1 Introduction

Fin-and-round tube heat exchangers have long been used as condensers in an air-conditioning system, and rigorous efforts have been made to improve the thermal performance of the heat exchangers. These include a usage of high performance fins, and of small diameter tubes, etc. However, fin-and-round tube heat exchangers have inherent short-comings such as contact resistance between fins and tubes, existence of low performance region behind tubes, etc. These short-comings may be mitigated if low-profile flat tubes are used, fins and tubes are soldered [1]. Brazed heat exchangers with aluminum louver fins and aluminum flat multi-port tubes satisfy the requirements, and have long been used as condensers of automotive air-conditioning units, and are being used as condensers of residential air-conditioners. For a proper design of the brazed aluminum heat exchanger, knowledge on condensation heat transfer and pressure drop characteristics in the flat multi-port tube is mandatory.

The literature shows fair amount of studies on condensation heat transfer and pressure drop in multi-port tubes. Existing studies are summarized in Table 1. Yang and Webb [2] and Webb and Yang [3] conducted condensation heat transfer tests using R-12 and R-134a in smooth (Dh = 2.64 mm) or micro-finned (Dh = 1.56 mm) multi-port tubes, both having 4 channels. The test range covered 400 ≤ G ≤ 1400 kg/m2 s, 4 ≤ q ≤ 12 kW/m2 at Tsat = 65 °C. They showed that the data were reasonably predicted by Akers et al. [21] correlation. The data were underpredicted by Shah [22] correlation. The heat transfer coefficients of the micro-finned tube were 10 to 60% higher than those of the smooth tube for the entire test range. The difference increased as vapor quality increased. They attributed the trend to the additional surface-tension drainage condensation on the fin surface, which is exposed to vapor at high vapor quality. Yang and Webb [4] reported pressure drop data for the same tubes tested by Yang and Webb [22]. The pressure drop of the micro-finned tube was higher than that of the smooth tube. They successfully correlated the pressure drop data using the equivalent mass velocity concept proposed by Akers et al. [21]. A semi-empirical model that accounts for the vapor shear and surface tension to predict the condensation heat transfer for micro-finned multiport tubes was proposed by Yang and Webb [23].

Table 1 Previous studies on condensation in multi-port flat tubes
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Chang et al. [5] tested four multi-port tubes (two smooth and two micro-finned) having 5 to 7 channels using R-22 (Tsat = 45 °C) and R-134a (Tsat = 40 °C). Their mass flux range, however, was rather low (30 ≤ G ≤ 100 kg/m2 s). Microfin tubes yielded 10 to 15% higher condensation heat transfer coefficients than smooth tubes for the entire test range. Wang et al. [6] tested a smooth multi-port tube of Dh = 1.46 mm having 10 channels using R-134a. The test range covered 150 ≤ G ≤ 750 kg/m2 s, 61.5 ≤ Tsat ≤ 66 °C. When the heat transfer data are compared with predictions of existing correlations, Akers et al. [21] correlation turned out to be the best. A new correlation, which covers both the annular and stratified flow regime, was proposed. Zhang and Webb [7] developed a friction correlation based on reduced pressure concept.

Kim et al. [8] provided condensation heat transfer data for the smooth (Dh = 1.41 mm, 7 channels) and micro-finned (Dh = 1.56 mm, 4 channels) multi-port tubes using R-22 and R-410A. The test range covered 200 ≤ G ≤ 600 kg/m2 s at q = 10 kW/m2 and Tsat = 45 °C.They noted that Akers et al. [21] correlation reasonably predicted the condensation heat transfer data. Heat transfer enhancement by microfins was dependent on the refrigerant. For R-22, the microfin tube yielded higher heat transfer coefficients. For R-410A, however, the reverse was true. The applicability of Akers et al. [21] correlation for multi-port tubes has further been confirmed by Webb and Ermis [9], where tubes having a wide range of hydraulic diameters (from 0.44 mm to 1.56 mm, smooth and microfin) were tested. Their test range covered 300 ≤ G ≤ 1000 kg/m2 s at q = 8 kW/m2 and Tsat = 60 °C. Koyama et al. [10] investigated R-134a condensation in smooth multi-port tubes with Dh = 1.36 mm (8 channels) and 0.8 mm (19 channels) for the test range 100 ≤ G ≤ kg/m2s at Tsat = 65 °C. Comparison of the data with Moser et al. [24] correlation based on equivalent Reynolds number was not satisfactory, and they developed a new correlation extending Haraguchi et al. [25] correlation. The friction data were well predicted by Mishima and Hibiki [26] correlation.

