- Email: [email protected]
Condensation heat transfer in rectangular microscale geometries
International Journal of Heat and Mass Transfer 100 (2016) 98–110
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Condensation heat transfer in rectangular microscale geometries Srinivas Garimella a,⇑, Akhil Agarwal b, Brian M. Fronk c a
Sustainable Thermal Systems Laboratory, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA SEPCO – Upstream Americas, 701 Poydras St., New Orleans, LA 70161, USA c School of Mechanical, Industrial and Manufacturing Engineering, Oregon State University, Corvallis, OR 97331, USA b
a r t i c l e
i n f o
Article history: Received 19 July 2015 Received in revised form 21 March 2016 Accepted 23 March 2016
Keywords: Condensation Heat transfer Microchannel Annular Intermittent
a b s t r a c t Heat transfer coefficients during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels are presented. A novel technique to accurately determine condensation heat duty and heat transfer coefficient in such microscale geometries at small Dx is used. Models in the literature that were developed for larger tubes are shown to under predict the data. A new model that accounts for the flow mechanisms during condensation at such small scales, and takes into account the effect of G, x, Tsat, Dh and AR, is developed. The model predicts 94% of the data in the intermittent, transition and annular flow regimes within ±25%. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
2. Prior work
Microchannels are increasingly being used by numerous industries to miniaturize heat transfer equipment, improve energy efficiency, and minimize heat transfer fluid inventory. A fundamental understanding of condensation at the microscales will yield far reaching benefits for the automotive and HVAC&R industries, and others that need miniaturized components, such as portable personal cooling devices, hazardous duty and high ambient air-conditioning, and medical devices. Garimella et al. [1] reported measurements and a unified model for intermittent and annular flow pressure drops during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels for a wide range of conditions (300 < G < 800 kg m2 s1, 0 < x < 1; 30 < Tsat < 60 °C). It was found that pressure drop and heat transfer coefficient increased with increasing vapor quality, increasing mass flux and decreasing saturation temperature. The pressure drop model of Garimella et al. [1] relied on an annular flow factor (AFF) that accounted for the relative predominance of annular or intermittent flow in these channels. In this paper, a substantial amount of additional heat transfer data are introduced and a new heat transfer model is proposed based on the data and using the insights on flow mechanisms and their influence on pressure drop reported in Garimella et al. [1].
Models for horizontal, in-tube condensation have primarily considered idealized gravity and shear-driven mechanisms and the relative importance of the two. Gravity driven models are most important in stratified and wavy flow regimes. As the channel diameter decreases, surface tension forces become increasingly important and the gravity driven stratified and wavy flow regimes become less prevalent, as evidenced by the condensation flow visualization studies of Coleman and Garimella [2,3]. Thus, modeling of condensation heat transfer in the shear-dominated annular and intermittent flow regimes are of primary importance. Shear-based annular flow models generally relate the interfacial shear stress to the analogous heat transfer across the condensate film. This approach was first introduced by Carpenter and Colburn [4] and later adapted by other researchers [5,6] with modifications made in the determination of the interfacial shear. Chen et al. [7] developed a general purpose annular flow correlation starting with asymptotic limits, and blending them through simple combinations of the terms at the respective limits. Several researchers have also used a two-phase multiplier approach, similar to that commonly used in two-phase pressure drop models. When used in the heat transfer models, the two-phase multiplier is applied to the corresponding single-phase heat transfer coefficient. It should be noted that shear-based models also use two-phase multipliers to determine the interfacial shear stress, and thus the two approaches are equivalent to each other. Several researchers, including, Dobson and Chato [8], Cavallini et al. [9], and Thome et al. [10], have analyzed condensation data
⇑ Corresponding author. E-mail address: [email protected] (S. Garimella). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.086 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
99
Nomenclature AR AFF Cp d D f G h J, j k L _ m N NUC Nu P Pr Q q00 R Re SLR T t U V W, w x z
aspect ratio annular flow factor specific heat (J kg1 K1) depth of microchannels (m) diameter (m) function of, friction factor mass flux (kg m2 s1) condensation heat transfer coefficient (W m2 K1) superficial velocity (m s1) conductivity (W m1 K1) length (m) mass flow rate (kg s1) number of parallel channels, number of segments in heat transfer analysis number of unit cells Nusselt Number = hD/k pressure (kPa) Prandtl number = l Cp/k heat duty (W) heat flux (W m2) thermal resistance (K W1) Reynolds number = qVD/l slug length ratio temperature (°C, K) thickness (m), time (s) average velocity (m s1) volume (m3), velocity (m s1) width (m) quality, length parameter length parameter
Greek symbols a void fraction, thermal diffusivity (k/qCp) b homogenous void fraction
from multiple researchers and developed heat transfer models spanning a wide range of mass fluxes, diameters, and fluids. The models attempt to account for flow-regime-specific heat transfer and pressure prop mechanisms. Most of these correlations classify the data into stratified/wavy or annular flows. Heat transfer models for intermittent and mist flow regimes have still not been successfully developed in these studies. Soliman [11] proposed a quasi-homogeneous model for the mist flow regime. Only a few researchers have reported heat transfer measurements and models for tubes of D < 3 mm. Webb and coworkers [12–16] conducted experiments to determine heat transfer coefficients in extruded aluminum tubes with multiple parallel ports of Dh < 3 mm. They attempted several different approaches to model the heat transfer coefficients including shear stress and equivalent mass flux models, but a reliable model that predicts and explains the variety of trends seen in these results has not yet been developed. Yang and Webb [15] explicitly account for surface tension forces in microchannels (with microfins) by computing the drainage of the liquid film from the microfin tips and the associated heat transfer enhancement when the fin tips are not flooded. Wang et al. [17,18] also proposed an analytical treatment for microchannels with D 1 mm that account for the combined influence of surface tension, shear and gravity in the condensation process. Garimella and Bandhauer [19] conducted heat transfer experiments in small diameter tubes (0.4 < Dh < 4.9 mm). They specifically addressed the experimental difficulty of accurately determining condensation heat transfer coefficients due to the
d D k
l q e g r
film thickness (m) change, difference eigen values dynamic viscosity (kg m1 s1) density (kg m3) emissivity fin efficiency surface tension (N m1)
Subscripts and superscripts 0 minimum ave average B bubble CL critical lower CU critical upper Cu copper exp experimental f film, film/bubble section fric frictional g gas phase h hydraulic i segment or node number in inlet l, L liquid, lower, laminar out outlet/exit refg refrigerant s slug sat saturation TS test section v vapor w water, wall @ X differential with respect to x ¼ @x @2 XX second order differential with respect to x ¼ @x 2
high heat transfer coefficients and low mass flow rates in microchannels by developing a novel thermal amplification technique. Bandhauer et al. [20] reported that during the condensation process, as the refrigerant quality decreases, the flow changes from mist to annular to intermittent flow with large overlaps in these types of flows. They developed an annular flow based model, because most of their data were either in the annular flow regime or in transition between the annular flow regime and other regimes. They noted that many of the available shear-driven models, though sound in formulation, led to poor predictions because of the inadequate calculation of shear stresses using pressure drop models that were not applicable to microchannels. Thus, their model is based on boundary layer analyses analogous to the development by Traviss et al. [6], but with the shear stress being calculated from the DP models of Garimella et al. [21] developed specifically for microchannels. Their model also indirectly accounts for surface tension through a surface tension parameter in the DP used for the shear stress calculation to yield accurate microchannel heat transfer predictions over a wide range of conditions.
3. Experimental methodology and results Evaluation of condensation heat transfer coefficient in rectangular channels was originally presented for a single geometry (200 100 lm, AR = 2) in a study by Agarwal and Garimella [22]. In the present study, a considerable amount of additional data over
100
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
a wide range of saturation temperatures, mass fluxes, and qualities were collected using three new tube geometries with different aspect ratios using the same test facility and data analysis procedure. Details of the fabrication of the refrigerant channels, experimental facility and data analysis procedure are available in Agarwal and Garimella [22]. A brief summary is presented here for reference.
Table 1 Test section details. Width wTS (lm)
Depth dTS (lm)
Hydraulic diameter Dh (mm)
Aspect ratio AR
No. of parallel channels, N
100 200 300 400
100 100 100 100
0.100 0.133 0.150 0.160
1 2 3 4
20 18 15 15
3.1. Test section and experimental facility The test section consists of a refrigerant R134a-to-water heat exchanger, wherein a refrigerant channel assembly is soldered between water channel blocks at the top and bottom as shown in Fig. 1. The refrigerant microchannels were fabricated on copper using X-ray lithography and electroforming processes, which provided excellent dimensional accuracy and minimal surface roughness (uncertainty of the channels is ±0.5 lm, with a surface roughness of 10–15 nm [23]). The taper in the vertical walls was less than 1°. Table 1 provides details of each test section geometry investigated here. Each water channel coolant block has five 1.5 cm long channels of 0.79 mm diameter drilled into the block. These water channel blocks are soldered to the refrigerant channel blocks as shown in Fig. 1. A thin flux-less solder foil was used to solder along the whole surface. The total heat transfer length along which the solder joint exists between the refrigerant channels and the water channel blocks is 1.5 cm. The width of the solder joint is equal to the total channel (device) width. A schematic of the experimental facility used in the current study is shown in Fig. 2. Subcooled refrigerant (state (1)) enters the pre-heater assembly, where heat is added until the desired two-phase condition (state 2) is achieved, by precisely controlling the applied electrical heat input. Thus, the heat input is varied based on the G, Tsat and xin of interest for a given data point. The test section inlet quality (state 2) is determined from the preheater energy balance, using the pre-heater inlet thermodynamic state (determined by the measured temperature and pressure) and the measured power input in the heater. Cooling water flows in counterflow through the water channel blocks coupled to the refrigerant channel assembly. The coolant flow rate and test section inlet temperature are varied with each data point to condense the refrigerant to the desired outlet quality. After exiting the test section (state 3), the refrigerant enters the post-heater where it is heated to a superheated state (4). The bulk temperature and pressure are measured at the exit of the heater to confirm the superheated state. The test section exit quality can then be calculated through an energy balance, and the condensation heat duty
Fig. 1. Test section assembly.
determined from the calculated test section inlet (state 2) and outlet (state 3) thermodynamic states. From the post-heater, the refrigerant flows to a total condenser, where it is subcooled (state 5) prior to being recirculated to the pre-heater via a positive displacement gear pump. Heat rejected to the coolant in the test-section is rejected ultimately in a unit consisting of a chiller and heater in series, which also serves as a pre-conditioning unit to maintain the desired coolant temperature in the test section. Further details of all the instruments along with measurement uncertainties are available in Agarwal and Garimella [22]. 3.2. Data analysis The refrigerant mass flow rate is calculated using the measured volumetric flow rate and the local density evaluated at the measured temperature and pressure. The refrigerant state can be determined completely by measuring the pressure and temperature in the sub-cooled (state 2 in Fig. 2) and super-heated (state 4 in Fig. 2) states. An energy balance on the pre-heater and post heater, including accounting for heat loss to the ambient, is used to determine the test section inlet/exit conditions and condensation heat duty. The average heat loss as a percent of total test section heat duty was approximately 6%. Frictional pressure drop data were obtained from the total measured pressure drop by accounting for expansion and contraction losses, and deceleration pressure gain, as detailed in Garimella et al. [1]. To accurately determine the local condensation heat transfer coefficient; it was necessary to account for the extended surface area provided by the channel walls and the effect of pressure drop within the channels on the local saturation temperatures. Thus, the conjugate effects due to the simultaneous coupled conduction and convection within the channels and the channel walls was addressed by dividing the test section into segments. Fig. 3 shows a sample coarse grid structure for the segmental heat transfer analysis. In the actual calculations, a much larger number of equal length segments (10 segments in each of the three sections) are used. The water temperature at each of the nodes is determined using a linear interpolation based on the length between the measured water inlet and exit temperatures. Refrigerant-side temperatures are determined based on the refrigerant saturation pressures at each of the nodes in the channel. Variation in saturation pressure along the length of channel was determined by taking into account the change in pressure gradient with variation of quality from inlet to exit. The refrigerant inlet enthalpy is known from the pre-heater energy balance and the enthalpy at the subsequent nodes is determined by subtracting the segmental condensation heat duty. Refrigerant quality is determined as a function of saturation pressure and enthalpy at the respective node. Fig. 3 also shows the thermal resistance network for the segmental heat transfer analysis. Each of the thermal resistances indicates a potential heat flow path. The heat flow between the adjacent nodes was thus defined based on the respective conduction or convection thermal resistances. The condensation heat transfer coefficient (hrefg) is assumed to be the same for all the segments. Even though hrefg also depends on quality, the measurements were not taken in
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
101
Fig. 2. Experimental facility schematic.
