Article pubs.acs.org/IECR
Investigation of Convective Heat Transfer with Liquids in Microtubes Ke-Jun Wu, Chun-Xia Zhao, Guo-Hua Xu, and Chao-Hong He* State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT: Convective heat transfer in microtubes has attracted considerable attention for almost three decades. It is wellknown that the solid−liquid interfacial tension plays an important role at the micrometer length scale. In this work, the characteristics of convective heat transfer in microtubes with inner diameters ranging from 0.353 to 1.045 mm were investigated; the effects of interfacial tension on heat transfer were also investigated using nine working fluids (including deionized water, 5 wt % ethanol aqueous solution, 50 wt % ethanol aqueous solution, ethanol, ethyl acetate, cyclohexane, n-hexane, cyclohexanone, and cyclopentanone). The experimental results showed that the performance of convective heat transfer of different working fluids in microtubes varied significantly and was different from that in conventional scale tubes. The results were correlated by an empirical correlation, in which the effect of solid−liquid interfacial tension was taken into account. The agreement between the correlation and the experimental data was found to be satisfactory.
1. INTRODUCTION Microscale heat transfer occurs widely in microreactors, microscale heat exchangers, microelectronic systems, space systems, and so on. Therefore, it is increasingly important to understand the mechanism of microscale heat transfer. Heat transfer at the conventional scale (characteristic dimensions greater than 1 mm)1 has been studied thoroughly, and it is wellknown that the Nusselt number is constant (equal to 4.36) for laminar flow in the case of constant-heat-flux boundary conditions.2 In recent years, heat transfer in microtubes has been investigated in comparison with the behavior in conventional tubes. However, there are significant discrepancies between different sets of published results. Adams et al.3 investigated the heat transfer of water flowing through microtubes with inner diameters ranging from 0.76 to 1.09 mm and found that the Nusselt numbers were higher than the conventional values. Lin et al.4 investigated the heat transfer of water in two stainless steel tubes with inner diameters of 0.123 and 0.962 mm. Their results were in good agreement with those predicted by conventional theory. Lelea et al.5 investigated the heat transfer of water in microtubes with inner diameters of 0.1, 0.3, and 0.5 mm and confirmed that the conventional or classic theories were applicable for water flowing through microchannels of these sizes. Celata et al.6,7 investigated the heat transfer in microtubes with diameters ranging from 0.05 to 0.528 mm, and in smaller tubes, the experimental Nusselt numbers were apparently lower than those predicted by conventional theory. The authors assumed that heat losses by peripheral conduction away from the test section were the reason for the deviations. Gao et al.8 investigated the heat transfer of water in microchannels with diameters ranging from 0.1 to 1 mm and found the experimental Nusselt numbers to be lower than those predicted by conventional theory for smaller tubes. The details of the experimental conditions of the aforementioned studies are listed in Table 1. The contradictions among these studies indicate that liquid heat transfer in microtubes is still not well understood. Moreover, water was used as the working fluid for © 2012 American Chemical Society
most experiments, which limited the application of the microscale heat-transfer technique. Thus, further systematic studies of heat-transfer behavior of various liquids in microtubes are needed. Furthermore, at the microscale, the effect of interfacial tension becomes increasingly important and even dominant.9 However, to the best of our knowledge, no studies have incorporated solid−liquid interfacial tension into the study of Nusselt numbers. In this work, the effect of solid−liquid interfacial tension on the convective heat transfer in stainless steel microtubes was experimentally investigated using nine working fluids with various surface tensions and polarities. Furthermore, an empirical equation was used to correlate the Nusselt numbers of convective heat transfer in the stainless steel microtubes.