Cavallini et al. [11] obtained condensation heat transfer and pressure drop data of R-134a and R-410A for a smooth multi-port tube (Dh = 1.4 mm, 13 channels) for the test range 200 ≤ G ≤ 1400 kg/m2 s at Tsat = 40 °C. R-410A yielded higher pressure drop and heat transfer coefficient compared with R-134a. They suggested a heat transfer model for condensation inside multi-port tubes, based on an analogy between heat and momentum transfer. Bandhauer et al. [12] tested three smooth multi-port tubes (0.51 ≤ Dh ≤ 1.52 mm) having circular cross-section using R-134a. The test range covered 150 ≤ G ≤ 750 kg/m2 s at Tsat = 55 °C. The condensation heat transfer coefficient increased as the tube hydraulic diameter decreased. Comparison of the data with available correlations revealed that Moser et al. [24] correlation performed best. A new heat transfer model based on boundary layer analysis was proposed. Their study was extended to non-circular multi-port tubes having various cross-sectional shapes (square, barrel, triangular, rectangular, W-shaped, N-shaped) by Agarwal et al. [13]. Park et al. [14] obtained condensation heat transfer coefficients of R-1234ze(E), R-134a and R-236fa in a smooth multi-port tube having Dh = 1.45 mm for the test range 100 ≤ G ≤ 250 kg/m2 s, 26 ≤ q ≤ 36 kW/m2 and 30 ≤ Tsat ≤ 70 °C. The heat transfer coefficients of R-1234ze(E) were 15 to 25% lower than those of R-134a, but relatively similar to those of R-236fa. Comparison of the data with existing correlations was not satisfactory, and a new correlation was proposed. Sakamatapan et al. [15] tested two smooth multi-port tubes (Dh = 1.1 and 1.2 mm) using R-134a for the test range 340 ≤ G ≤ 680 kg/m2 s, 15 ≤ q ≤ 25 kW/m2 at 35 ≤ Tsat ≤ 45 °C. Comparison of the data with Webb et al. [27] correlation was satisfactory. Illan-Gomez et al. [16] compared condensation heat transfer coefficients and pressure drops of R-1234yf and R-134a in a smooth multi-port tube having Dh = 1.16 mm. The test range covered 275 ≤ G ≤ 940 kg/m2 s, 25 ≤ Tsat ≤ 55 °C. R-134a yielded higher heat transfer coefficient and higher pressure drops. Lopez-Belchi et al. [17] extended the study to R-32 and R-410A. R-32 yielded higher heat transfer coefficient and higher pressure drops. Additional studies on condensation heat transfer of CO2 in multi-port tubes were reported by Park and Hrnjak [18] and Heo et al. [19]. Fernando et al. [20] investigated the condensation heat transfer performance of propane in a multi-port tube. A review on the prediction methods for the pressure drop in a multi-port tube was provided by Yun and Jeong [28].