Fig. 3. Segmental heat transfer analysis schematic.
enough detail to enable a segment-wise evaluation of the refrigerant heat transfer coefficient. The water-side heat transfer coefficient was determined using the Dittus–Bolter correlation. A conservative ±25% uncertainty is assumed in the hwater for the overall uncertainty analysis. As the refrigerant channels are coupled to coolant channels at both the top and bottom, all the heat flow definitions were appropriately modified to account for the symmetric nature of the problem. The energy balance for each of the nodes was then defined based on the fixed sign convention, by equating the heat coming into the node, to the heat going out of the node. The total condensation heat duty on the refrigerant side was equated to the measured test-section heat duty, determined from the pre- and post-heater energy balance. All the energy balance equations for each of the nodes, and the equations for each of the thermal resistances are solved simultaneously to determine the average refrigerant-side heat transfer coefficient.
Based on a representative case, Agarwal and Garimella [22] demonstrated that this heat transfer data analysis methodology was able to successfully account for the heat transfer contribution due to the additional extended surface areas. From a sensitivity analysis, a total of 30 segments (10 segments in each of the two fins and 10 segments in the central heat exchange section) were chosen for analyzing the data from this study. Using the representative case, Agarwal and Garimella [22] also demonstrated that most of the heat is transferred in the central heat exchanger section, but the contribution of the extended surfaces could not be neglected. Due to this variation in heat duty, the average refrigerant or refrigerant-side wall temperatures were determined from a weighted average based on segmental heat duties (Qrefg,i). Similarly, in calculating the refrigerant-side effective heat transfer area (needed to calculate resistance ratios) the difference in the fin effect on the inlet and exit side was taken into account.
102
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
The data analysis discussed above was conducted using Engineering Equation Solver [24] software, which also enables computation of the respective uncertainties based on the approach of Taylor and Kuyatt [25]. The most important contributor to the uncertainty in hrefg is the assumed 25% uncertainty in hwater. As hrefg increases, Rratio ðRrefg =ðRTotal Rrefg ÞÞ decreases, leading to higher uncertainties in hrefg due to the increased importance of the hwater uncertainties. The other major contributors to the uncertainties in the hrefg are the uncertainties in the determination of heat losses to the ambient in the pre-heater and post-heater and uncertainties in the measurement of pressures and temperatures in the test section. For the same test section and saturation temperature, the uncertainty increases as the mass flux decreases, primarily because the test section heat transfer rates are smaller at the lower mass fluxes. 3.3. Experimental results and comparison with literature Tests were conducted on channels of four different sizes and shapes detailed in Table 1. Each of these tubes has a different aspect ratio and hydraulic diameter. Detailed results for the 200 100 lm were presented in Agarwal and Garimella [22], with experiments on the additional test sections conducted for this study. Fig. 4 shows representative results for the 300 100 lm test section at different saturation temperatures and mass fluxes. All of the heat transfer data correspond to the pressure drop results presented by Garimella et al. [1], i.e., heat transfer and pressure drop measurements were conducted simultaneously. Thus, many of the trends observed for pressure drop are also seen in the heat transfer results. The condensation heat transfer coefficient increases as the G and/or x increases due to an increase in flow velocities. The slight increase in heat transfer coefficient with decreasing x at low qualities observed for G = 800 kg m2 s1 and T = 30 °C (Fig. 4) is likely an artifact of experimental uncertainty in these points. Additionally, the probability of occurrence of
annular flow is higher at higher G and higher x leading to higher hrefg. As the Tsat decreases, the vapor-to-liquid density ratio decreases, leading to an increase in flow velocities and interfacial shear. This increase in flow velocities leads to increases in hrefg. The increase in interfacial shear has also been related to increases in hrefg by several researchers [4,6,9,20]. Also, as the Tsat decreases from 60 °C to 30 °C, the latent heat of evaporation increases from 139 to 173 kJ kg1, leading to higher hrefg at lower Tsat. Fig. 5 presents a comparison of the heat transfer coefficients observed in different tubes under similar flow conditions. For most of the cases, the differences in the data for different tubes under similar flow conditions are within the error bands. However, it appears that for most cases, the hrefg are the highest for the 400 100 lm channels (which have the highest aspect ratio), followed by hrefg for the 100 100 lm, 200 100 lm and 300 100 lm channels. These trends are similar to those observed in the pressure gradients for these channels by Garimella et al. [1]. The average uncertainty in the calculated hrefg for the 100 100 lm, 200 100 lm, 300 100 lm and 400 100 lm channels are 17, 16, 19 and 25% respectively. Overall, 82% of the data points have uncertainties in the hrefg of less than 25%. The most important contributor to the uncertainty in hrefg is the assumed 25% uncertainty in the water-side heat transfer coefficients, particularly when the ratio of the refrigerant resistance to the remaining resistances is low. A summary of a comparison of the data with existing multiregime or annular flow correlations in the literature [5,6,8– 10,20,26,27] is presented in Table 2. In general, all models significantly under predict the heat transfer coefficients reported here. Unlike the results of the comparison with pressure drop models from the literature by Garimella et al. [1], even empirical correlations using large experimental databases are not able to predict the heat transfer coefficients for the small diameter tubes accurately, due to a lack of accurate heat transfer data for small diameter channels in those databases. Predictions of the shear flow
Fig. 4. Heat transfer results for 300 100 lm channels.
103
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 5. Heat transfer results for the same data set and different tubes.
Table 2 Average deviation for various heat transfer correlations. Channel Width (lm) Correlation
Average deviation (%) 100
200
300
400
All
Shah [26] Soliman [5] Traviss et al. [6] Dobson and Chato [8] Moser et al. [27] Thome et al. [10] Cavallini et al. [9] Bandhauer et al. [20] Kim and Mudawar [28]
66 77 61 62 71 73 41 53 73
59 73 54 56 65 69 33 46 68
66 77 63 63 71 74 49 57 73
70 79 66 68 74 78 53 62 77
66 77 61 63 70 74 46 56 73
Average absolute deviation (%)
models based on the boundary layer analysis suggested by Traviss et al. [6] seem to provide better predictions but the results are strongly dependent on the flow models used to determine the film
66 77 61 63 70 74 46 56 73
thickness and the applicable shear stress. Improved shear stress predictions would also improve heat transfer predictions. Also, most of the models assume annular flow throughout and do not
104
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 6. Intermittent flow unit cell.
account for the existence of intermittent flow, which is significant in the channels under investigation here. The film thickness determined by using an annular flow assumption could be larger than the actual film thickness in the bubble section of intermittent flow with correspondingly lower heat transfer coefficients.
4. Model development The two-phase flow regime, pressure drop and heat transfer mechanisms are strongly coupled. Thus, the heat transfer model developed here relies on the previously developed pressure drop model by Garimella et al. [1]. In this model, the microchannel flow is assumed to be either intermittent or annular, with annular flow corresponding to an infinitely long bubble. The concept of a unit cell, shown schematically in Fig. 6 is introduced. Each unit cell contains a transition between the liquid film and slug regions. By correlating the number of unit cells (Nuc) based on the local flow conditions (i.e., Reynolds number, tube aspect ratio and fluid properties), the pressure drop contribution in the bubble region, liquid slug and transition between bubble/slug (due to the acceleration of the liquid in the film to the slug velocity) were each modeled separately and combined to determine an overall pressure drop. The model predicted the pressure drop for 95% of the data within ±25% of the experimental values [1]. During condensation, the vapor quality of the refrigerant decreases along the length of the tube. Thus, as condensation proceeds, the size of the vapor bubbles decreases due to vapor condensation and the size of the slug increases, leading to an overall decrease in the length of the unit cell. Garimella et al. [1] showed that the number of unit cells and the slug length ratio increases as the vapor quality decreases. At a fixed location along the length of tube, the process proceeds as follows: first a liquid slug passes (without any entrained vapor), leaving behind a thin film of liquid, and then an elongated bubble with a uniform liquid film along its circumference passes. The liquid film around the bubble is formed by the initial film left by the passing slug and the condensing vapor. The process continues upon arrival of the next slug. Fig. 7 shows a schematic of the condensation process as modeled here. The dashed line is an exaggerated representation of the change in film thickness from the leading to trailing edge as
the bubble moves down the channel and condenses. As in the pressure drop model, the heat transfer model is composed of models for the slug section and the film/bubble section. The slug region is treated as single-phase liquid flow, while for the film/bubble section, a condensation heat transfer model is developed. The flow velocities (Ububble and Uslug), and the average film thickness (dave), are known from the pressure drop model. The film/bubble section heat transfer coefficient is referred to as the film heat transfer coefficient (hf) in this section. For heat transfer model development, it is assumed that hs and hf do not vary along the length of the slug (lslug) or within the bubble section (lbubble) in a unit cell. Analysis of the data yields the time-averaged effective refrigerant-side heat transfer coefficient (hrefg) as a function of slug and film heat transfer coefficients. The heat flux could vary with time as the slug or the bubble passes by, but the overall average heat flux may be calculated using the time-averaged refrigerantside heat transfer coefficient. It is assumed that subcooling effects in the liquid in the slug or the film are negligible; thus, both liquid and vapor are assumed to be at the refrigerant saturation temperature. The slug Reynolds number (Reslug) for the data under consideration ranges from 2194 to 12,187. Churchill [29] proposed the correlation shown in Eq. (1) for the determination of Nusselt number (Nu) which is valid for the laminar, transition and turbulent regions. In this correlation, Nul is the laminar Nusselt number, Nulc is the laminar Nusselt number at a critical Reynolds number of 2100 where the transition region begins, and Nu0 is the asymptotic value of the laminar Nusselt number as Pr ? 0 and Re ? 2100.
Nu10 ¼ Nu10 l 2 6e þ4
Nu2lc
0 þ @Nu0 þ
qffiffi
f 8 4=5 5=6
0:079 Re ð1 þ Pr
Þ
Pr
12 35 A 7 5
ð1Þ
Churchill [29] suggested that for the uniform heat flux case, Nu0 = 6.3. Shah and Bhatti [30] recommended that for uniform wall heat flux during laminar flow in rectangular channels, a correlation based on aspect ratio, given in Eq. (2), should be used.
Nul ¼ 8:235
1 2:0421
1 AR
þ 3:0853
1 2
þ1:5078
AR
2:4765
1 4 AR
1 3
0:1861
!
AR
1 5 AR
ð2Þ Eq. (2) is used to determine the laminar Nusselt number based on the hydraulic diameter, and since in the laminar region, Nul remains constant, Nulc = Nul. For turbulent Reynolds numbers, Bhatti and Shah [31] recommended that the circular tube correlation (Eq. (1)) can also be used for the rectangular channels with Nu and Re defined on the basis of hydraulic diameter. The slug heat transfer coefficient is then defined as follows:
hs ¼ Nus
Fig. 7. Schematic of condensation process.
ð2200ReÞ=365
kl D
ð3Þ
For the determination of the film heat transfer coefficient, consider the bubble section shown in Fig. 7. The pressure drop model assumed a uniform film thickness in the film/bubble section as shown by the dashed line in Fig. 7 and determined an average film thickness (dave). As the slug passes by, it leaves a thin liquid film behind, and along the length of the elongated bubble, the film thickness increases due to the condensation process. Consider a small section of length dz along the length of the bubble, with, dt the time taken by this section to pass through a particular point and dd the increase in film thickness during this time. Assuming
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
105
the heat flux to be q00f =b and equating the heat leaving the tube to the latent heat of evaporation of additional condensed liquid film thickness yields:
q00f =b 2pR dz dt ¼ qL 2pðR dÞ dd dz hfg
ð4Þ
Eq. (4) is rearranged to obtain the variation in film thickness with time, where the film thickness is assumed to be much smaller than the tube radius, i.e., ðR dÞ R. Integrating the result yields Eq. (5).