2. EXPERIMENTAL SETUP AND PROCEDURE The experimental setup is presented schematically in Figure 1. Circulation of the fluid was driven in a microtube by a microsyringe pump (Harvard Pump 33), with the rate ranging from 1.22 × 10−7 to 53.35 mL/min with an accuracy of ±0.35%. A filter was placed before the test section to prevent the blockage of the flow by small particles. The liquid temperature at the microtube inlet was kept at a prescribed value using a constant-temperature bath (Shanghai Rongfeng 501A) with a measurement uncertainty of ±0.05 K. The micropump, filter, and valve were connected by a steel tube with an inner diameter of 3 mm. The microtube was placed inside a vacuum chamber that was evacuated by a vacuum pump to create an environment free of natural convection. In the setup, the vacuum pressure was less than 100 Pa, so the heat loss to the surroundings through convection can be considered nonexistent. The heat loss due to radiation was evaluated by considering the formula for two concentric cylindrical surfaces, simplified for the limiting case Received: January 18, 2012 Accepted: June 20, 2012 Published: June 20, 2012 9386
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Table 1. Details of Experimental Conditions in the Literature ref
inner diameters (mm)
Adams et al.3 Lin et al.4 Lelea et al.5 Celeta et al.6,7 Gao et al.8
0.76 and 1.09 0.123 and 0.962 0.1, 0.3, and 0.5 0.05, 0.12, 0.259, 0.325, and 0.528 0.1−1.0
boundary condition constant constant constant constant constant
heat heat heat heat heat
flux flux flux flux flux
Reynolds numbers
measurement technique
(0.26−2.3) × 104 (0.010−1.1) × 104 laminar range up to 8.0 × 102 (0.114−3.14) × 103 (0.10−8.0) × 103
T-type thermocouple liquid-crystal thermography K-type thermocouple K-type thermocouple T-type thermocouple
Figure 1. Schematic diagram of the experimental setup.
where the surface of the internal body is much smaller than the external, concave surface (the inner diameter of the vacuum chamber was 100 mm) Q rad = C0Aε0[(Ti /100)4 − (Te/100)4 ]
(1)
The extent of this loss can be calculated from eq 1. Celata et al.7 reported that the extent of this loss can be considered negligible if it is on the order of 0.01%. In our work, the inner wall temperature of the vacuum chamber was equal to the ambient temperature of about 293.15−298.15 K, and the highest outer wall temperature of the tube was about 313.15 K. Even in the case of the highest temperature difference between the two surfaces, the heat loss was less than 0.01% of the total heat input and was thus assumed to be negligible. At the two ends of the tested microtube, two sumps were fabricated to connect the tube. On the microscale, it is a difficult task to measure the fluid temperature in a microtube. Any insertion-type measurement methods have an effect on the flow and heat transfer in the microtube. Thus, in our work and some other studies in the literature,10,11 sumps were used to install thermocouples or thermal resistances for use in measuring fluid temperatures. The structure of the sumps is shown in Figure 2. The sumps were covered with thermal insulating foam and also placed in the vacuum chamber. Two Pt100 resistance temperature detectors (RTDs; SMWZPM201) were embedded at the two ends of the sump to measure the inlet and outlet temperatures of the fluid flowing through the test section. The temperature of the inner wall is hard to measure, so we employed the same approach as used by Lelea et al.,5 Celata et al.,7 and Li and Zhong:12 Several 50-μm K-type
Figure 2. Structure of the sumps: (A) connector to the 3-mm steel tube, (B) connector to the Pt100 RTD, (C) connector to the microtube, and (D) connector to the pressure sensor.
thermocouples (RKC ST-50) were installed on the outside of the tube wall surface along the flow direction as shown in Figure 3a to measure the temperature of the outer wall. Then, the temperature of the inner wall was obtained from the onedimensional heat conduction equation using the temperature of the outer wall. The temperature measurement is very important in our experiments; therefore, the method proposed by Karwa et al.13 and du Plessis et al.14 was used to calibrate the temperature sensors: The K-type thermocouples and Pt100 RTDs were calibrated by comparing their response with a standard mercury thermometer (better than ±0.1 K) at the same temperature from 283.15 to 348.15 K. The constant-temperature bath was used to control the temperature. The Pt100 RTDs were immersed into the constant-temperature bath directly, whereas the K-type thermocouples were adhered to the inner wall of 9387
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Figure 3. (a) Installation of the thermocouple on the outside tube wall surface and (b) SEM image of the cross section.