As mentioned, brazed aluminum heat exchangers which use multi-port tubes have long been used as condensers of automotive air-conditioning units, where R-134a is a refrigerating fluid. Thus, most of the condensation studies have been conducted using R-134a. Recently, brazed aluminum heat exchangers are commonly used as condensers of residential air-conditioning systems, where R-410A is a refrigerating fluid. Table 1 shows that only three studies [8, 11, 17] are available for condensation of R-410A in a multi-port tube. These studies are, however, limited to Dh larger than 1 mm, relatively large mass flux (200 ≤ G ≤ 1400 kg/m2 s) and relatively large heat flux (q ≅ 10 kW/m2). Typical mass flux and heat flux of the residential condenser are 100 ≤ G ≤ 400 kg/m2 s and q ≅ 3 kW/m2 [29]. In addition, recent trend of the multi-port tube is to use reduced hydraulic diameter tubes. In this study, R-410A condensation tests were conducted in four smooth multi-port tubes having a range of hydraulic diameter (0.78 ≤ Dh ≤ 0.95 mm). The effect of hydraulic diameter on condensation heat transfer of R-410A is discussed. The test range covered the mass flux from 100 to 400 kg/m2 s and the heat flux was fixed at 3 kW/m2. The saturation temperature was maintained at 45o C.

2 Experimental apparatus

Schematic drawings of the multi-port tubes tested in this study are shown in Fig. 1, and geometric details are listed in Table 2. Tubes are designated by a set of hydraulic diameter - number of channels. For example, 0.78–18 tube has 0.78 mm hydraulic diameter and 18 channels. Four tubes have different hydraulic diameter, number of channels and aspect ratios. As mentioned, the hydraulic diameter ranged from 0.78 to 0.95 mm. Numbers of channels of the four tubes are widely different - from 9 to 18. Aspect ratios of 0.95–16 and 0.78–18 tube are approximately the same, and those of 0.80–9 and 0.83–11 tube are approximately similar. This complexity of tube dimensions was unavoidable due to the availability of the tubes from the market.

Fig. 1
figure 1

Flat multi-port tubes tested in this study

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Table 2 Geometric details of the multi-port flat tubes
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A schematic drawing of the apparatus is shown in Fig. 2, and a detailed drawing of the test section and photos of the tube are shown in Fig. 3. The same apparatus and test section by Kim et al. [8] were used in the present study. The apparatus has been successfully used to obtain the smooth tube condensation data [30]. The test section comprises of a flat tube and an annular channel with a length of 455 mm. Fig 3a shows the cross-sectional view of the test section. The refrigerant flows inside of the tube and the cooling water flows in the annular channel. Fig. 3b shows a photo of the tube with brazed transition sections. Fig. 3c shows the end view of the tube sample, where thin wire of 0.3 mm was wrapped with 3.0 mm pitch around the tube. Each tube had separate transition sections due to different tube width. Smooth transition from the flat tube and the round tube of 6.0 mm diameter was accomplished by the transition tube [Fig. 3d] of 20 mm diameter and 150 mm length. For an accurate measurement of tube-side condensation heat transfer coefficient, it is important to minimize the thermal resistance of the annular side. This was accomplished by increasing the annular-side water velocity, and by wrapping thin wire around the tube. To increase the water velocity, the annular gap was maintained small (1.0 mm).

Fig. 2
figure 2

Schematic drawing of the experimental apparatus

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Fig. 3
figure 3

Detailed drawing of the test section

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As illustrated in Fig. 2, the refrigerant flows into the test section at a known quality and partly condenses in the test section by an annular-side cooling water. Two-phase refrigerant mixture from the test section enters the separator, where the liquid drains down to the receiver and the vapor flows into the upper shell-and-tube condenser. The condensed liquid drains down to the receiver. The sub-cooled liquid passes through the magnetic gear pump, mass flow meter, and enters the pre-heater. The refrigerant flow rate was controlled by by-passing an appropriate amount of liquid. The vapor quality into the test section was controlled by the heat input supplied to the pre-heater. The heat flux to the flat tube was controlled by changing the temperature of the cooling water. The flow rate of the cooling water was fixed at 1.0 l per minute throughout the test.