Z
Z
d
q00f =b
t
dd ¼
qL hfg
0
d0
dt ) d ¼ d0 þ
q00f =b
qL hfg
t
ð5Þ
Because the slug-bubble interface moves at velocity Ububble [1], the time t ¼ U z , where z is the distance along the bubble length Bubble
from the beginning of the bubble where the film thickness is a minimum. Also, the heat flux in the film/bubble section q00f =b ¼ hf ðT T wall Þ, which when substituted in Eq. (5) yields:
hf ðT T wall Þ z ) d ¼ d0 þ U bubble qL hfg
ð6Þ
Eq. (6) provides the film thickness as a function of distance from the nose of a bubble. Integrating the above equation over the length of the bubble, lBubble, the average film thickness is found by Eq. (7).
dave ¼
Z
1 lBubble
lBubble
d dz ¼ d0 þ
0
hf ðT T wall Þ lBubble qL hfg 2 U bubble
ð7Þ
An expression for the film heat transfer coefficient (Eq. (7)) can be found by rearranging Eq. (8).
hf ¼ ðdave d0 Þ
Fig. 8. Variation of d0 =dave with quality.
2 qL hfg U bubble ðT T wall Þ lB
ð8Þ
Other than minimum film thickness (d0), all the parameters on the right hand side of Eq. (8) are known. Inspection of Eq. (8) reveals that higher heat transfer coefficients are expected at lower saturation temperatures due to the higher latent heat of vaporization (hfg) at lower saturation temperatures. Similarly, at higher mass fluxes, the heat transfer coefficients are expected to be higher due to increased bubble velocities (Ububble). The effect of aspect ratio is accounted for through the length of the bubble, lBubble ½lBubble ¼ ðLtube =N UC Þð1 SLRÞ. For higher aspect ratio channels, the numbers of unit cells are higher [1], leading to lower bubble lengths, which in turn yields higher heat transfer coefficients. (It should be noted that NUC is the total number of unit cells in a fixed length of tube, so as the NUC increases, the same tube length is divided into more number of segments, thus leading to correspondingly lower slug and bubble lengths.) These trends were also observed in the experimentally obtained refrigerant heat transfer coefficients. Since both the slug and the bubble are moving with the same apparent velocity UBubble, the time taken by the film/bubble section to pass by (tf) and the time taken by the slug to pass by (ts) are proand portional to their lengths, i.e., t f ¼ lbubble =U Bubble t s ¼ lslug =U Bubble . The time-averaged hrefg can thus be determined by averaging the two heat transfer coefficients weighted by the time taken by the slug and bubble to pass by as shown in Eq. (9).
hrefg ðt f þ t s Þ ¼ hs t s þ hf tf
periodically varying convective boundary condition (due to the changing heat transfer coefficient) on the refrigerant side were neglected. This assumption is justified in the following section. In the analysis presented above, the only remaining unknown is the minimum film thickness (d0), which is required to determine the film heat transfer coefficient, hf. The data are used to obtain the values of minimum film thickness (d0). Substituting the experimental refrigerant heat transfer coefficient (hrefg) and the slug heat transfer coefficient (hs) into Eq. (10), the film heat transfer coefficient, hf can be obtained. The SLR is known from the pressure drop model [1] and hs is determined from Eq. (3). These film heat transfer coefficient (hf) values are then substituted into Eq. (8) to determine the minimum film thickness (d0). All other parameters (dave, qL, hfg, Ububble, lBubble, T, Twall) are known from the pressure drop model and data analysis. Fig. 8 shows the variation of the ratio of minimum film thickness and the average film thickness for the data obtained in the present study. As the quality increases and the diameter decreases, the d0 =dave decreases. For all the data points, the film thickness is 4.4–4.6% of the channel diameter. As the tube hydraulic diameter increases, the film thickness increases, leading to increased thermal resistance, i.e., a decrease in heat transfer coefficient. This decrease in heat transfer coefficient leads to lower condensation rates and hence the difference between the average and minimum film thickness decreases. As the quality increases, the slug length ratio decreases, leading to a decrease in the thickness of the film that it leaves behind. As the
ð9Þ
Substituting tf and ts yields:
hrefg ¼ hs
lslug lslug þ lbubble
þ hf 1
lslug lslug þ lbubble
ð10Þ
Sun et al. [32] also used a similar approach to determine the effective hrefg in intermittent flow. By determining the timeaveraged hrefg in the above manner, transient effects due to the
Fig. 9. Comparison of experimental and predicted hrefg.
106
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 10. Experimental and predicted hrefg for 300 100 lm channels.
diameter increases, the effect of a change in diameter diminishes and the change in d0 =dave with quality decreases. Based on this trend, beyond a certain diameter, an increase in hydraulic diameter will no longer affect the film thickness ratio. Based on the above observations, the correlation given in Eq. (11) is proposed for determining d0 =dave ; where Dref is the maximum diameter investigated here, (Dref = 160 lm).
x 0:424 2:82DD d0 ref ¼ 1 0:25 e 1x dave
ð11Þ
In the above correlation, the first term on the right hand side is a constant 1, which implies that for a quality of zero, the minimum film thickness and the maximum film thickness are the same and no condensation occurs. The second term captures the dependence on quality and diameter. The dependence on diameter was explained above. The dependence on quality is observed because a higher vapor quality implies that there is less liquid in the slug, and hence, as the slug passes by, it leaves a thinner liquid film behind. Fig. 9 shows a comparison of the heat transfer coefficients
predicted by the above heat transfer model and the experimentally determined values. The proposed heat transfer model predicts refrigerant heat transfer coefficients for 94% of the data within ±25% of the experimental values. The agreement is illustrated in Fig. 10 for the 300 100 lm data. The proposed model captures the trends in the data from the present study well. The average absolute deviations for the 100 100 lm, 200 100 lm, 300 100 lm and 400 100 lm tubes are 5%, 8%, 9% and 18% respectively, with an overall average absolute deviation of 11%. 4.1. Transient analysis In the above analysis for determining the effective hrefg, the transient effects due to the periodically varying refrigerant-side heat transfer coefficient were neglected. To validate this assumption, a one-dimensional transient analysis was conducted with a periodically varying convection condition on one side and a constant heat flux boundary condition on the other side as shown in 2
Fig. 11, where T X ¼ @T and T XX ¼ @@xT2 . The detailed solution of this @x transient heat transfer phenomenon is provided in Agarwal [33]. The temperature profile in the wall is given by Eq. (12), with the functions Tf and Ts given in Eqs. (13), (14), respectively.
Tðx; tÞ ¼
T f ðx; tÞ
0 < t < tf
T s ðx; t t f Þ t f < t < ðt f þ ts Þ
T f ðx; tÞ ¼ T R þ q00
X 1 1 1 þ Bf;n cosðkf;n xÞ ðL xÞ þ k hf n¼1
expðk2f;n a tÞ
T s ðx; tÞ ¼ T R þ q00 Fig. 11. Schematic for wall transient problem with periodic convection.
ð12Þ
ð13Þ
X 1 1 1 þ ðL xÞ þ Bs;n cosðks;n xÞ k hs n¼1
expðk2s;n a tÞ
ð14Þ
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 12. Temperature profile in wall for representative case.
The experimentally obtained hrefg and the theoretically calculated hs are used to obtain values for the hf. The thickness of the wall (L) was assumed to be equal to 1 mm. In the transient analysis over one complete cycle, the refrigerant-side wall temperature varies; however, the average refrigerant-side wall temperature was assumed to be equal to the wall temperatures determined in the data analysis. The effective film heat transfer coefficient, hf, determined using the results from this transient analysis were within 1% deviation of the hf determined using Eq. (34) for 89% of the data, with a maximum deviation of 2.5% for all the data points. Thus, the assumption of neglecting the transient effects in the wall and using a simple weighted average of the slug and film/bubble heat transfer coefficients is justified. Fig. 12 shows a plot for the wall temperature profile for the same representative case of eighteen 200 100 lm channels for G = 606 kg m2 s1, Tsat = 60.5 °C and xave = 0.39. For this case, the experimental refrigerant-side heat transfer coefficient (hrefg) is 21.7 kW m2 K1, while the average refrigerant-side wall temperature is 58.1 °C. The lengths of the bubble and slug are 1.96 and 0.24 mm, respectively. Slug (hs) and film (hf) heat transfer coefficients are determined to be 9.2 and 23.2 kW m2 K1 respectively. With a bubble velocity (Ububble) of 3.7 m s1, the time taken by the bubble (tf) and slug (ts) to pass a particular point are 535 and 66 ls, respectively. In Fig. 12, the refrigerant side is shown towards the front (L = 1 mm) and the constant heat flux side is shown towards the back (L = 0 mm). Only one time cycle is shown, where from time 0 to tf, the bubble passes by and from time tf to (tf + ts), the slug passes by.
107
The temperature variation along the thickness of the wall is approximately linear for low Biot numbers (hL/k). For all cases in the present study, the Biot Number is less than 0.3. Since the hf are higher than the hs and heat flux at the other end is assumed to be constant, during the time when a bubble passes, more heat is transferred to the wall due to which the temperature of the wall rises and as the slug passes, the wall temperature again rapidly drops. This can be explained further as follows. The rate at which heat is flowing out of the wall to the coolant is fixed due to the constant heat flux boundary condition on the coolant side. But, the rate at which the heat enters the wall from the refrigerant side is higher when the heat transfer coefficient is higher. Thus, when the bubble passes, more heat is transferred into the wall leading to a rise in temperature of the wall due to thermal storage. During the slug phase, the rate of heat transfer into the wall is much less, hence the stored thermal energy in the wall decreases, leading to a decrease in the wall temperature. Fig. 13 shows the variation in wall temperature with time at various depths (for the representative case discussed above) and further compares these variations with the overall temperature difference between the refrigerant side and the coolant side. These two plots together show that while the above analysis captures the temperature variations in the wall with time, they are not particularly significant compared to the overall driving temperature difference between the refrigerant and the coolant for the cases investigated here.
5. Parametric analysis In this section, the heat transfer model described above is used to illustrate the effects of various parameters such as hydraulic diameter, aspect ratio, mass flux and saturation temperature on heat transfer coefficient. (Only one parameter is varied at a time.) Fig. 14 shows the effect of variation of mass flux and quality for D = 130 lm, AR = 3, T = 50 °C and T Twall = 2 °C, G varying from 200 to 800 kg m2 s1. The hrefg increases with increasing mass flux due to an increase in the Reynolds number. Film heat transfer coefficients are in general higher than the slug heat transfer coefficients. Fig. 15 shows a representative comparison of the variation in slug and film heat transfer coefficients with mass flux and quality as predicted by the model. Garimella et al. [1] demonstrated that as the x increases, the slug length ratio decreases, leading to a higher contribution of the film heat transfer coefficient towards the time-averaged hrefg. Also, as the x increases, the flow velocities increase due to an increase in vapor velocity, which in turn leads to a corresponding increase in slug velocity. Thus, the slug heat
Fig. 13. Variation in wall temperature with time at various depths.