Figure 3b). Both ends of the tubes were measured to make the results more precise, and the deviations of the diameter results for the two ends were less than 1% for all three tubes. Thus, we assumed the tubes to have perfectly circular cross sections. The inner diameters of the stainless steel tubes were also measured by the weighing method, which showed good agreement with the results obtained by the microscopy method. All tests were performed under steady-state conditions. If the temperatures of the inlet and outlet fluids varied by less than 0.1 K/15 min, we considered the system to have reached a steady state. In general, this process required 60−120 min depending on the fluid and volume rates. To avoid errors caused by fluctuations in the temperature measurements, the mean temperature for 10−15 min after the system reached steady state was used to calculate the Nusselt numbers. For ease of calculation, the thermodynamic properties of the fluids were considered to be constant by averaging the temperatures of the inlet and outlet fluids as the reference temperature.15 Deionized water, 5 wt % ethanol aqueous solution, 50 wt % ethanol aqueous solution, ethanol, ethyl acetate, cyclohexane, nhexane, cyclohexanone, and cyclopentanone were chosen as working fluids because of their significantly different surface tensions and polarities. The permittivities at 293.15 K and other physical properties of these fluids at the minimum and maximum experimental temperatures are listed in Table 3.
thin-wall vessels and then immersed into the constanttemperature bath to avoid interference. Each calibration at different temperatures lasted for 1 h. Thus, the uncertainty in the temperature measurements was estimated to be ±0.1 K. The thermocouples were adhered to the tube wall with Loctite 416 cyanoacrylate instant adhesive and then covered with thermal insulating foam. The microtube was heated by Joule heating with an electrical power supply (GWinstek PSS2005) that provided a constant current over the test tube with an accuracy better than 0.1% + 5 mA. The two electrodes were soldered directly to the outer wall with a contact resistance of less than 0.5 mΩ. Thus, the entire wall thickness was heated through the Joule effect, and the heat due to contact resistance can be neglected. The electrodes at the ends of the heated section of the solid pipe were installed near the sumps to reduce the temperature changes between them. All data from the RTDs, thermocouples, and power supply were collected by a digital acquisition system (Advantech, USB 4718). Stainless steel tubes were used, and their dimensions are listed in Table 2. The diameters of different scale tubes were Table 2. Dimensions of the Tubes Tested material stainless steel stainless steel stainless steel
inner diameter (mm)
outer diameter (mm)
length (mm)
heating length (mm)
heat input (W)
0.353
0.609
250
167
0.8−4
0.597
0.875
250
163
0.5−3
1.045
1.29
200
112
0.5−3
3. EXPERIMENTAL DATA REDUCTION For each different set of experimental runs, the data including flow rates, inlet and outlet temperatures of the liquid, outer wall temperatures of the microtube, and input voltages and currents were continuously measured and recorded through a digital acquisition system based on a Dell personal computer. The local value of Nu was calculated using the equation
measured from enlarged images taken by scanning electron microscopy (SEM) and optical microscopy (OM) (see, e.g., Table 3. Physical Properties of the Working Fluids
density (g/cm3)
a
viscosity (cP)
surface tension (mN/m)
system
permittivity (293.15 K)
max Ta
min Tb
max Ta
min Tb
max Ta
min Tb
deionized water 5 wt % ethanol aqueous solution 50 wt % ethanol aqueous solution ethanol cyclohexanone cyclopentanone ethyl acetate cyclohexane n-hexane
80.10 62.30 30.30 25.30 16.10 13.58 6.08 2.22 2.08
0.994 0.983 0.899 0.773 0.936 0.942 0.883 0.765 0.648
0.997 0.987 0.909 0.784 0.942 0.952 0.898 0.774 0.659
0.664 0.828 1.299 0.713 1.829 1.013 0.379 0.792 0.275