Temperatures were measured at five locations; refrigerant temperatures at inlet and outlet of the test tube, cooling water temperatures at inlet and outlet of the annular channel and a sub-cooled refrigerant temperature at the inlet of the pre-heater. Thermowells having five thermocouples each were used to measure local temperatures. Two absolute pressures were measured - one at the inlet of the test section, and the other at the inlet of the pre-heater. These absolute pressures were used to check the state (sub-cooled or saturated) of the refrigerant. A differential pressure transducer was used to measure the pressure drop across the test section. The refrigerant and water flow rates were measured by mass flow meters (Micromotion: DN25S-SS-1) with ±1.5 × 10−6 m3/s accuracy.

The saturation pressure corresponding to the saturation temperature 45 °C is 27 bar and the apparatus was checked for leak-tight. Leak tests were conducted by a soap bubble technique followed by a halogen leak detection. The leakage resulted in a decrease in pressure less than 0.5 kPa per hour. The condensation test started at the maximum mass flux. After the system was stabilized, quality (from 0.2 to 0.8) and mass flux (from 100 to 400 kg/m2 s) were sequentially varied, all in a decreasing manner. The heat flux was fixed at 3 kW/m2 and the saturation temperature was maintained at 45o C.

3 Data reduction

From the measured data, the overall heat transfer coefficient U o is obtained from the following equations. Here, m w is the water flow rate, c pw is the specific heat of water, T w,out and T w,in is the water outlet and inlet temperature, T r,out and T r,in is the refrigerant outlet and inlet temperature

$$ Q={m}_w{c}_{pw}\left({T}_{w, out}-{T}_{w, in}\right)={U}_o{A}_o\varDelta {T}_{lm} $$
(1)
$$ \varDelta {T}_{lm}=\frac{\left({T}_{w, in}-{T}_{r, in}\right)-\left({T}_{w, out}-{T}_{r, out}\right)}{\ln \frac{\left({T}_{w, in}-{T}_{r, in}\right)}{\left({T}_{w, out}-{T}_{r, out}\right)}} $$
(2)

The tube-side condensation coefficient h i is determined from Eq. (3) using the overall heat transfer coefficient U o and the annular-side heat transfer coefficient h o . Here, A i is the internal surface area, A o is the outer surface area, A m is the area at the middle plane of tube wall (excluding internal web).

$$ {h}_i={\left[\left(\frac{1}{U_o}-\frac{1}{h_o}\right)\frac{A_i}{A_o}-\frac{tA_i}{kA_m}\right]}^{-1} $$
(3)

The annular-side heat transfer coefficient h o was determined from the Wilson plot [31]. Wilson plot method is graphically illustrated in Fig. 4. In the Wilson plot method, tests are conducted increasing tube-side velocity (V), and 1/UA values are drawn as a function of V -0.8. At an infinite velocity, which is obtained by extrapolating the 1/UA line, tube-side thermal resistance 1/h i A i becomes zero and 1/UA equals 1/h o A o  + t/kA m , where h o is readily calculated. One thing to be careful when running a Wilson plot test is that the annulus-side water temperature should be kept constanst, which is realized by controlling the water temperatue out of the constant water temperature bath. Other important thing to run a Wilson plot test is to maintain both tube-side and annulus-side turbulent. To promote turbulence at the annulus-side, thin wire of 0.3 mm diameter was wrapped around the tube at 2.0 mm pitch. Heat balances between the tube-side and annulus-side were checked during Wilson plot tests, which were less than ±3.0% for the entire test range. Tests were conducted changing annulus-side water velocity and temperature to obtain exponents of Reynolds number and Prandtl number. The resulting annulus-side forced convection equation is as follows.

