed by Mortean et al. [19] (Eq. (3)) is implemented for developing laminar flow, while the value of 3.63 is used for fully developed laminar flow and the results from Gnielinski [32] (Eq. (12)), for
(10)
In the recent study performed by Gnielinski [32], the author suggests replacing Eq. (10) by the one proposed by Konakov [33]:
f=
(12)
(11)
According to Rosa et al. [23], the applicability of the classical Nusselt number correlations proposed by Dittus and Boelter [24], Petukhov [25] and Gnielinski [26] to micro/mini channels, still needs to be studied. Barik et al. [34] performed a numerical study of turbulent flow inside a circular duct and compared their results with experimental data of Sleicher and Rouse [35] and with the correlation of Gnielinski [26]. The numerical results agreed well with the experimental data, however the theoretical model underestimated the experimental 516
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the transition regime. A heat exchanger was manufactured and tested with two streams operation condition: hot water and air at ambient temperature, for Reynolds numbers ranging from 360 to 2500. The theoretical and experimental data were compared and a new Nusselt number correlation for the transition region, based on the asymptotic correlation method was proposed.
3. Heat exchanger and experimental facility 3.1. Heat exchanger manufacture The cut-plate heat exchanger was produced with a new manufacturing method, presented for the first time by Mortean et al. [9]. This technology employs the water jet cutting process to produce minichannels on flat plates and diffusion bonding process to join the heat exchanger core. It should be noted that Mortean et al. [9] first produced a diffusion bonded compact heat exchanger prototype with C-12000 copper as base metal, but later, aiming industrial applications, stainless steel AISI 316L heat exchanger was produced and the data regarding its performance are used in the present work. Their manufacturing process consists of alternately stacking comb shape machined plates, produced with water jet cutting process, and non-machined flat plates. The sandwich between flat non-machined plates and comb shape plates forms one layer of the heat exchanger core (Fig. 1). The flat plates are responsible for separating the flows, and the comb shape machined plates are responsible for creating the channels sidewalls. After completing the layer stacking process, the assembly is inserted into a hot press furnace and diffusion bonded, producing the heat exchanger core. Diffusion bonding is a solid-state joining technique, where the assembly is submitted to uniaxial pressure and high temperature, in a vacuum atmosphere. Due to high temperatures and interface contact, the union of the two solid materials occurs through atomic diffusion. The main parameters involved in the process are: temperature, pressure and time [42]. Besides using basically the same stacking and diffusion bonding process, the present technology differs from the usual printed circuit heat exchangers in the sense that the geometry of the channels can be easily controlled by the design of the grooves of the middle plate and of the thickness of the sheets. The channels of the present heat exchanger was design to have square cross section of 3 mm of edge and rectangular cross section solid metal side walls (fins), 2 mm wide and 3 mm high. To separate the layers, 0.5 mm thickness flat plates were used (see Fig. 1). Boarder plates of 3 mm of thickness closes the core. The core was produced with 15 layers and 450 square channels for each flow, in a cross flow configuration. The complete heat exchanger was 167 mm wide, 167 mm long and 111 mm high (Fig. 1). The entire assembly was bonded at 1050 °C for 60 min under a constant pressure of 18 MPa. Finishing the manufacturing process, headers were welded on the sides of the heat exchanger. The complete heat exchanger is shown in Fig. 2.
Fig. 2. Compact heat exchanger with headers.
Fig. 3. Core sample.
To evaluate the joint quality of the heat exchanger, a metallography analysis of the bonded interface was performed in an optical microscope. The studied region is shown in Fig. 3. The metallography analysis was carried out on the top and bottom part of the channel. In total, four channels were investigated. Fig. 4 presents the result. As it can be seen in Fig. 4, the sample presented a small deformation, resulting in some failures in the bonding interface, as evidenced by horizontal dark lines in images I, J, L, M and N. The main effect of these failures in the bonding interface is to reduce the maximum working pressure that the heat exchanger can bear. In the current work, the core was tested at low pressures, between 1.5 and 3 bar; therefore the failures did not influence the results. To produce a core without failures and able to work with high pressures, it is recommended the conduction
Fig. 1. Stainless steel heat exchanger: (a) stacking process, (b) complete core and (c) core geometric details. 517
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Fig. 4. Metallography result.