108
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 14. Model predictions: effect of mass flux and quality.
transfer coefficients increase as the x increases. For the lowest G case, the slug heat transfer coefficient remains almost constant due laminar Reslug. In the laminar region, Nu is constant for single-phase flow. Also, as the quality increases, the minimumto-average film thickness ratio decreases leading to an increase in film heat transfer coefficients. Thus, as the quality increases, the hrefg increase, due to an increase in both hslug and hf. Fig. 16 shows the effect of the variation of Tsat on the heat transfer coefficient for D = 130 lm, AR = 3, G = 600 kg m2 s1 and T Twall = 2 °C. The Tsat varies from 30 to 60 °C. As the Tsat decreases, the gas-to-liquid phase density ratio ðqg =ql Þ decreases (qg/ql = 0.083 @ 60 °C and qg/ql = 0.032 @ 30 °C), leading to an increase in void fraction (a = 0.74 @ 60 °C and a = 0.80 @ 30 °C). As the Tsat decreases, the hrefg increases due to two factors. Firstly, as mentioned above, at lower Tsat, the flow velocities are higher, leading to an increase in the hrefg. Also, as the Tsat decreases, the slug length ratio decreases; thus, the contribution of the film heat transfer coefficient toward the time-averaged hrefg is higher at lower Tsat. Fig. 17 shows the effect of a variation in channel aspect ratio (AR) on the heat transfer coefficient for D = 130 lm, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The AR was varied
Fig. 16. Model predictions: effect of saturation temperature.
Fig. 17. Model predictions: effect of channel aspect ratio.
Fig. 15. Variation in hslug and hfilm with mass flux and quality.
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 18. Model predictions: effect of diameter.
Fig. 19. Model predictions: effect of driving temperature difference.
109
from 1 to 4, without varying the hydraulic diameter. It should be noted that unlike the actual channels tested in present study, the hydraulic diameter is kept constant here and only the AR varies. As the AR increases, the channel depth decreases and the width increases. The hs are higher for higher aspect ratio channels if the slug flow is laminar. Also, according to the model, hf is inversely proportional to the bubble length. Larger number of unit cells at higher AR [1] lead to shorter bubble lengths, yielding higher film heat transfer coefficients. Thus, as the aspect ratio increases, the hrefg increases. Physically this can be understood as follows. Each time the slug passes by, it breaks the liquid film boundary layer and due to fluid passing from film to slug and then from slug to film, there is more mixing of the fluid leading to higher heat transfer coefficients. The higher the number of unit cells, the more frequent is this turbulent mixing due to the passing slugs, leading to higher heat transfer coefficients in high aspect ratio channels. Fig. 18 shows the effect of variation of channel hydraulic diameter on the heat transfer coefficient for AR = 3, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The hydraulic diameter varies from 100 lm to 160 lm. The hrefg also increases with a decrease in diameter due to the increase in hf with decreasing diameters. The hf increase due to a decrease in film thickness with the decrease in tube diameter. Fig. 19 illustrates the effect of driving temperature difference on the refrigerant heat transfer coefficients for AR = 3, G = 600 kg m2 s1, T = 50 °C, xave = 0.5 and T Twall = 2 °C. As the driving temperature difference decreases, the rate of condensation decreases, leading to a thinner liquid film on the channel walls, which in turn yields higher heat transfer coefficients. The effect of a change in driving temperature difference diminishes with increasing driving temperature difference. The average vapor quality for the data obtained in the present study varies between 0.2 and 0.8. Fig. 20 shows the heat transfer coefficient predictions if the proposed model is extrapolated to vapor qualities varying from 0.05 to 0.95 for D = 130 lm, AR = 3, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The single-phase liquid only and vapor-only heat transfer coefficient (hrefg,LO = 2.2 kW m2 K1 and hrefg,VO = 3.1 kW m2 K1) are also shown for this case. As the quality approaches zero, the heat transfer coefficient also approaches the single-phase heat transfer coefficient. As the quality approaches zero, the contribution of slug heat transfer coefficient in the time-averaged heat transfer coefficient increases, which leads to the approach to the single phase liquid value. The heat transfer coefficient continues to increase as the quality approaches one due to a progressively thinner liquid film. (The model does not account for inlet superheat, which might be the situation in an actual condenser.) The model proposed here is able to predict the trends in the heat transfer data based on a physical basis. The effect of surface tension is not explicitly captured, even though it might play an important role in microchannels. The behavior at very low (x < 0.2) and very high (x > 0.8) quality is also not addressed in great detail, because of the lack of data in these regions. The effect of fluid property variation is also not addressed; this would require additional data with different refrigerants. The model proposed here also assumes uniform distribution of flow in all channels and steady state conditions. Flow maldistribution and instabilities could affect the heat transfer coefficients in condensers using tubes with multiple parallel ports. Future studies with varying inlet and outlet header designs can be used to investigate these effects.
6. Conclusions
Fig. 20. Extrapolation of proposed model.
A study of condensation heat transfer in microchannels was conducted. A technique first introduced by Agarwal and Garimella
110
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
[22] was used accurately determine pressure drop and heat transfer coefficients during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels for wide ranging flow conditions (300 < G < 800 kg m2 s1; 0 < x < 1; 30 < Tsat < 60 °C). It was found that hrefg increased with increasing x, increasing G and decreasing Tsat. The hrefg were the highest for the 400 100 lm channels, followed by 100 100 lm, 200 100 lm, and 300 100 lm channels. It was also found that most of the models from the literature significantly under-predicted the heat transfer data from the present study. The flow velocities and other parameters determined for the pressure drop model [1] are used as inputs for the heat transfer models. The slug and bubble regions were analyzed separately to determine the slug and film heat transfer coefficients. A timeaveraged condensation heat transfer coefficient was determined by combining the slug and film heat transfer coefficients according to their transit times through the channel. The proposed heat transfer models predict 94% of the data within ±25%. In addition, the effect of the variation of heat transfer coefficient in intermittent flows from the slug to bubble regions was investigated in detail using a transient conjugate analysis. The proposed model was also used to analyze the effect of various parameters such as mass flux, saturation temperature, hydraulic diameter, and aspect ratio. Observed trends were explained based on the relative predominance of the intermittent or annular flow regime. As the G increases, the hrefg increase due to an increase in flow velocities. As the Tsat decreases, the void fraction increases due to a decrease in the vapor-to-liquid density ratio, which increases velocities and interfacial shear, and in turn, leads to an increase in hrefg. As the channel hydraulic diameter decreases, the hrefg increase due to a decrease in film thickness and channel diameter. As the AR increases, the hrefg increase due to an increased occurrence of slugs. The proposed model may be used by engineers for analyzing condensing two-phase flow in microchannel geometries and in the design of more efficient heat transfer equipment. References [1] S. Garimella, A. Agarwal, B.M. Fronk, The intermittent and annular flow condensation continuum: pressure drops at the microscale, Int. J. Multiphase Flow (2016), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.03.023. [2] J.W. Coleman, S. Garimella, Two-phase flow regime transitions in microchannel tubes: the effect of hydraulic diameter, Am. Soc. Mech. Eng. Heat Transfer Div. 366 (2000) 71–83 (Publication) HTD. [3] J.W. Coleman, S. Garimella, Two-phase flow regimes in round, square and rectangular tubes during condensation of refrigerant R134a, Int. J. Refrig 26 (1) (2003) 117–128. [4] F.G. Carpenter, A.P. Colburn, The effect of vapor velocity on condensation inside tubes, ASME Proceedings of the General Discussion of Heat Transfer, 1951, pp. 20–26. [5] H.M. Soliman, J.R. Schuster, P.J. Berenson, A general heat transfer correlation for annular flow condensation, J. Heat Transfer 90 (2) (1968) 267–276. [6] D.P. Traviss, W.M. Rohsenow, A.B. Baron, Forced-Convection condensation inside tubes: a heat transfer equation for condenser design, ASHRAE Trans. 79 (Part 1) (1973) 157–165. [7] S.L. Chen, F.M. Gerner, C.L. Tien, General film condensation correlations, Exp. Heat Transfer 1 (2) (1987) 93–107.
[8] M.K. Dobson, J.C. Chato, Condensation in smooth horizontal tubes, J. Heat Transfer 120 (1) (1998) 193–213. [9] A. Cavallini, G. Censi, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, Condensation of halogenated refrigerants inside smooth tubes, HVAC&R Res. 8 (4) (2002) 429–451. [10] J.R. Thome, J. El Hajal, A. Cavallini, Condensation in horizontal tubes, part 2: new heat transfer model based on flow regimes, Int. J. Heat Mass Transfer 46 (18) (2003) 3365–3387. [11] H.M. Soliman, Mist-annular transition during condensation and its influence on the heat transfer mechanism, Int. J. Multiphase Flow 12 (2) (1986) 277– 288. [12] R.L. Webb, K. Ermis, Effect of hydraulic diameter on condensation of R-134A in flat, extruded aluminum tubes, J. Enhanced Heat Transfer 8 (2) (2001) 77–90. [13] C.Y. Yang, R.L. Webb, Friction pressure drop of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins, Int. J. Heat Mass Transfer 39 (4) (1996) 801–809. [14] C.Y. Yang, R.L. Webb, Condensation of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins, Int. J. Heat Mass Transfer 39 (4) (1996) 791–800. [15] C.Y. Yang, R.L. Webb, A predictive model for condensation in small hydraulic diameter tubes having axial micro-fins, J. Heat Transfer 119 (4) (1997) 776– 782. [16] M. Zhang, R.L. Webb, Correlation of two-phase friction for refrigerants in small-diameter tubes, Exp. Thermal Fluid Sci. 25 (3–4) (2001) 131–139. [17] H.S. Wang, J.W. Rose, Film condensation in horizontal triangular section microchannels: a theoretical model, Proceedings of the Second International Conference on Microchannels and Minichannels (ICMM2004), June 17–19 2004, American Society of Mechanical Engineers, New York, NY 10016-5990, United States, Rochester, NY, United States, 2004, pp. 661–666. [18] H.S. Wang, J.W. Rose, H. Honda, A theoretical model of film condensation in square section horizontal microchannels, Chem. Eng. Res. Des. 82 (4) (2004) 430–434. [19] S. Garimella, T.M. Bandhauer, Measurement of condensation heat transfer coefficients in microchannel tubes, 2001 ASME International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, New York, NY, United States, 2001, pp. 243–249. [20] T.M. Bandhauer, A. Agarwal, S. Garimella, Measurement and modeling of condensation heat transfer coefficients in circular microchannels, J. Heat Transfer 128 (10) (2006) 1050–1059. [21] S. Garimella, A. Agarwal, J.D. Killion, Condensation pressure drop in circular microchannels, Heat Transfer Eng. 26 (3) (2005) 28–35. [22] A. Agarwal, S. Garimella, Representative results for condensation measurements at hydraulic diameters similar to 100 microns, J. Heat Transfer 132 (4) (2010) 1–12. [23] T. Christenson, Email Communication Dated August 19th 2005, in: A. Agarwal (Ed.), Atlanta, 2005. [24] S.A. Klein, Engineering Equation Solver, in, F-Chart Software, 2006. [25] B.N. Taylor, C.E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, National Institute of Standards and Technology, Washington, DC, USA, 1994. [26] M.M. Shah, A general correlation for heat transfer during film condensation inside pipes, Int. J. Heat Mass Transfer 22 (4) (1979) 547–556. [27] K.W. Moser, R.L. Webb, B. Na, A new equivalent Reynolds number model for condensation in smooth tubes, J. Heat Transfer 120 (2) (1998) 410–417. [28] S.M. Kim, I. Mudawar, Universal approach to predicting heat transfer coefficient for condensing mini/micro-channel flow, Int. J. Heat Mass Transfer 56 (1–2) (2013) 238–250. [29] S.W. Churchill, Comprehensive correlating equations for heat mass and momentum-transfer in fully developed flow in smooth tubes, Ind. Eng. Chem. Fundam. 16 (1) (1977) 109–116. [30] R.K. Shah, M.S. Bhatti, Laminar convective heat transfer in ducts, in: S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, NY, 1987. [31] M.S. Bhatti, R.K. Shah, Turbulent and transitions flow convective heat transfer in ducts, in: S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, NY, 1987. [32] G. Sun, G.F. Hewitt, V.V. Wadekar, A heat transfer model for slug flow in a horizontal tube, Int. J. Heat Mass Transfer 47 (12–13) (2004) 2807–2816. [33] A. Agarwal, Heat Transfer and Pressure Drop During Condensation in Microchannels PhD Dissertation, Georgia Institute of Technology, Atlanta, 2006.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Condensation heat transfer in rectangular microscale geometries Srinivas Garimella a,⇑, Akhil Agarwal b, Brian M. Fronk c a
Sustainable Thermal Systems Laboratory, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA SEPCO – Upstream Americas, 701 Poydras St., New Orleans, LA 70161, USA c School of Mechanical, Industrial and Manufacturing Engineering, Oregon State University, Corvallis, OR 97331, USA b
a r t i c l e
i n f o
Article history: Received 19 July 2015 Received in revised form 21 March 2016 Accepted 23 March 2016
Keywords: Condensation Heat transfer Microchannel Annular Intermittent
a b s t r a c t Heat transfer coefficients during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels are presented. A novel technique to accurately determine condensation heat duty and heat transfer coefficient in such microscale geometries at small Dx is used. Models in the literature that were developed for larger tubes are shown to under predict the data. A new model that accounts for the flow mechanisms during condensation at such small scales, and takes into account the effect of G, x, Tsat, Dh and AR, is developed. The model predicts 94% of the data in the intermittent, transition and annular flow regimes within ±25%. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
2. Prior work
Microchannels are increasingly being used by numerous industries to miniaturize heat transfer equipment, improve energy efficiency, and minimize heat transfer fluid inventory. A fundamental understanding of condensation at the microscales will yield far reaching benefits for the automotive and HVAC&R industries, and others that need miniaturized components, such as portable personal cooling devices, hazardous duty and high ambient air-conditioning, and medical devices. Garimella et al. [1] reported measurements and a unified model for intermittent and annular flow pressure drops during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels for a wide range of conditions (300 < G < 800 kg m2 s1, 0 < x < 1; 30 < Tsat < 60 °C). It was found that pressure drop and heat transfer coefficient increased with increasing vapor quality, increasing mass flux and decreasing saturation temperature. The pressure drop model of Garimella et al. [1] relied on an annular flow factor (AFF) that accounted for the relative predominance of annular or intermittent flow in these channels. In this paper, a substantial amount of additional heat transfer data are introduced and a new heat transfer model is proposed based on the data and using the insights on flow mechanisms and their influence on pressure drop reported in Garimella et al. [1].