Fig. 4
figure 4

Graphical interpretation of the Wilson plot [31]

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$$ {Nu}_{Dh}=0.141{\mathrm{Re}}_{Dh}^{0.70}{\mathrm{Pr}}_w^{0.4}\left(0.78-18\right) $$
(4)
$$ {Nu}_{Dh}=0.173{\mathrm{Re}}_{Dh}^{0.68}{\mathrm{Pr}}_w^{0.4}\left(0.89-9\right) $$
(5)
$$ {Nu}_{Dh}=0.058{\mathrm{Re}}_{Dh}^{0.82}{\mathrm{Pr}}_w^{0.4}\left(0.83-11\right) $$
(6)
$$ {Nu}_{Dh}=0.041{\mathrm{Re}}_{Dh}^{0.86}{\mathrm{Pr}}_w^{0.4}\left(0.96-16\right) $$
(7)

Eqs. (4) to (7) are applicable for 2400 ≤ Re Dh  ≤ 5200.To obtain reliable heat transfer data from tests using Wilson plot, it is important to minimize the Wilson-plot-side thermal resistance. For the present test, approximately 1/3 of total thermal resistance was on the annular-side. This portion may be reduced by increasing the flow velocity. In that case, however, the temperature difference between inlet and outlet becomes small, and the uncertainty on the heat supplied to the test section increases. An approximate optimum annular-side water flow rate, which balanced the water temperature difference and the thermal resistance, was found (1.0 l per minute) by trial and error with corresponding Reynolds number of approximately 4600. At this water flow rate, the water temperature difference from inlet to outlet was larger than 0.7 °C, which was obtained at the lowest mass flux (G = 100 kg/m2 s), and the annular-side thermal resistance was less than 1/3 of the total resistance for the entire test range. The average vapor quality in the test tube is determined from Eq. (8).

$$ {x}_{ave}={x}_{in}-\varDelta x/2 $$
(8)
$$ \varDelta x=\frac{m_w{c}_{pw}\left({T}_{w, out}-{T}_{w, in}\right)}{m_r{i}_{fg}} $$
(9)

Here, Δ x is the change of vapor quality across the test section, m r is the refrigerant flow rate, i fg is the latent heat of the refrigerant. The amount of Δ x ranged from 0.13 to 0.52. It increased as mass flux decreased. In Eq. (10), Q p is the heat supplied to the pre-heater, T p,in is the liquid refrigerant temperature into the pre-heater and c pr is the specific heat of the liquid refrigerant. Sub-cooled liquid out of the pump was supplied to the pre-heater.

$$ {x}_{in}=\frac{1}{i_{fg}}\left[\frac{Q_p}{m_r}-{c}_{pr}\left({T}_{sat}-{T}_{p, in}\right)\right] $$
(10)

For two-phase flow, the total pressure gradient comprises of three terms - frictional, accelerational and gravitational pressure gradient.

$$ \frac{dP}{dz}={\left.\frac{dP}{dz}\right)}_f+{\left.\frac{dP}{dz}\right)}_a+{\left.\frac{dP}{dz}\right)}_g $$
(11)

Then, the frictional pressure gradient may be obtained by subtracting the accelerational pressure gradient from the measured total pressure gradient. In the present setup, gravitational pressure gradient is zero because the test section is under horizontal position. The accelerational pressure gradient obtained from Collier and Thome [32] is

$$ {\left.\frac{dP}{dz}\right)}_a={G}^2\frac{d}{dz}\left[\frac{x^2{v}_g}{\alpha }+\frac{{\left(1-x\right)}^2{v}_f}{\left(1-\alpha \right)}\right] $$
(12)

where the void fraction α is obtained from Zivi [33]. According to Kline and McClintock [34], when a parameter ‘R’ is a function of measured variables (x 1, x 2,  ⋯ , x n ), the uncertainty on the parameter ‘w R ’ is obtained from the following equation.

$$ {w}_R={\left[{\left(\frac{\partial R}{\partial {x}_1}{w}_1\right)}^2+{\left(\frac{\partial R}{\partial {x}_2}{w}_2\right)}^2+\cdots +{\left(\frac{\partial R}{\partial {x}_n}{w}_n\right)}^2\right]}^{1/2} $$
(13)