minimum to the maximum (c1 to c6). This same procedure was performed for all flow levels, resulting in 30 tests. In the tests, the air flow ranged from laminar to transition region: 364 < Re < 2496. Table 3 presents the inlet and outlet temperatures, as well as the Reynolds numbers for both flows, the experimental heat transfer rate (qexp) and the experimental effectiveness (ε exp), this last defined as the rate between the experimental heat transfer rate and the maximum heat transfer rate that is thermodynamically possible. As it can be seen in Table 3, the heat transfer rate varied from 0.44 to 2.23 kW. It is worth noting that the high temperature of the outlet air is close to the water inlet temperature, evidencing the high effectiveness of the heat exchanger, ranging between 0.77 and 0.96. The analytical model, used to perform the thermal design of the heat exchanger, was the same presented by Mortean et al. [10]. The authors use the core geometric parameters and ε-NTU method to predict the thermal behavior of the compact heat exchanger. This method depends on the Nusselt number to estimate the convection heat transfer coefficient. For more details of the design procedure, see Mortean et al. [10]. In the current work, the Nusselt number model proposed by Mortean et al. [19] (see Eq. (3)) was used for the laminar flow under thermal development. For fully developed laminar flow, the Nu number was taken as 3.63. For the transition regime, the correlation proposed by Gnielinski [32] (see Eq. (12)) was used. The fan used did not allowed to reach turbulent flow regimes and so Eq. (8) proposed by Gnielinski [26] was not applied. Fig. 6 shows the heat transfer rate (q) as a function of the cold stream Reynolds number (Rec), for the test set data h1_c1 to h1_c6 (see Table 1). In this case, the hot side mass flow rate (water) was kept constant at the lowest level (h1), while the cold side mass flow rate
of further analysis of the diffusion bonding process for compact heat exchanger applications to establish the optimal bonding parameters of stainless steel AISI 316L. 3.2. Experimental tests An experimental apparatus was developed with the objective of evaluating the thermal performance of the cut-plate heat exchanger and verifying the application of Nusselt number correlations for these heat transfer flows. This setup was designed to work with air at ambient temperature and water at high temperature. On the hot side, the water is stored in a vessel and heated by resistors, up to the desired working temperature. On the cold side, a centrifugal fan is responsible for the air flow at ambient temperature. The fan power allows the heat exchanger to be tested from the laminar flow regime to the beginning of the transition flow. Fig. 5 shows the experimental apparatus. It is basically composed of tubes, a centrifugal fan, a vessel, a pump and a bank of electrical resistances. Type T thermocouples and RTD temperature sensors monitor the temperatures. Liquid turbine flow meter FTB-1316 and an anemometer measure the liquid and air mass flow rates. Besides, among other equipment, the setup counts with two manometers, a data acquisition system National Instrument NI cDAQ -9178 and a computer. The tests are performed varying the mass flow rates: five levels for the hot side stream (water) and six levels for the cold side stream (air), as shown in Table 1. During the tests, the inlet air temperature was maintained at constant levels: ambient temperature for the cold side and 70 °C for the hot side (water). The absolute pressure ranged from 195 kPa to 230 kPa, for the hot side, and from 103.5 kPa to 105.5 kPa, for the cold side. The following parameters were measured: inlet and outlet temperatures of the hot and cold flows (Th,in, Tc,in, Th,out, and Tc,out), inlet pressure, mass flow rates on the hot side and cold air velocity (Ph,in, Pc,in, ṁ h and uair , respectively). Based on the air velocity, measured with the anemometer, the mass flow rate was calculated. The data uncertainty is presented in Tables 1 and 2. 4. Results and discussion As already noted, in order to study the influence of the Reynolds number on the heat transfer, the tests were performed by varying the mass flow of one stream while the other one was kept constant. Due to fan power limitations, it was not possible to achieve turbulent flow inside the channels. Initially, the hot side was kept constant at h1 (0.348 kg/s) and the cold flow was varied from the lowest (c1 = 0.0095 kg/s) to the highest value (c6 = 0.0645 kg/s). Then, the hot flow was set to the second level, h2 (0.607 kg/s) and the cold flow was again varied from the
Fig. 5. Experimental facility. 518
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Table 1 Mass flow rate. Cold side (kg/s) - air
Theoretical model Experimental data
2.5
Hot side (kg/s) - water Uncertainty
ṁ h
Uncertainty
c1 = 0.0095 c2 = 0.0211 c3 = 0.0314 c4 = 0.0422 c5 = 0.0539 c6 = 0.0645
0.0015 0.0022 0.0055 0.0062 0.0069 0.0075
h1 = 0.348 h2 = 0.607 h3 = 0.850 h4 = 1.122 h5 = 1.349 –
0.04 0.04 0.04 0.04 0.04 –
2.0
q (kW)
ṁ c
1.5 1.0 0.5 0.0 0
Table 2 Uncertainty parameters. Parameter Uncertainty
Th, in (°C) 0.18
500
1000
1500
2000
2500
3000
Re_c Th, out (°C) 0.18
Tc, in (°C) 0.08
Tc, out (°C) 0.18
Fig. 6. Comparison between theoretical and experimental heat transfer rate for the test set h1_c1 to h1_c6 (Reh = 640).