Models for horizontal, in-tube condensation have primarily considered idealized gravity and shear-driven mechanisms and the relative importance of the two. Gravity driven models are most important in stratified and wavy flow regimes. As the channel diameter decreases, surface tension forces become increasingly important and the gravity driven stratified and wavy flow regimes become less prevalent, as evidenced by the condensation flow visualization studies of Coleman and Garimella [2,3]. Thus, modeling of condensation heat transfer in the shear-dominated annular and intermittent flow regimes are of primary importance. Shear-based annular flow models generally relate the interfacial shear stress to the analogous heat transfer across the condensate film. This approach was first introduced by Carpenter and Colburn [4] and later adapted by other researchers [5,6] with modifications made in the determination of the interfacial shear. Chen et al. [7] developed a general purpose annular flow correlation starting with asymptotic limits, and blending them through simple combinations of the terms at the respective limits. Several researchers have also used a two-phase multiplier approach, similar to that commonly used in two-phase pressure drop models. When used in the heat transfer models, the two-phase multiplier is applied to the corresponding single-phase heat transfer coefficient. It should be noted that shear-based models also use two-phase multipliers to determine the interfacial shear stress, and thus the two approaches are equivalent to each other. Several researchers, including, Dobson and Chato [8], Cavallini et al. [9], and Thome et al. [10], have analyzed condensation data
⇑ Corresponding author. E-mail address: [email protected] (S. Garimella). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.086 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
99
Nomenclature AR AFF Cp d D f G h J, j k L _ m N NUC Nu P Pr Q q00 R Re SLR T t U V W, w x z
aspect ratio annular flow factor specific heat (J kg1 K1) depth of microchannels (m) diameter (m) function of, friction factor mass flux (kg m2 s1) condensation heat transfer coefficient (W m2 K1) superficial velocity (m s1) conductivity (W m1 K1) length (m) mass flow rate (kg s1) number of parallel channels, number of segments in heat transfer analysis number of unit cells Nusselt Number = hD/k pressure (kPa) Prandtl number = l Cp/k heat duty (W) heat flux (W m2) thermal resistance (K W1) Reynolds number = qVD/l slug length ratio temperature (°C, K) thickness (m), time (s) average velocity (m s1) volume (m3), velocity (m s1) width (m) quality, length parameter length parameter
Greek symbols a void fraction, thermal diffusivity (k/qCp) b homogenous void fraction
from multiple researchers and developed heat transfer models spanning a wide range of mass fluxes, diameters, and fluids. The models attempt to account for flow-regime-specific heat transfer and pressure prop mechanisms. Most of these correlations classify the data into stratified/wavy or annular flows. Heat transfer models for intermittent and mist flow regimes have still not been successfully developed in these studies. Soliman [11] proposed a quasi-homogeneous model for the mist flow regime. Only a few researchers have reported heat transfer measurements and models for tubes of D < 3 mm. Webb and coworkers [12–16] conducted experiments to determine heat transfer coefficients in extruded aluminum tubes with multiple parallel ports of Dh < 3 mm. They attempted several different approaches to model the heat transfer coefficients including shear stress and equivalent mass flux models, but a reliable model that predicts and explains the variety of trends seen in these results has not yet been developed. Yang and Webb [15] explicitly account for surface tension forces in microchannels (with microfins) by computing the drainage of the liquid film from the microfin tips and the associated heat transfer enhancement when the fin tips are not flooded. Wang et al. [17,18] also proposed an analytical treatment for microchannels with D 1 mm that account for the combined influence of surface tension, shear and gravity in the condensation process. Garimella and Bandhauer [19] conducted heat transfer experiments in small diameter tubes (0.4 < Dh < 4.9 mm). They specifically addressed the experimental difficulty of accurately determining condensation heat transfer coefficients due to the
d D k
l q e g r
film thickness (m) change, difference eigen values dynamic viscosity (kg m1 s1) density (kg m3) emissivity fin efficiency surface tension (N m1)
Subscripts and superscripts 0 minimum ave average B bubble CL critical lower CU critical upper Cu copper exp experimental f film, film/bubble section fric frictional g gas phase h hydraulic i segment or node number in inlet l, L liquid, lower, laminar out outlet/exit refg refrigerant s slug sat saturation TS test section v vapor w water, wall @ X differential with respect to x ¼ @x @2 XX second order differential with respect to x ¼ @x 2
high heat transfer coefficients and low mass flow rates in microchannels by developing a novel thermal amplification technique. Bandhauer et al. [20] reported that during the condensation process, as the refrigerant quality decreases, the flow changes from mist to annular to intermittent flow with large overlaps in these types of flows. They developed an annular flow based model, because most of their data were either in the annular flow regime or in transition between the annular flow regime and other regimes. They noted that many of the available shear-driven models, though sound in formulation, led to poor predictions because of the inadequate calculation of shear stresses using pressure drop models that were not applicable to microchannels. Thus, their model is based on boundary layer analyses analogous to the development by Traviss et al. [6], but with the shear stress being calculated from the DP models of Garimella et al. [21] developed specifically for microchannels. Their model also indirectly accounts for surface tension through a surface tension parameter in the DP used for the shear stress calculation to yield accurate microchannel heat transfer predictions over a wide range of conditions.
3. Experimental methodology and results Evaluation of condensation heat transfer coefficient in rectangular channels was originally presented for a single geometry (200 100 lm, AR = 2) in a study by Agarwal and Garimella [22]. In the present study, a considerable amount of additional data over
100
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
a wide range of saturation temperatures, mass fluxes, and qualities were collected using three new tube geometries with different aspect ratios using the same test facility and data analysis procedure. Details of the fabrication of the refrigerant channels, experimental facility and data analysis procedure are available in Agarwal and Garimella [22]. A brief summary is presented here for reference.
Table 1 Test section details. Width wTS (lm)
Depth dTS (lm)
Hydraulic diameter Dh (mm)
Aspect ratio AR
No. of parallel channels, N
100 200 300 400
100 100 100 100
0.100 0.133 0.150 0.160
1 2 3 4
20 18 15 15
3.1. Test section and experimental facility The test section consists of a refrigerant R134a-to-water heat exchanger, wherein a refrigerant channel assembly is soldered between water channel blocks at the top and bottom as shown in Fig. 1. The refrigerant microchannels were fabricated on copper using X-ray lithography and electroforming processes, which provided excellent dimensional accuracy and minimal surface roughness (uncertainty of the channels is ±0.5 lm, with a surface roughness of 10–15 nm [23]). The taper in the vertical walls was less than 1°. Table 1 provides details of each test section geometry investigated here. Each water channel coolant block has five 1.5 cm long channels of 0.79 mm diameter drilled into the block. These water channel blocks are soldered to the refrigerant channel blocks as shown in Fig. 1. A thin flux-less solder foil was used to solder along the whole surface. The total heat transfer length along which the solder joint exists between the refrigerant channels and the water channel blocks is 1.5 cm. The width of the solder joint is equal to the total channel (device) width. A schematic of the experimental facility used in the current study is shown in Fig. 2. Subcooled refrigerant (state (1)) enters the pre-heater assembly, where heat is added until the desired two-phase condition (state 2) is achieved, by precisely controlling the applied electrical heat input. Thus, the heat input is varied based on the G, Tsat and xin of interest for a given data point. The test section inlet quality (state 2) is determined from the preheater energy balance, using the pre-heater inlet thermodynamic state (determined by the measured temperature and pressure) and the measured power input in the heater. Cooling water flows in counterflow through the water channel blocks coupled to the refrigerant channel assembly. The coolant flow rate and test section inlet temperature are varied with each data point to condense the refrigerant to the desired outlet quality. After exiting the test section (state 3), the refrigerant enters the post-heater where it is heated to a superheated state (4). The bulk temperature and pressure are measured at the exit of the heater to confirm the superheated state. The test section exit quality can then be calculated through an energy balance, and the condensation heat duty
Fig. 1. Test section assembly.
determined from the calculated test section inlet (state 2) and outlet (state 3) thermodynamic states. From the post-heater, the refrigerant flows to a total condenser, where it is subcooled (state 5) prior to being recirculated to the pre-heater via a positive displacement gear pump. Heat rejected to the coolant in the test-section is rejected ultimately in a unit consisting of a chiller and heater in series, which also serves as a pre-conditioning unit to maintain the desired coolant temperature in the test section. Further details of all the instruments along with measurement uncertainties are available in Agarwal and Garimella [22]. 3.2. Data analysis The refrigerant mass flow rate is calculated using the measured volumetric flow rate and the local density evaluated at the measured temperature and pressure. The refrigerant state can be determined completely by measuring the pressure and temperature in the sub-cooled (state 2 in Fig. 2) and super-heated (state 4 in Fig. 2) states. An energy balance on the pre-heater and post heater, including accounting for heat loss to the ambient, is used to determine the test section inlet/exit conditions and condensation heat duty. The average heat loss as a percent of total test section heat duty was approximately 6%. Frictional pressure drop data were obtained from the total measured pressure drop by accounting for expansion and contraction losses, and deceleration pressure gain, as detailed in Garimella et al. [1]. To accurately determine the local condensation heat transfer coefficient; it was necessary to account for the extended surface area provided by the channel walls and the effect of pressure drop within the channels on the local saturation temperatures. Thus, the conjugate effects due to the simultaneous coupled conduction and convection within the channels and the channel walls was addressed by dividing the test section into segments. Fig. 3 shows a sample coarse grid structure for the segmental heat transfer analysis. In the actual calculations, a much larger number of equal length segments (10 segments in each of the three sections) are used. The water temperature at each of the nodes is determined using a linear interpolation based on the length between the measured water inlet and exit temperatures. Refrigerant-side temperatures are determined based on the refrigerant saturation pressures at each of the nodes in the channel. Variation in saturation pressure along the length of channel was determined by taking into account the change in pressure gradient with variation of quality from inlet to exit. The refrigerant inlet enthalpy is known from the pre-heater energy balance and the enthalpy at the subsequent nodes is determined by subtracting the segmental condensation heat duty. Refrigerant quality is determined as a function of saturation pressure and enthalpy at the respective node. Fig. 3 also shows the thermal resistance network for the segmental heat transfer analysis. Each of the thermal resistances indicates a potential heat flow path. The heat flow between the adjacent nodes was thus defined based on the respective conduction or convection thermal resistances. The condensation heat transfer coefficient (hrefg) is assumed to be the same for all the segments. Even though hrefg also depends on quality, the measurements were not taken in
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
101
Fig. 2. Experimental facility schematic.