Here, ‘w 1 ’ is the uncertainty on variable ‘x 1 ’, ‘w 2 ’ is the uncertainty on variable ‘x 2 ’, etc. As an example, the uncertainty on evaporation heat transfer coefficient is obtained as follows. From Eqs. (1), (3) and (13),

$$ \frac{w_{hi}}{h_i}={\left[{\left(\frac{h_o}{h_o-{U}_o}\frac{w_{U_o}}{U_o}\right)}^2+{\left(\frac{-{U}_o}{h_o-{U}_o}\frac{w_{h_o}}{h_o}\right)}^2\right]}^{1/2} $$
(14)
$$ \frac{w_{U_o}}{U_o}={\left[{\left(\frac{w_q}{q}\right)}^2+{\left(\frac{w_{\varDelta {T}_{lm}}}{\varDelta {T}_{lm}}\right)}^2\right]}^{1/2} $$
(15)
$$ \frac{w_{\varDelta {T}_{lm}}}{\varDelta {T}_{lm}}=\frac{1}{\varDelta {T}_{lm}}{\left[{\left(\frac{w_{T_{r, in}}}{2}\right)}^2+{\left(\frac{w_{T_{r, out}}}{2}\right)}^2+{\left(\frac{w_{T_{w, in}}}{2}\right)}^2+{\left(\frac{w_{T_{w, out}}}{2}\right)}^2\right]}^{1/2} $$
(16)

Introducing the measurement uncertainties listed in Table 3 and assuming an uncertainty on Wilson plot w ho /h o to be ±10.0%, the uncertainties on the heat transfer coefficients are ±11.1 ~ ±13.0%, and those of pressure drops are ±0.2 ~ ±4.3%. All the uncertainties are based on 95% confidence level. The uncertainty increased as the mass flux or vapor quality decreased.

Table 3 Experimental uncertainties
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4 Results and discussions

In Fig. 5, condensation heat transfer coefficients of the four tubes are shown as a function of mass flux. At the lowest mass flux of 100 kg/m2 s, low quality data missing due to difficulty of obtaining the data. At the low quality, the exit quality of the test section was negative (liquid). Error bars for the heat transfer coefficients and qualities are also drawn in the figure. Fig. 5 shows that the heat transfer coefficient increases as mass flux or quality increases. This is a general trend of condensation heat transfer. As mass flux or quality increases, the flow momentum increases, which results in higher heat transfer coefficient.

Fig. 5
figure 5

Condensation heat transfer coefficients of the tested tubes showing the effect of mass flux

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For two-phase flow, the heat transfer coefficient and the pressure drop are significantly affected by the flow pattern. Thus, it is necessary to identify the flow regime for the entire test range. In Fig. 6, tested mass fluxes and qualities are marked in the flow pattern map of Garimella [35]. Garimella map was made based on the flow visualization of R-134a in a 1.0 mm circular channel. Garimella noted that the flow pattern in a rectangular channel, which is of interest in the present study, was approximately the same as that in a circular channel. Although the present study was conducted using R-410A, the flow pattern would be approximately the same as that obtained using R-134a, considering that the liquid and gas density ratio of R-410A is not significantly different from that of R-134a (8.0 for R-410A and 20.2 for R-134a at 45 °C saturation temperature). The liquid and gas density ratio may be the most influential parameter, which determines the flow pattern. Also worth noting is that Garimella map was based on a single tube. At present, no flow pattern map is available for a multi-port tube. Fig. 6 shows that the flow pattern of the present study belongs to annular even at the lowest mass flux of 100 kg/m2 s. In Fig. 7, the frictional pressure gradients of the four tubes are shown along with error bars. As expected, the pressure gradient increased as mass flux or quality increased.