presented similar behaviors, within the range of experimental uncertainty. The mean difference between the theoretical and experimental data, taking into account all tests, was approximately 12%. Also, it can be observed that for low Reynolds numbers (the first four test points: 350 < Re < 1650), within the laminar region, the theoretical and experimental results presented a very similar trend, with an average difference of approximately 8%. However, for higher Reynolds numbers, the difference tends to increase, reaching approximately 19.8% for Re > 1650 (the last 2 test points). This difference between theoretical and experimental data can be imputed to the fact that the data is located within the beginning of the transition region, i.e., from laminar to turbulent flow.
varied from the lowest (c1) to highest value (c6). Fig. 7 shows the results for the last set of tests: h5_c1 to h5_c6, where hot side mass flow rate was kept constant at level h5, while the cold side mass flow rate varied from c1 to c6. In the first three experimental points (Rec ≈ 320, 820 and 1220), the cold side flow was at the fully developed laminar regime, sinceL∗ > 0.06204 (see Eq. (2)). In the next two experimental points (Rec ≈ 1660 and 2110), the flow was in the limit of the laminar condition and the flow is considered under development since L∗ ⩽ 0.06204 . The last experimental point (Rec ≈ 2490), the flow was in the transition region. As already explained, in the theoretical model, the appropriate Nusselt number correlation was used according to each of these flow conditions. From Figs. 6 and 7, one can see that, for the range of Reynolds numbers studied, the theoretical model curves and experimental data
Table 3 Parameters collected during the tests. Test
Reh
Rec
Th,in (°C)
Th,out (°C)
Tc,in (°C)
Tc,out (°C)
qexp (kW)
εexp
c1_h1 c2_h1 c3_h1 c4_h1 c5_h1 c6_h1
638 644 638 636 644 635
364 821 1216 1658 2083 2493
70.12 70.22 70.42 70.35 71.78 70.54
69.93 69.84 69.73 69.48 70.65 69.25
22.27 22.13 21.17 21.85 24.32 26.62
68.29 62.78 60.16 59.43 60.21 59.58
0.44 0.87 1.22 1.61 1.94 2.14
0.96 0.84 0.79 0.77 0.75 0.75
c1_h2 c2_h2 c3_h2 c4_h2 c5_h2 c6_h2
1104 1131 1115 1112 1113 1116
365 824 1213 1662 2112 2487
70.24 70.37 70.53 70.39 70.40 70.45
70.13 70.17 70.16 69.92 69.82 69.74
22.14 22.26 20.95 22.14 24.47 26.62
68.48 63.33 60.68 60.13 60.15 60.35
0.44 0.88 1.24 1.63 1.95 2.18
0.96 0.85 0.80 0.78 0.77 0.76
c1_h3 c2_h3 c3_h3 c4_h3 c5_h3 c6_h3
1561 1560 1562 1560 1557 1564
363 818 1246 1674 2112 2487
70.07 70.14 70.41 70.18 70.24 70.38
69.99 69.99 70.17 69.82 69.81 69.87
21.87 22.25 20.81 22.26 24.34 26.58
68.41 63.34 60.81 60.24 60.37 60.60
0.44 0.87 1.28 1.65 1.97 2.20
0.96 0.85 0.80 0.79 0.78 0.77
c1_h4 c2_h4 c3_h4 c4_h4 c5_h4 c6_h4
2040 2055 2068 2064 2059 2061
364 815 1229 1603 2071 2492
69.98 70.07 70.22 70.16 70.17 70.20
69.91 69.95 70.03 69.91 69.87 69.84
21.36 22.33 20.74 22.33 24.23 26.56
68.23 63.41 60.80 60.45 60.42 60.73
0.45 0.87 1.26 1.58 1.94 2.22
0.96 0.86 0.80 0.79 0.78 0.78
c1_h5 c2_h5 c3_h5 c4_h5 c5_h5 c6_h5
2473 2467 2480 2471 2469 2472
355 797 1227 1598 2093 2496
69.92 69.91 70.15 70.18 70.12 70.22
69.87 69.80 70.00 69.98 69.88 69.91
20.71 21.90 20.48 22.24 24.36 26.58
68.13 63.29 60.75 60.53 60.56 60.86
0.44 0.86 1.27 1.58 1.96 2.23
0.96 0.86 0.81 0.79 0.79 0.78
519
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16
2.5
Theoretical model
14
Experimental data
12 10 Nu
q (kW)
2.0
Mortean et al. [19] - Eq. (3) Gnielinski [26] - Eq. (8) Proposed - Eq. (18) Gnielinski [32] - Eq. (12)
1.5
8 6
1.0
4
0.5
2
0.0
0 1000
0
500
1000
1500
2000
2500
3000
Re_c
1500
2000
2500 Re
3000
3500
4000
Fig. 8. Nusselt number as a function of Reynolds number.