0.954 1.101 2.350 1.059 2.027 1.426 0.435 0.896 0.299
69.72 54.01 26.82 20.92 33.72 32.65 22.10 23.10 16.97
72.38 55.62 27.90 21.93 34.61 33.70 23.73 24.18 17.85
At maximum experimental temperature. bAt minimum experimental temperature. 9388
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Nu =
Article
−
2R ih kf
Nu =
(2)
where h is the local heat-transfer coefficient, defined as qi h= Twn − Tbn
(3)
πDi Lnqi Mc p
(4)
L π (R o 2 − R i 2 )
(6)
with Q defined as the heat absorbed by the fluid, given by (7)
In our experiments, the test-section heat duty was calculated from the fluid temperature difference between the inlet and outlet; thus, this fluid temperature difference should not be too small because of the uncertainty in the temperature measurements. The measurement uncertainties of the difference (Tout − Tin) are discussed in section 4. According to the boundary conditions
r = Ro ,
′ T = Twn
r = Ri,
ks
S R 2 − R i2 Q dT = qi = = c o dr 2R iπL 2 Ri
⎤ Sc ⎡ 2 ⎛ R o ⎞ ⎢R o ln⎜ ⎟ − (R o 2 − R i 2)⎥ ⎥⎦ 4ks ⎢⎣ ⎝ Ri ⎠
Tin + Tout 2
(9)
(10)
(11)
Thus, the local Nusselt number was obtained as
Nu =
qiDi k f (Twn − Tbn)
maximum uncertainty (%)
Di M Tout − Tin L
0.60 0.35 2.86 0.14
parameter
maximum uncertainty (%)
Twn − Tbn h Nu
5.68 6.40 6.43
(14)
where ηb is the viscosity at the bulk temperature, ηw is the viscosity at the wall temperature, and Nucp is the Nusselt number for a constant-property fluid. If the temperature difference between the wall and the bulk fluid is large, this effect should be taken into account. However, in this work, this effect can be basically ignored because of the small temperature difference between the wall and the bulk. The viscosity data at the temperatures of the inner wall and bulk fluid under experimental conditions with the maximum temperature difference are presented in the Table 5. 5.2. Effects of Entrance Region and Axial Heat Conduction. Many effects that are negligible at the conventional scale need to be taken into account in microscale applications, such as the effects of hydrodynamic and thermal entrance regions and axial heat conduction. In laminar flow, the hydrodynamic entrance length (Le) is calculated as Le ≈ 0.05ReDi (15)
(8)
The mean value of the bulk temperatures, which was taken as the reference temperature, was calculated from the equation Tbn,avg =
parameter
Nu/Nucp = (ηb /ηw )0.14
the local temperature at the inside portion of the microtube wall was obtained as ′ − Twn = Twn
(13)
5. RESULTS AND DISCUSSION Liquid flowing in microtubes usually has a low velocity and is considered to be in laminar flow. For the sake of identical analysis, we controlled the Reynolds number, Re, in a low range (52−527) in our experiments. 5.1. Effects of Temperature-Dependent Fluid Properties on Heat Transfer. The effects of temperature-dependent fluid properties on heat transfer have been well reported in the literature.17−19 For most fluids, the physical properties of specific heat, thermal conductivity, and density are relatively independent of temperature, but the viscosity decreases markedly with temperature. Traditionally, the effect of temperature on heat transfer is expressed as
Q
Q = Mc p(Tout − Tin) = ηI 2R
Nu dz
(5)
where Sc is the internal heat source generated inside the microtube wall by the electric power, defined as Sc =
Lfd
Table 4. Measurement Uncertainties
Twn is the local wall temperature at the inside of the tube and was obtained from the one-dimensional heat conduction equation using the measured wall temperature at the outside of the microtube S S d2T 1 dT 1 d ⎛⎜ dT ⎞⎟ r + + c = + c =0 2 ⎝ ⎠ r dr ks r dr dr ks dr
∫0
4. EXPERIMENTAL UNCERTAINTIES The uncertainties in the experimental results were estimated following the recommendations and method described by Moffat.16 The uncertainty analysis of the Nusselt number consists of the uncertainties in M, (Tout − Tin), Di, L, and (Twn − Tbn). The maximum uncertainty in the Nusselt number was estimated as about 6.43%. The details of the related parameter uncertainties are listed in Table 4.