Fig. 3. Segmental heat transfer analysis schematic.
enough detail to enable a segment-wise evaluation of the refrigerant heat transfer coefficient. The water-side heat transfer coefficient was determined using the Dittus–Bolter correlation. A conservative ±25% uncertainty is assumed in the hwater for the overall uncertainty analysis. As the refrigerant channels are coupled to coolant channels at both the top and bottom, all the heat flow definitions were appropriately modified to account for the symmetric nature of the problem. The energy balance for each of the nodes was then defined based on the fixed sign convention, by equating the heat coming into the node, to the heat going out of the node. The total condensation heat duty on the refrigerant side was equated to the measured test-section heat duty, determined from the pre- and post-heater energy balance. All the energy balance equations for each of the nodes, and the equations for each of the thermal resistances are solved simultaneously to determine the average refrigerant-side heat transfer coefficient.
Based on a representative case, Agarwal and Garimella [22] demonstrated that this heat transfer data analysis methodology was able to successfully account for the heat transfer contribution due to the additional extended surface areas. From a sensitivity analysis, a total of 30 segments (10 segments in each of the two fins and 10 segments in the central heat exchange section) were chosen for analyzing the data from this study. Using the representative case, Agarwal and Garimella [22] also demonstrated that most of the heat is transferred in the central heat exchanger section, but the contribution of the extended surfaces could not be neglected. Due to this variation in heat duty, the average refrigerant or refrigerant-side wall temperatures were determined from a weighted average based on segmental heat duties (Qrefg,i). Similarly, in calculating the refrigerant-side effective heat transfer area (needed to calculate resistance ratios) the difference in the fin effect on the inlet and exit side was taken into account.
102
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
The data analysis discussed above was conducted using Engineering Equation Solver [24] software, which also enables computation of the respective uncertainties based on the approach of Taylor and Kuyatt [25]. The most important contributor to the uncertainty in hrefg is the assumed 25% uncertainty in hwater. As hrefg increases, Rratio ðRrefg =ðRTotal Rrefg ÞÞ decreases, leading to higher uncertainties in hrefg due to the increased importance of the hwater uncertainties. The other major contributors to the uncertainties in the hrefg are the uncertainties in the determination of heat losses to the ambient in the pre-heater and post-heater and uncertainties in the measurement of pressures and temperatures in the test section. For the same test section and saturation temperature, the uncertainty increases as the mass flux decreases, primarily because the test section heat transfer rates are smaller at the lower mass fluxes. 3.3. Experimental results and comparison with literature Tests were conducted on channels of four different sizes and shapes detailed in Table 1. Each of these tubes has a different aspect ratio and hydraulic diameter. Detailed results for the 200 100 lm were presented in Agarwal and Garimella [22], with experiments on the additional test sections conducted for this study. Fig. 4 shows representative results for the 300 100 lm test section at different saturation temperatures and mass fluxes. All of the heat transfer data correspond to the pressure drop results presented by Garimella et al. [1], i.e., heat transfer and pressure drop measurements were conducted simultaneously. Thus, many of the trends observed for pressure drop are also seen in the heat transfer results. The condensation heat transfer coefficient increases as the G and/or x increases due to an increase in flow velocities. The slight increase in heat transfer coefficient with decreasing x at low qualities observed for G = 800 kg m2 s1 and T = 30 °C (Fig. 4) is likely an artifact of experimental uncertainty in these points. Additionally, the probability of occurrence of
annular flow is higher at higher G and higher x leading to higher hrefg. As the Tsat decreases, the vapor-to-liquid density ratio decreases, leading to an increase in flow velocities and interfacial shear. This increase in flow velocities leads to increases in hrefg. The increase in interfacial shear has also been related to increases in hrefg by several researchers [4,6,9,20]. Also, as the Tsat decreases from 60 °C to 30 °C, the latent heat of evaporation increases from 139 to 173 kJ kg1, leading to higher hrefg at lower Tsat. Fig. 5 presents a comparison of the heat transfer coefficients observed in different tubes under similar flow conditions. For most of the cases, the differences in the data for different tubes under similar flow conditions are within the error bands. However, it appears that for most cases, the hrefg are the highest for the 400 100 lm channels (which have the highest aspect ratio), followed by hrefg for the 100 100 lm, 200 100 lm and 300 100 lm channels. These trends are similar to those observed in the pressure gradients for these channels by Garimella et al. [1]. The average uncertainty in the calculated hrefg for the 100 100 lm, 200 100 lm, 300 100 lm and 400 100 lm channels are 17, 16, 19 and 25% respectively. Overall, 82% of the data points have uncertainties in the hrefg of less than 25%. The most important contributor to the uncertainty in hrefg is the assumed 25% uncertainty in the water-side heat transfer coefficients, particularly when the ratio of the refrigerant resistance to the remaining resistances is low. A summary of a comparison of the data with existing multiregime or annular flow correlations in the literature [5,6,8– 10,20,26,27] is presented in Table 2. In general, all models significantly under predict the heat transfer coefficients reported here. Unlike the results of the comparison with pressure drop models from the literature by Garimella et al. [1], even empirical correlations using large experimental databases are not able to predict the heat transfer coefficients for the small diameter tubes accurately, due to a lack of accurate heat transfer data for small diameter channels in those databases. Predictions of the shear flow
Fig. 4. Heat transfer results for 300 100 lm channels.
103
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 5. Heat transfer results for the same data set and different tubes.
Table 2 Average deviation for various heat transfer correlations. Channel Width (lm) Correlation
Average deviation (%) 100
200
300
400
All
Shah [26] Soliman [5] Traviss et al. [6] Dobson and Chato [8] Moser et al. [27] Thome et al. [10] Cavallini et al. [9] Bandhauer et al. [20] Kim and Mudawar [28]
66 77 61 62 71 73 41 53 73
59 73 54 56 65 69 33 46 68
66 77 63 63 71 74 49 57 73
70 79 66 68 74 78 53 62 77
66 77 61 63 70 74 46 56 73
Average absolute deviation (%)
models based on the boundary layer analysis suggested by Traviss et al. [6] seem to provide better predictions but the results are strongly dependent on the flow models used to determine the film
66 77 61 63 70 74 46 56 73
thickness and the applicable shear stress. Improved shear stress predictions would also improve heat transfer predictions. Also, most of the models assume annular flow throughout and do not
104
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 6. Intermittent flow unit cell.
account for the existence of intermittent flow, which is significant in the channels under investigation here. The film thickness determined by using an annular flow assumption could be larger than the actual film thickness in the bubble section of intermittent flow with correspondingly lower heat transfer coefficients.
4. Model development The two-phase flow regime, pressure drop and heat transfer mechanisms are strongly coupled. Thus, the heat transfer model developed here relies on the previously developed pressure drop model by Garimella et al. [1]. In this model, the microchannel flow is assumed to be either intermittent or annular, with annular flow corresponding to an infinitely long bubble. The concept of a unit cell, shown schematically in Fig. 6 is introduced. Each unit cell contains a transition between the liquid film and slug regions. By correlating the number of unit cells (Nuc) based on the local flow conditions (i.e., Reynolds number, tube aspect ratio and fluid properties), the pressure drop contribution in the bubble region, liquid slug and transition between bubble/slug (due to the acceleration of the liquid in the film to the slug velocity) were each modeled separately and combined to determine an overall pressure drop. The model predicted the pressure drop for 95% of the data within ±25% of the experimental values [1]. During condensation, the vapor quality of the refrigerant decreases along the length of the tube. Thus, as condensation proceeds, the size of the vapor bubbles decreases due to vapor condensation and the size of the slug increases, leading to an overall decrease in the length of the unit cell. Garimella et al. [1] showed that the number of unit cells and the slug length ratio increases as the vapor quality decreases. At a fixed location along the length of tube, the process proceeds as follows: first a liquid slug passes (without any entrained vapor), leaving behind a thin film of liquid, and then an elongated bubble with a uniform liquid film along its circumference passes. The liquid film around the bubble is formed by the initial film left by the passing slug and the condensing vapor. The process continues upon arrival of the next slug. Fig. 7 shows a schematic of the condensation process as modeled here. The dashed line is an exaggerated representation of the change in film thickness from the leading to trailing edge as
the bubble moves down the channel and condenses. As in the pressure drop model, the heat transfer model is composed of models for the slug section and the film/bubble section. The slug region is treated as single-phase liquid flow, while for the film/bubble section, a condensation heat transfer model is developed. The flow velocities (Ububble and Uslug), and the average film thickness (dave), are known from the pressure drop model. The film/bubble section heat transfer coefficient is referred to as the film heat transfer coefficient (hf) in this section. For heat transfer model development, it is assumed that hs and hf do not vary along the length of the slug (lslug) or within the bubble section (lbubble) in a unit cell. Analysis of the data yields the time-averaged effective refrigerant-side heat transfer coefficient (hrefg) as a function of slug and film heat transfer coefficients. The heat flux could vary with time as the slug or the bubble passes by, but the overall average heat flux may be calculated using the time-averaged refrigerantside heat transfer coefficient. It is assumed that subcooling effects in the liquid in the slug or the film are negligible; thus, both liquid and vapor are assumed to be at the refrigerant saturation temperature. The slug Reynolds number (Reslug) for the data under consideration ranges from 2194 to 12,187. Churchill [29] proposed the correlation shown in Eq. (1) for the determination of Nusselt number (Nu) which is valid for the laminar, transition and turbulent regions. In this correlation, Nul is the laminar Nusselt number, Nulc is the laminar Nusselt number at a critical Reynolds number of 2100 where the transition region begins, and Nu0 is the asymptotic value of the laminar Nusselt number as Pr ? 0 and Re ? 2100.
Nu10 ¼ Nu10 l 2 6e þ4
Nu2lc
0 þ @Nu0 þ
qffiffi
f 8 4=5 5=6
0:079 Re ð1 þ Pr
Þ
Pr
12 35 A 7 5
ð1Þ
Churchill [29] suggested that for the uniform heat flux case, Nu0 = 6.3. Shah and Bhatti [30] recommended that for uniform wall heat flux during laminar flow in rectangular channels, a correlation based on aspect ratio, given in Eq. (2), should be used.
Nul ¼ 8:235
1 2:0421
1 AR
þ 3:0853
1 2
þ1:5078
AR
2:4765
1 4 AR
1 3
0:1861
!
AR
1 5 AR
ð2Þ Eq. (2) is used to determine the laminar Nusselt number based on the hydraulic diameter, and since in the laminar region, Nul remains constant, Nulc = Nul. For turbulent Reynolds numbers, Bhatti and Shah [31] recommended that the circular tube correlation (Eq. (1)) can also be used for the rectangular channels with Nu and Re defined on the basis of hydraulic diameter. The slug heat transfer coefficient is then defined as follows:
hs ¼ Nus
Fig. 7. Schematic of condensation process.
ð2200ReÞ=365
kl D
ð3Þ
For the determination of the film heat transfer coefficient, consider the bubble section shown in Fig. 7. The pressure drop model assumed a uniform film thickness in the film/bubble section as shown by the dashed line in Fig. 7 and determined an average film thickness (dave). As the slug passes by, it leaves a thin liquid film behind, and along the length of the elongated bubble, the film thickness increases due to the condensation process. Consider a small section of length dz along the length of the bubble, with, dt the time taken by this section to pass through a particular point and dd the increase in film thickness during this time. Assuming
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
105
the heat flux to be q00f =b and equating the heat leaving the tube to the latent heat of evaporation of additional condensed liquid film thickness yields:
q00f =b 2pR dz dt ¼ qL 2pðR dÞ dd dz hfg
ð4Þ
Eq. (4) is rearranged to obtain the variation in film thickness with time, where the film thickness is assumed to be much smaller than the tube radius, i.e., ðR dÞ R. Integrating the result yields Eq. (5).