Fig. 7. Comparison between theoretical and experimental heat transfer rate for the test set h5_c1 to h5_c6 (Reh = 2472).
Nulam for Re → 0 Nutran = ⎧ ⎨ ⎩ Nuturb for Re → ∞
(16)
4.1. Transition region analysis
Nutran = [(Nulam )n + (Nuturb )n]1
According to the literature, the transition region between the laminar and turbulent flows is characterized when a critical Reynolds number (Recrit) [21] value is exceeded. For circular channels and fully developed flows, the critical Reynolds number is approximately 2300 [21]. However, some researches concerning mini and micro channels presented smaller transition Reynolds numbers. Wang and Peng [43] investigated water and methanol flow in a rectangular micro channel and reported that the turbulent flow started at Reynolds number close to 1500. Harms et al. [29] also verified that the transition region occurred for Reynolds number equal to 1500. Lee et al. [31] verified a change in the laminar flow behavior for Reynolds number between 1500 and 2000. The experimental and numerical results of Sahar et al. [44] showed that the transition from laminar to turbulent occurred for 1600 < Re < 2000. Natrajan and Christensen [45] attributed the change of regime flows for Reynolds numbers of 1800. Mylavarapu [46] studied the flow in mini channels and also reported the change from laminar to turbulent regime for Reynolds number equal to 1800. One can see that in most studies the transition from laminar to turbulent regime occurred for Reynolds numbers lower than 2300 and near to 1800. In the current work, this change behavior occurred for Reynolds number between 1650 and 2050. So, based on the literature reports, in the present work, the critical Reynolds number is defined as 1800. However, Eq. (12) [32] and the existing correlations for the transition region, as proposed by Abraham et al. [28], are not valid for Reynolds numbers smaller than 2300. Therefore, it is not possible to use these correlations in the comparison with experimental data in the transition region, considered to start at 1800. In order to better predict the heat exchanger thermal behavior in the transition region, a Nusselt number correlation is proposed (Nutrans ) , based on the asymptotic correlation method proposed by Churchill and Usagi [47]. The model presents an interpolation method that mathematically associates two distinct equation behaviors, for the same dependent dimensionless parameter y, which are functions of the independent dimensionless parameter z. For the case of the following functions:
y = Bz p for z → 0 y=⎧ 0 q ⎨ ⎩ y∞ = Cz for z → ∞
n
(17)
where Nulam and Nuturb are the Nusselt numbers for the developing laminar flow, Eq. (3), and turbulent flow, Eq. (8), respectively. This method proposes a correlation that presents a smooth transition between the laminar and turbulent flow regimes, able to capture better the observed experimental results. To estimate the parameter n, the difference between Eqs. (3) and (8), for Re = 1800 and Re = 4000, respectively, and those estimated by Eq. (17), should be the lowest possible value. Based on the mentioned criteria, the value of n, which minimizes the root mean square error comparing the theoretical data by Eq. (17) with the experimental results of Table 3, was estimated to be 6. Therefore, the following model is proposed:
Nutran = [(Nulam )6 + (Nuturb )6]1
6
(18)
Fig. 8 shows the Nusselt number based on Eqs. (3), (8), (12) and (18), i.e., using the developing laminar flow correlation developed by Mortean et al. [19], the turbulent flow correlation of Gnielinski [26], the transition region correlation based on the work of Gnielinski [32] and finally the model proposed on the present work (Eq. (18)), respectively. As expected, the proposed solution provides a smooth transition from the developing laminar flow to the turbulent flow. Comparing the experimental data from the transition region (presented in Table 3) with the theoretical model using Eq. (18), the mean difference between the results is approximately 13.9%, lower than the 19.8% previous calculated with Eq. (12) [32]. In order to check the efficiency of the proposed model, additional tests were carried out, focusing on the transition region. During the tests, the water mass flow rate was kept constant at 0.853 kg/s and air flow was varied from 0.037 kg/s (Rec = 1440) to 0.069 kg/s (Rec = 2690). Fig. 9 shows the comparison between the results, using Eq. (12) [32], the proposed correlation (Eq. (18)), and the experimental data. As can it be seen in Fig. 9, the theoretical model compare quite well with the experimental data. The mean difference between experimental data and theoretical model, for transition region, using Eq. (12) [32] was approximately 21.6%; however, with the proposed model (Eq. (18)), the difference reduced to 16.3%. Besides, the model was able to reproduce all the data trends observed. Therefore, based on the present results, it is suggested the use of the proposed correlation to the calculation of the Nusselt number (and so the convective heat transfer coefficients) for flows in the laminar to turbulent transition regions, in compact heat exchangers operating in similar conditions of the device tested in the current work.
(14)
where B and C are dimensionless coefficients, one has, for q > p :
y = [(y0 )n + (y∞ )n]1
n
(15)
where n is a superposition parameter determined by comparison with experimental data. For the current case, the model takes the form: 520
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2.5 q (kW)
2.0
1.5 1.0
Gnielinski [32] - Eq. (12) Experimental data Proposed - Eq. (18)
0.5
0.0 1300
1550
1800
2050
2300
2550
2800
Re_c Fig. 9. Comparison between proposed model and experimental data of additional tests.
5. Conclusion In this work, a theoretical and experimental study of the thermal performance of cut plate heat exchanger was performed, for the equipment operating in the laminar and transition regimes, aiming to verify the applicability of the existing models and correlations for this range of operation. The theoretical model and the main correlations for laminar and transition regimes were presented. The compact heat exchanger, produced in 316L stainless steel and with 450 square channels of 3 mm edge, was used in the experimental tests. An experimental facility was developed to operate with air at ambient temperature and hot water at 70 °C. Initially 30 tests were performed, for laminar and transition flows. To measure the heat exchanger thermal performance, the analytical model presented by Mortean et al. [10] was applied. The Nusselt number correlation proposed by Mortean et al. [19] (Eq. (3)), for developing laminar flow, and the correlation of Gnielinski [32] (Eq. (12)), for transition regime, were used. The theoretical and experimental results showed similar behaviors, with the theoretical data laying within the experimental uncertainty range. However, the difference between the results tended to increase as the Reynolds number increased, presenting an average difference of 19.8% for Reynolds numbers higher than 1650, showing limitations of the of the existing correlations. Therefore, a new Nusselt number correlation for air flows in the transition regime, and water under close to constant liquid temperature conditions, using 1800 as the critical Reynolds number, was proposed for compact heat exchanger. The mean difference between the theoretical model, using the proposed correlation, and experimental data, from the transition region, reduced to approximately 13.9%. Also, the new correlation was able to follow all the trends observed in the data. Aiming to confirm the applicability of the new correlation, an additional set of test was performed, in which the Reynolds number was varied from Rec = 1440 to Rec = 2690, therefore in the transition region. Comparing the new experiment data with the theoretical model, it was verified that the new correlation presented better results than those tested before, as the mean difference between experimental data and theoretical model, using the proposed correlation, was 16.3%. Finally, the present authors propose the use of this model to calculate the Nusselt number for the laminar to turbulent transition region, for compact heat exchangers, especially those with square cross section mini channels. Due to the limitations of the experiments, further study still needs to be performed in order to reduce the error and increase the accuracy of the proposed equation. References [1] Z.L. An, W.-L. Luan, F.Z. Xuan, S.T. Tu, High temperature performance of 316L-SS joint produced by diffusion bonding, Key Eng. Mater. 297–300 (2005) 2795–2799,
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