In eq 3, Tbn is the local bulk temperature of the liquid at each point of the wall temperature measurement, and it was calculated from the heat balance equation Tbn = Tin +
1 Lfd
where Di is the inner diameter of the tubes and Re is the Reynolds number. In our experiments, with Re and Di ranging from 52 to 527 and from 0.353 to 1.045 mm, respectively, the hydrodynamic entrance length was no longer than 10.8 mm (Di = 1.045 mm, Re = 206) and can thus be neglected. The thermal entrance length is a transitory region where the flow at uniform temperature develops under the influence of uniformly heated
(12)
Because the mean Nusselt numbers in the fully developed region are important for engineering applications, they were calculated from the local Nusselt numbers in the fully developed region as 9389
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Table 5. Viscosity Data at the Temperatures of the Inner Wall and Bulk Fluid under Experimental Conditions with the Maximum Temperature Difference system
maximum temperature difference (K)
wall temperature (K)
bulk fluid temperature (K)
ηw (cP)
ηb (cP)
viscosity ratio ηb/ηw
deionized water 5 wt % ethanol aqueous solution 50 wt % ethanol aqueous solution ethanol cyclohexanone cyclopentanone ethyl acetate cyclohexane n-hexane
2.72 4.35 3.02 5.37 3.37 3.26 3.15 3.04 4.06
303.17 307.22 309.86 305.13 301.07 300.05 307.45 306.92 303.43
300.45 302.87 306.85 299.76 297.70 296.79 304.30 302.88 299.37
0.797 0.868 1.472 0.970 1.937 1.042 0.387 0.806 0.286
0.853 0.989 1.701 1.088 2.031 1.091 0.399 0.840 0.298
1.07 1.14 1.16 1.12 1.05 1.05 1.03 1.04 1.04
Table 6. Gz and M Values for Different Reynolds Numbers in Different Tubes inner diameter (mm) 0.597 0.597 0.597 0.597 0.597 0.597 0.597 0.597 1.045 1.045 1.045 0.353 0.353 0.353 0.353 0.353 0.353 0.353 0.353
system deionized water 5 wt % ethanol aqueous solution 50 wt % ethanol aqueous solution ethanol ethyl acetate cyclohexane n-hexane cyclohexanone deionized water 5 wt % ethanol aqueous solution ethanol deionized water 5 wt % ethanol aqueous solution ethanol cyclohexanone cyclopentanone 50 wt % ethanol aqueous solution ethyl acetate n-hexane
Gz
M × 10−6
4.61−9.27 5.40−6.77
66−132 95−118
8.65−9.45
114−124
7.97−11.19 7.88−9.47 10.28−13.7 9.50−12.32 7.52−9.03 9.55−19.36 10.14−25.44
202−284 277−332 200−267 262−340 336−404 66−131 52−131
24.32−40.73 4.10−6.50 3.49−4.34
114−190 48−77 86−110
5.69−7.11 8.02−9.62 7.69−8.56 5.71−7.61
182−227 149−179 154−171 56−75
3.10−4.13 3.30−4.39
113−150 131−174
walls to a steady-state profile of temperature that depends on the fluid velocity and conductivity. Whether a heated flow has reached this steady thermal state can be deduced from the value of the Graetz number Re × PrDi Gz = (16) z
Figure 4. Experimental values of local Nusselt numbers for the microtube with an inner diameter of 0.597 mm: (A) deionized water, (B) 5 wt % ethanol aqueous solution, (C) 50 wt % ethanol aqueous solution, (D) ethanol, (E) ethyl acetate, (F) cyclohexane, (G) nhexane, (H) cyclohexanone.
For Gz < 10, a thermally fully developed profile is assumed to have been achieved.6 For low values of the Reynolds number, the mean value of the Nusselt number coincides with the fully developed value, because the entrance region is very short and the effects of viscous dissipation are not important in this regime of flow. In contrast, at low Reynolds numbers, the effects of conjugate heat transfer on the mean value of the Nusselt number can become very important because the conduction along the channel walls becomes a competitive mechanism of heat transfer with respect to internal convection. Maranzana et al.20 defined a criterion for the region of significance of axial conduction in the walls in the
heat-transfer problem that was also used by Morini.21 For a circular tube, it can be calculated as ⎛ k ⎞⎛ D 2 − Di 2 ⎞ 1 ⎟ M = ⎜ w ⎟⎜ o ⎝ k f ⎠⎝ Di L ⎠ Re × Pr
(17)
where kw is the thermal conductivity of the wall material and kf is the thermal conductivity of the liquid. For M > 10−2, the 9390
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Figure 5. Experimental values of local Nusselt numbers for the microtube with an inner diameter of 1.045 mm: (A) deionized water, (B) 5 wt % ethanol aqueous solution, (C) ethanol.