Fig. 6
figure 6

Flow pattern of the present study shown in the map of Garimella [35]

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Fig. 7
figure 7

Frictional pressure gradients of the tested tubes showing the effect of mass flux

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In Fig. 8, the heat transfer coefficients are re-grouped to show the effect of tube geometry on heat transfer coefficient. For all mass fluxes, the heat transfer coefficient increased as tube hydraulic diameter decreased. The difference was more evident at a lower mass flux. Also shown in the figure are heat transfer coefficients obtained by Kim et al. [8] for the multi-port tube of 1.41 mm hydraulic diameter using the same refrigerant R-410A. One thing to be noted is that Kim et al.’s [8] data were obtained at 10 kW/m2 heat flux, which is much higher than the present heat flux of 3 kW/m2. The original Kim et al. data (shown as a dark solid line) lie between those of Dh = 0.80 mm (0.80–9) and 0.83 mm (0.83–11). The effect of heat flux on condensation heat transfer coefficient in multi-port tubes were investigated by Yang and Webb [2], who reported that the heat transfer coefficient increased as the heat flux increased with the heat flux dependency of q0.2. Kim et al’s [8] data were adjusted to the lower heat flux of 3 kW/m2 using the heat flux dependency of q0.2, and the results are shown in Fig. 8 as a dotted line. The adjusted data are indeed lower than those of the present tubes, supporting the argument that the heat transfer coefficient decreased as the hydraulic diameter increased. In Fig. 9, the frictional pressure gradients are re-drawn to elucidate the effect of tube diameter. Generally, pressure gradient increased as hydraulic diameter decreased. The effect is most pronounced for the tube having the smallest hydraulic diameter (0.78–18).

Fig. 8
figure 8

Condensation heat transfer coefficients of the tested tubes showing the effect of tube geometry

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Fig. 9
figure 9

Frictional pressure gradients of the tested tubes showing the effect of tube geometry

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For condensation in a multi-port tube, flow mal-distribution between the ports may occur. The flow mal-distribution increases the pressure drop and reduces the heat transfer. The effect of flow mal-distribution on condensation heat transfer has been theoretically studied by Lopez-Belchi et al. [36], who showed that the heat transfer could be significantly reduced (up to 70%) by the mal-distribution. They obtained possible two-phase flow profiles which satisfied the equal pressure drop constraint, and compared the heat transfer rates. Pressure drop increase by the mal-distribution, however, was not assessed. The effect of flow mal-distribution on pressure drop may be assessed if a mal-distribution profile is given. As an extreme case, consider a single phase liquid and assume that all the liquid flows into a single port. Then, the velocity in that port will be N times (N: number of port) of the average value. The pressure drop increase will approximately be N2 whereas the heat transfer increase is approximately N0.8. Then, we may roughly conclude that the effect of flow mal-distribution is more severe on pressure drop than on the heat transfer. This may explain the different hydraulic diameter behavior between heat transfer and pressure drop (Figs. 8 and 9). As shown in Fig. 1, Dh = 0.78 mm (0.78–18) and 0.95 mm (0.95–16) tubes have larger number of ports than other tubes (0.80–9 and 0.83–11). Considering that mal-distribution will be larger for the tubes having larger number of ports, the pressure drop increase of the 0.78–18 and 0.95–16 tubes over other tubes will be more apparent than that of the heat transfer counterpart. This may explain the exceptionally large pressure drop for the 0.78–18 tube and approximately equal (with 0.80–9 and 0.83–11) pressure drop of the 0.96–16 tube (Fig. 9), and the gradual increase of the heat transfer coefficients with decrease of the hydraulic diameter (Fig. 8).