Z
Z
d
q00f =b
t
dd ¼
qL hfg
0
d0
dt ) d ¼ d0 þ
q00f =b
qL hfg
t
ð5Þ
Because the slug-bubble interface moves at velocity Ububble [1], the time t ¼ U z , where z is the distance along the bubble length Bubble
from the beginning of the bubble where the film thickness is a minimum. Also, the heat flux in the film/bubble section q00f =b ¼ hf ðT T wall Þ, which when substituted in Eq. (5) yields:
hf ðT T wall Þ z ) d ¼ d0 þ U bubble qL hfg
ð6Þ
Eq. (6) provides the film thickness as a function of distance from the nose of a bubble. Integrating the above equation over the length of the bubble, lBubble, the average film thickness is found by Eq. (7).
dave ¼
Z
1 lBubble
lBubble
d dz ¼ d0 þ
0
hf ðT T wall Þ lBubble qL hfg 2 U bubble
ð7Þ
An expression for the film heat transfer coefficient (Eq. (7)) can be found by rearranging Eq. (8).
hf ¼ ðdave d0 Þ
Fig. 8. Variation of d0 =dave with quality.
2 qL hfg U bubble ðT T wall Þ lB
ð8Þ
Other than minimum film thickness (d0), all the parameters on the right hand side of Eq. (8) are known. Inspection of Eq. (8) reveals that higher heat transfer coefficients are expected at lower saturation temperatures due to the higher latent heat of vaporization (hfg) at lower saturation temperatures. Similarly, at higher mass fluxes, the heat transfer coefficients are expected to be higher due to increased bubble velocities (Ububble). The effect of aspect ratio is accounted for through the length of the bubble, lBubble ½lBubble ¼ ðLtube =N UC Þð1 SLRÞ. For higher aspect ratio channels, the numbers of unit cells are higher [1], leading to lower bubble lengths, which in turn yields higher heat transfer coefficients. (It should be noted that NUC is the total number of unit cells in a fixed length of tube, so as the NUC increases, the same tube length is divided into more number of segments, thus leading to correspondingly lower slug and bubble lengths.) These trends were also observed in the experimentally obtained refrigerant heat transfer coefficients. Since both the slug and the bubble are moving with the same apparent velocity UBubble, the time taken by the film/bubble section to pass by (tf) and the time taken by the slug to pass by (ts) are proand portional to their lengths, i.e., t f ¼ lbubble =U Bubble t s ¼ lslug =U Bubble . The time-averaged hrefg can thus be determined by averaging the two heat transfer coefficients weighted by the time taken by the slug and bubble to pass by as shown in Eq. (9).
hrefg ðt f þ t s Þ ¼ hs t s þ hf tf
periodically varying convective boundary condition (due to the changing heat transfer coefficient) on the refrigerant side were neglected. This assumption is justified in the following section. In the analysis presented above, the only remaining unknown is the minimum film thickness (d0), which is required to determine the film heat transfer coefficient, hf. The data are used to obtain the values of minimum film thickness (d0). Substituting the experimental refrigerant heat transfer coefficient (hrefg) and the slug heat transfer coefficient (hs) into Eq. (10), the film heat transfer coefficient, hf can be obtained. The SLR is known from the pressure drop model [1] and hs is determined from Eq. (3). These film heat transfer coefficient (hf) values are then substituted into Eq. (8) to determine the minimum film thickness (d0). All other parameters (dave, qL, hfg, Ububble, lBubble, T, Twall) are known from the pressure drop model and data analysis. Fig. 8 shows the variation of the ratio of minimum film thickness and the average film thickness for the data obtained in the present study. As the quality increases and the diameter decreases, the d0 =dave decreases. For all the data points, the film thickness is 4.4–4.6% of the channel diameter. As the tube hydraulic diameter increases, the film thickness increases, leading to increased thermal resistance, i.e., a decrease in heat transfer coefficient. This decrease in heat transfer coefficient leads to lower condensation rates and hence the difference between the average and minimum film thickness decreases. As the quality increases, the slug length ratio decreases, leading to a decrease in the thickness of the film that it leaves behind. As the
ð9Þ
Substituting tf and ts yields:
hrefg ¼ hs
lslug lslug þ lbubble
þ hf 1
lslug lslug þ lbubble
ð10Þ
Sun et al. [32] also used a similar approach to determine the effective hrefg in intermittent flow. By determining the timeaveraged hrefg in the above manner, transient effects due to the
Fig. 9. Comparison of experimental and predicted hrefg.
106
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 10. Experimental and predicted hrefg for 300 100 lm channels.
diameter increases, the effect of a change in diameter diminishes and the change in d0 =dave with quality decreases. Based on this trend, beyond a certain diameter, an increase in hydraulic diameter will no longer affect the film thickness ratio. Based on the above observations, the correlation given in Eq. (11) is proposed for determining d0 =dave ; where Dref is the maximum diameter investigated here, (Dref = 160 lm).
x 0:424 2:82DD d0 ref ¼ 1 0:25 e 1x dave
ð11Þ
In the above correlation, the first term on the right hand side is a constant 1, which implies that for a quality of zero, the minimum film thickness and the maximum film thickness are the same and no condensation occurs. The second term captures the dependence on quality and diameter. The dependence on diameter was explained above. The dependence on quality is observed because a higher vapor quality implies that there is less liquid in the slug, and hence, as the slug passes by, it leaves a thinner liquid film behind. Fig. 9 shows a comparison of the heat transfer coefficients
predicted by the above heat transfer model and the experimentally determined values. The proposed heat transfer model predicts refrigerant heat transfer coefficients for 94% of the data within ±25% of the experimental values. The agreement is illustrated in Fig. 10 for the 300 100 lm data. The proposed model captures the trends in the data from the present study well. The average absolute deviations for the 100 100 lm, 200 100 lm, 300 100 lm and 400 100 lm tubes are 5%, 8%, 9% and 18% respectively, with an overall average absolute deviation of 11%. 4.1. Transient analysis In the above analysis for determining the effective hrefg, the transient effects due to the periodically varying refrigerant-side heat transfer coefficient were neglected. To validate this assumption, a one-dimensional transient analysis was conducted with a periodically varying convection condition on one side and a constant heat flux boundary condition on the other side as shown in 2
Fig. 11, where T X ¼ @T and T XX ¼ @@xT2 . The detailed solution of this @x transient heat transfer phenomenon is provided in Agarwal [33]. The temperature profile in the wall is given by Eq. (12), with the functions Tf and Ts given in Eqs. (13), (14), respectively.
Tðx; tÞ ¼
T f ðx; tÞ
0 < t < tf
T s ðx; t t f Þ t f < t < ðt f þ ts Þ
T f ðx; tÞ ¼ T R þ q00
X 1 1 1 þ Bf;n cosðkf;n xÞ ðL xÞ þ k hf n¼1
expðk2f;n a tÞ
T s ðx; tÞ ¼ T R þ q00 Fig. 11. Schematic for wall transient problem with periodic convection.
ð12Þ
ð13Þ
X 1 1 1 þ ðL xÞ þ Bs;n cosðks;n xÞ k hs n¼1
expðk2s;n a tÞ
ð14Þ
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 12. Temperature profile in wall for representative case.
The experimentally obtained hrefg and the theoretically calculated hs are used to obtain values for the hf. The thickness of the wall (L) was assumed to be equal to 1 mm. In the transient analysis over one complete cycle, the refrigerant-side wall temperature varies; however, the average refrigerant-side wall temperature was assumed to be equal to the wall temperatures determined in the data analysis. The effective film heat transfer coefficient, hf, determined using the results from this transient analysis were within 1% deviation of the hf determined using Eq. (34) for 89% of the data, with a maximum deviation of 2.5% for all the data points. Thus, the assumption of neglecting the transient effects in the wall and using a simple weighted average of the slug and film/bubble heat transfer coefficients is justified. Fig. 12 shows a plot for the wall temperature profile for the same representative case of eighteen 200 100 lm channels for G = 606 kg m2 s1, Tsat = 60.5 °C and xave = 0.39. For this case, the experimental refrigerant-side heat transfer coefficient (hrefg) is 21.7 kW m2 K1, while the average refrigerant-side wall temperature is 58.1 °C. The lengths of the bubble and slug are 1.96 and 0.24 mm, respectively. Slug (hs) and film (hf) heat transfer coefficients are determined to be 9.2 and 23.2 kW m2 K1 respectively. With a bubble velocity (Ububble) of 3.7 m s1, the time taken by the bubble (tf) and slug (ts) to pass a particular point are 535 and 66 ls, respectively. In Fig. 12, the refrigerant side is shown towards the front (L = 1 mm) and the constant heat flux side is shown towards the back (L = 0 mm). Only one time cycle is shown, where from time 0 to tf, the bubble passes by and from time tf to (tf + ts), the slug passes by.
107
The temperature variation along the thickness of the wall is approximately linear for low Biot numbers (hL/k). For all cases in the present study, the Biot Number is less than 0.3. Since the hf are higher than the hs and heat flux at the other end is assumed to be constant, during the time when a bubble passes, more heat is transferred to the wall due to which the temperature of the wall rises and as the slug passes, the wall temperature again rapidly drops. This can be explained further as follows. The rate at which heat is flowing out of the wall to the coolant is fixed due to the constant heat flux boundary condition on the coolant side. But, the rate at which the heat enters the wall from the refrigerant side is higher when the heat transfer coefficient is higher. Thus, when the bubble passes, more heat is transferred into the wall leading to a rise in temperature of the wall due to thermal storage. During the slug phase, the rate of heat transfer into the wall is much less, hence the stored thermal energy in the wall decreases, leading to a decrease in the wall temperature. Fig. 13 shows the variation in wall temperature with time at various depths (for the representative case discussed above) and further compares these variations with the overall temperature difference between the refrigerant side and the coolant side. These two plots together show that while the above analysis captures the temperature variations in the wall with time, they are not particularly significant compared to the overall driving temperature difference between the refrigerant and the coolant for the cases investigated here.
5. Parametric analysis In this section, the heat transfer model described above is used to illustrate the effects of various parameters such as hydraulic diameter, aspect ratio, mass flux and saturation temperature on heat transfer coefficient. (Only one parameter is varied at a time.) Fig. 14 shows the effect of variation of mass flux and quality for D = 130 lm, AR = 3, T = 50 °C and T Twall = 2 °C, G varying from 200 to 800 kg m2 s1. The hrefg increases with increasing mass flux due to an increase in the Reynolds number. Film heat transfer coefficients are in general higher than the slug heat transfer coefficients. Fig. 15 shows a representative comparison of the variation in slug and film heat transfer coefficients with mass flux and quality as predicted by the model. Garimella et al. [1] demonstrated that as the x increases, the slug length ratio decreases, leading to a higher contribution of the film heat transfer coefficient towards the time-averaged hrefg. Also, as the x increases, the flow velocities increase due to an increase in vapor velocity, which in turn leads to a corresponding increase in slug velocity. Thus, the slug heat
Fig. 13. Variation in wall temperature with time at various depths.
108
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 14. Model predictions: effect of mass flux and quality.
transfer coefficients increase as the x increases. For the lowest G case, the slug heat transfer coefficient remains almost constant due laminar Reslug. In the laminar region, Nu is constant for single-phase flow. Also, as the quality increases, the minimumto-average film thickness ratio decreases leading to an increase in film heat transfer coefficients. Thus, as the quality increases, the hrefg increase, due to an increase in both hslug and hf. Fig. 16 shows the effect of the variation of Tsat on the heat transfer coefficient for D = 130 lm, AR = 3, G = 600 kg m2 s1 and T Twall = 2 °C. The Tsat varies from 30 to 60 °C. As the Tsat decreases, the gas-to-liquid phase density ratio ðqg =ql Þ decreases (qg/ql = 0.083 @ 60 °C and qg/ql = 0.032 @ 30 °C), leading to an increase in void fraction (a = 0.74 @ 60 °C and a = 0.80 @ 30 °C). As the Tsat decreases, the hrefg increases due to two factors. Firstly, as mentioned above, at lower Tsat, the flow velocities are higher, leading to an increase in the hrefg. Also, as the Tsat decreases, the slug length ratio decreases; thus, the contribution of the film heat transfer coefficient toward the time-averaged hrefg is higher at lower Tsat. Fig. 17 shows the effect of a variation in channel aspect ratio (AR) on the heat transfer coefficient for D = 130 lm, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The AR was varied
Fig. 16. Model predictions: effect of saturation temperature.