the Nusselt number on the Reynolds number for all tubes with inner diameters ranging from 120 to 528 μm. Compared to the 120- and 528-μm tubes, a weaker dependence of the Nusselt number on the Reynolds number was found for 259- and 325μm tubes. Liu et al.23 studied the average Nusselt number depending on the Reynolds number for water in quartz glass tubes with inner diameters of 45, 92, and 141 μm for the range of Reynolds number 100 < Re < 3000. According to their experimental results, the experimental Nussult numbers were less than predicted by classical laminar correlations at lower Reynolds numbers, and with increasing Reynolds number, the experimental data sharply increased and were higher than the values predicted by classical transitional correlations. Hetsroni et al.24 studied the behavior of the dependence of the average Nusselt number on the Reynolds number for water and water− surfactant solutions in a pipe with a inner diameter of 1.07 mm for the range of Reynolds numbers 10 < Re < 450. According to their experimental results, an obvious dependence of the Nusselt number, Nu, on the Reynolds number, Re, was observed for both water and water−surfactant solution, and the average Nusselt number increased with the Reynolds number. Nguyen et al.25 investigated the heat transfer in trapezoid-shaped channels and reported two approximate functions of the Nusselt number for laminar and turbulent flows:
effect of axial heat conduction cannot be neglected. For large values of M in a microtube with an imposed external heat flux, the fluid bulk temperature increases at the inlet with an exponential law, and after this initial region, the temperature difference between the fluid and the walls becomes constant, as is typical for constant-heat-flux boundary conditions. In contrast, for low values of the conductance number M, the axial bulk temperature distribution along the microtube tends to become linear.22 Table 6 lists the Gz and M values of each experimental system. The z in the Gz expression is the distance from the entrance to the first temperature measurement point. It can be seen from Table 6 that M is much smaller than the threshold value for all systems. 5.3. Results for the 0.597-mm-Inner-Diameter Tube. In Figure 4, the experimental results for the local value of Nu are plotted as a function of the axial location for the respective experimental conditions; the value of the Nusselt number for fully thermally developed flow of conventional theory is always given as a reference. Deionized water, 5 wt % ethanol aqueous solution, 50 wt % ethanol aqueous solution, ethanol, ethyl acetate, cyclohexane, n-hexane, and cyclohexanone were used as the working fluids because of their different surface tensions and polarities. As can be seen from the values of Gz in Table 6, for the inner diameter of 0.597 mm, nearly all flows were thermally developed. It can be seen from Figure 4 that the local Nusselt numbers exhibited a slight axial dependence. Meanwhile, we found a distinct dependence of the local Nusselt numbers on the Reynolds number, as reported in many studies in the literature,6,7,23−25 Celeta et al.6,7 found a dependence of
Nu = 8.39Re1/2 − 1.33Re 2/3
(18)
in the laminar regime and 9391
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Figure 7. Calculated results of the empirical correlation compared to experimental data for mean Nusselt numbers.
from the values of Gz in Table 6, for the inner diameter of 1.045 mm, some of the flows were still in the thermal developing region. Only nearest to the exit and at low Reynolds numbers did these flows approach the thermally developed state. The results for Nu in the thermal developing region were higher than those in the fully developed region because of the thinner boundary layer of the former. It can be seen from Figure 5, compared with the results for the tube with an inner diameter of 0.597 mm, there was an increase in the local Nusselt number at equal Reynolds numbers and a stronger axial differentiation for the 1.045-mm-diameter tube. 5.5. Results for the 0.353-mm-Inner-Diameter Tube. With a decrease of the tube diameter, we noticed a decrease in the local Nusselt numbers in comparison with those in tubes with larger diameters (Figure 6). In addition, the dependence of the local Nusselt numbers on the Reynolds number and the axial dependence were weaker. Figures 4−6 show that the polar liquids (εr >15),26 such as deionized water and ethanol aqueous solution, usually had higher Nusselt numbers than the nonpolar liquids (εr