Fig. 17. Model predictions: effect of channel aspect ratio.
Fig. 15. Variation in hslug and hfilm with mass flux and quality.
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
Fig. 18. Model predictions: effect of diameter.
Fig. 19. Model predictions: effect of driving temperature difference.
109
from 1 to 4, without varying the hydraulic diameter. It should be noted that unlike the actual channels tested in present study, the hydraulic diameter is kept constant here and only the AR varies. As the AR increases, the channel depth decreases and the width increases. The hs are higher for higher aspect ratio channels if the slug flow is laminar. Also, according to the model, hf is inversely proportional to the bubble length. Larger number of unit cells at higher AR [1] lead to shorter bubble lengths, yielding higher film heat transfer coefficients. Thus, as the aspect ratio increases, the hrefg increases. Physically this can be understood as follows. Each time the slug passes by, it breaks the liquid film boundary layer and due to fluid passing from film to slug and then from slug to film, there is more mixing of the fluid leading to higher heat transfer coefficients. The higher the number of unit cells, the more frequent is this turbulent mixing due to the passing slugs, leading to higher heat transfer coefficients in high aspect ratio channels. Fig. 18 shows the effect of variation of channel hydraulic diameter on the heat transfer coefficient for AR = 3, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The hydraulic diameter varies from 100 lm to 160 lm. The hrefg also increases with a decrease in diameter due to the increase in hf with decreasing diameters. The hf increase due to a decrease in film thickness with the decrease in tube diameter. Fig. 19 illustrates the effect of driving temperature difference on the refrigerant heat transfer coefficients for AR = 3, G = 600 kg m2 s1, T = 50 °C, xave = 0.5 and T Twall = 2 °C. As the driving temperature difference decreases, the rate of condensation decreases, leading to a thinner liquid film on the channel walls, which in turn yields higher heat transfer coefficients. The effect of a change in driving temperature difference diminishes with increasing driving temperature difference. The average vapor quality for the data obtained in the present study varies between 0.2 and 0.8. Fig. 20 shows the heat transfer coefficient predictions if the proposed model is extrapolated to vapor qualities varying from 0.05 to 0.95 for D = 130 lm, AR = 3, G = 600 kg m2 s1, T = 50 °C and T Twall = 2 °C. The single-phase liquid only and vapor-only heat transfer coefficient (hrefg,LO = 2.2 kW m2 K1 and hrefg,VO = 3.1 kW m2 K1) are also shown for this case. As the quality approaches zero, the heat transfer coefficient also approaches the single-phase heat transfer coefficient. As the quality approaches zero, the contribution of slug heat transfer coefficient in the time-averaged heat transfer coefficient increases, which leads to the approach to the single phase liquid value. The heat transfer coefficient continues to increase as the quality approaches one due to a progressively thinner liquid film. (The model does not account for inlet superheat, which might be the situation in an actual condenser.) The model proposed here is able to predict the trends in the heat transfer data based on a physical basis. The effect of surface tension is not explicitly captured, even though it might play an important role in microchannels. The behavior at very low (x < 0.2) and very high (x > 0.8) quality is also not addressed in great detail, because of the lack of data in these regions. The effect of fluid property variation is also not addressed; this would require additional data with different refrigerants. The model proposed here also assumes uniform distribution of flow in all channels and steady state conditions. Flow maldistribution and instabilities could affect the heat transfer coefficients in condensers using tubes with multiple parallel ports. Future studies with varying inlet and outlet header designs can be used to investigate these effects.
6. Conclusions
Fig. 20. Extrapolation of proposed model.
A study of condensation heat transfer in microchannels was conducted. A technique first introduced by Agarwal and Garimella
110
S. Garimella et al. / International Journal of Heat and Mass Transfer 100 (2016) 98–110
[22] was used accurately determine pressure drop and heat transfer coefficients during condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 lm) rectangular (1 < AR < 4) channels for wide ranging flow conditions (300 < G < 800 kg m2 s1; 0 < x < 1; 30 < Tsat < 60 °C). It was found that hrefg increased with increasing x, increasing G and decreasing Tsat. The hrefg were the highest for the 400 100 lm channels, followed by 100 100 lm, 200 100 lm, and 300 100 lm channels. It was also found that most of the models from the literature significantly under-predicted the heat transfer data from the present study. The flow velocities and other parameters determined for the pressure drop model [1] are used as inputs for the heat transfer models. The slug and bubble regions were analyzed separately to determine the slug and film heat transfer coefficients. A timeaveraged condensation heat transfer coefficient was determined by combining the slug and film heat transfer coefficients according to their transit times through the channel. The proposed heat transfer models predict 94% of the data within ±25%. In addition, the effect of the variation of heat transfer coefficient in intermittent flows from the slug to bubble regions was investigated in detail using a transient conjugate analysis. The proposed model was also used to analyze the effect of various parameters such as mass flux, saturation temperature, hydraulic diameter, and aspect ratio. Observed trends were explained based on the relative predominance of the intermittent or annular flow regime. As the G increases, the hrefg increase due to an increase in flow velocities. As the Tsat decreases, the void fraction increases due to a decrease in the vapor-to-liquid density ratio, which increases velocities and interfacial shear, and in turn, leads to an increase in hrefg. As the channel hydraulic diameter decreases, the hrefg increase due to a decrease in film thickness and channel diameter. As the AR increases, the hrefg increase due to an increased occurrence of slugs. The proposed model may be used by engineers for analyzing condensing two-phase flow in microchannel geometries and in the design of more efficient heat transfer equipment. References [1] S. Garimella, A. Agarwal, B.M. Fronk, The intermittent and annular flow condensation continuum: pressure drops at the microscale, Int. J. Multiphase Flow (2016), http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.03.023. [2] J.W. Coleman, S. Garimella, Two-phase flow regime transitions in microchannel tubes: the effect of hydraulic diameter, Am. Soc. Mech. Eng. Heat Transfer Div. 366 (2000) 71–83 (Publication) HTD. [3] J.W. Coleman, S. Garimella, Two-phase flow regimes in round, square and rectangular tubes during condensation of refrigerant R134a, Int. J. Refrig 26 (1) (2003) 117–128. [4] F.G. Carpenter, A.P. Colburn, The effect of vapor velocity on condensation inside tubes, ASME Proceedings of the General Discussion of Heat Transfer, 1951, pp. 20–26. [5] H.M. Soliman, J.R. Schuster, P.J. Berenson, A general heat transfer correlation for annular flow condensation, J. Heat Transfer 90 (2) (1968) 267–276. [6] D.P. Traviss, W.M. Rohsenow, A.B. Baron, Forced-Convection condensation inside tubes: a heat transfer equation for condenser design, ASHRAE Trans. 79 (Part 1) (1973) 157–165. [7] S.L. Chen, F.M. Gerner, C.L. Tien, General film condensation correlations, Exp. Heat Transfer 1 (2) (1987) 93–107.
[8] M.K. Dobson, J.C. Chato, Condensation in smooth horizontal tubes, J. Heat Transfer 120 (1) (1998) 193–213. [9] A. Cavallini, G. Censi, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, Condensation of halogenated refrigerants inside smooth tubes, HVAC&R Res. 8 (4) (2002) 429–451. [10] J.R. Thome, J. El Hajal, A. Cavallini, Condensation in horizontal tubes, part 2: new heat transfer model based on flow regimes, Int. J. Heat Mass Transfer 46 (18) (2003) 3365–3387. [11] H.M. Soliman, Mist-annular transition during condensation and its influence on the heat transfer mechanism, Int. J. Multiphase Flow 12 (2) (1986) 277– 288. [12] R.L. Webb, K. Ermis, Effect of hydraulic diameter on condensation of R-134A in flat, extruded aluminum tubes, J. Enhanced Heat Transfer 8 (2) (2001) 77–90. [13] C.Y. Yang, R.L. Webb, Friction pressure drop of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins, Int. J. Heat Mass Transfer 39 (4) (1996) 801–809. [14] C.Y. Yang, R.L. Webb, Condensation of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins, Int. J. Heat Mass Transfer 39 (4) (1996) 791–800. [15] C.Y. Yang, R.L. Webb, A predictive model for condensation in small hydraulic diameter tubes having axial micro-fins, J. Heat Transfer 119 (4) (1997) 776– 782. [16] M. Zhang, R.L. Webb, Correlation of two-phase friction for refrigerants in small-diameter tubes, Exp. Thermal Fluid Sci. 25 (3–4) (2001) 131–139. [17] H.S. Wang, J.W. Rose, Film condensation in horizontal triangular section microchannels: a theoretical model, Proceedings of the Second International Conference on Microchannels and Minichannels (ICMM2004), June 17–19 2004, American Society of Mechanical Engineers, New York, NY 10016-5990, United States, Rochester, NY, United States, 2004, pp. 661–666. [18] H.S. Wang, J.W. Rose, H. Honda, A theoretical model of film condensation in square section horizontal microchannels, Chem. Eng. Res. Des. 82 (4) (2004) 430–434. [19] S. Garimella, T.M. Bandhauer, Measurement of condensation heat transfer coefficients in microchannel tubes, 2001 ASME International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, New York, NY, United States, 2001, pp. 243–249. [20] T.M. Bandhauer, A. Agarwal, S. Garimella, Measurement and modeling of condensation heat transfer coefficients in circular microchannels, J. Heat Transfer 128 (10) (2006) 1050–1059. [21] S. Garimella, A. Agarwal, J.D. Killion, Condensation pressure drop in circular microchannels, Heat Transfer Eng. 26 (3) (2005) 28–35. [22] A. Agarwal, S. Garimella, Representative results for condensation measurements at hydraulic diameters similar to 100 microns, J. Heat Transfer 132 (4) (2010) 1–12. [23] T. Christenson, Email Communication Dated August 19th 2005, in: A. Agarwal (Ed.), Atlanta, 2005. [24] S.A. Klein, Engineering Equation Solver, in, F-Chart Software, 2006. [25] B.N. Taylor, C.E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, National Institute of Standards and Technology, Washington, DC, USA, 1994. [26] M.M. Shah, A general correlation for heat transfer during film condensation inside pipes, Int. J. Heat Mass Transfer 22 (4) (1979) 547–556. [27] K.W. Moser, R.L. Webb, B. Na, A new equivalent Reynolds number model for condensation in smooth tubes, J. Heat Transfer 120 (2) (1998) 410–417. [28] S.M. Kim, I. Mudawar, Universal approach to predicting heat transfer coefficient for condensing mini/micro-channel flow, Int. J. Heat Mass Transfer 56 (1–2) (2013) 238–250. [29] S.W. Churchill, Comprehensive correlating equations for heat mass and momentum-transfer in fully developed flow in smooth tubes, Ind. Eng. Chem. Fundam. 16 (1) (1977) 109–116. [30] R.K. Shah, M.S. Bhatti, Laminar convective heat transfer in ducts, in: S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, NY, 1987. [31] M.S. Bhatti, R.K. Shah, Turbulent and transitions flow convective heat transfer in ducts, in: S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, NY, 1987. [32] G. Sun, G.F. Hewitt, V.V. Wadekar, A heat transfer model for slug flow in a horizontal tube, Int. J. Heat Mass Transfer 47 (12–13) (2004) 2807–2816. [33] A. Agarwal, Heat Transfer and Pressure Drop During Condensation in Microchannels PhD Dissertation, Georgia Institute of Technology, Atlanta, 2006.