To deduce the effect of hydraulic diameter on heat transfer coefficient, the data were normalized by those of the largest hydraulic diameter (Dh = 0.95 mm). The correlating variables are Dh, N, G and x. Listed below is the derived correlation.

$$ \frac{h}{h_{0.95}}=2.19{D}_h^{-4.03}{G}^{-0.27}{x}^{0.03}{N}^{0.07} $$
(17)

Fig. 10a shows that the correlation predicted the experimental results within ±10%. Eq. (17) reveals that hydraulic diameter dependency of the heat transfer coefficient is very strong. The effect of hydraulic diameter on condensation heat transfer in a multi-port tube has been investigated by Webb and Ermis [9] for the test range 300 ≤ G ≤ 1000 kg/m2 s at q = 8 kW/m2 and Tsat = 60 °C using R-134a. Their multi-port tubes had a range of hydraulic diameter (from 0.44 to 1.56 mm) with 4 to 12 sub-channels. From the test results, they recommended Akers et al. [21] correlation to predict the condensation heat transfer in a multiport tube. The Akers et al. [21] correlation suggests heat transfer coefficient to be proportional to Dh -0.2. This hydraulic diameter dependency is much weaker than the present results (Eq. 10). The range of the hydraulic diameter of the present study is rather narrow (from 0.78 to 0.95 mm), and the present results should be considered with caution. Addition of Kim et al. [8] data of Dh = 1.41 mm mitigates the hydraulic diameter dependency to Dh -0.5.

Fig. 10
figure 10

Present data compared with Eqs. (10) and (11)

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Similar attempt was made on the pressure drop. The derived correlation is as follows.

$$ \frac{{\left. dP/ dz\right)}_f}{{\left. dP/ dz\right)}_{f,0.95}}=0.40{D}_h^{-3.75}{G}^{-0.06}{x}^{0.01}{N}^{0.22} $$
(18)

Fig. 10b shows that the correlation predicted most of the experimental results within ±10%. Eq. (18) reveals that hydraulic diameter dependency of the pressure drop is also very strong. In addition, the exponent of N (number of ports) is not small. This value was quite small for heat transfer couterpart [Eq. (14)]. Considering that the exponent of N imples the amount of flow mal-distribution in the multi-port tube, larger exponent of N for the pressure drop correlation further supports the argument proposed in the previous section that the effect of flow mal-distribution is more severe on pressure drop than on heat transfer.

The literature survey reveals many condensation heat transfer and pressure drop correlations. The correlations may be grouped to those for macro-channel tubes and those for mini-channel tubes. Heat transfer correlations for macro-channel tubes include Akers et al. [21], Shah [37], Dobson and Chato [38], Cavallini et al. [39] and Thome et al. [40]. Heat transfer correlations for mini-channel tubes include Wang et al. [6], Koyama et al. [10], Cavallini et al. [11], Park et al. [14], Shah [41], Kim and Mudawar [42]. All these correlations are applicable to annular flow regime, which is the flow regime of this study. This paper addresses three Shah [22, 37, 41] correlations. Shah [37] updated his original version [22] to cover new refrigerants including R-410A. Recently, a separate correlation for condensation heat transfer in a mini-channel was proposed [41]. Experimental heat transfer coefficients are compared with the predictions of the correlations in Fig. 11, and RMS errors are summarized in Table 4. Fig. 11a shows that some correlations for macro-channel tubes, Cavallini et al. [39], Akers et al. [21], Thome et al. [40] and Shah [37] reasonably predict the data. Especially, Cavallini et al. [39] correlation predicts the data with 25.5% RMS error. Of the mini-channel correlations, Wang et al. [6] and Koyama et al. [10] correlations reasonably predict the data, with 23.1% RMS error by Wang et al. [6] correlation.

Fig. 11
figure 11

Present heat transfer coefficients compared macro- and mini-channel tube correlations

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Table 4 RMS errors of the correlations
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Predictions of all the correlations showed minimal effect of hydraulic diameter on heat transfer coefficient. The heat transfer coefficients were approximately proportional to D h -0.2. When the exponent (−0.2) was applied to the present range of hydraulic diameter (0.78 to 0.95 mm), the difference of heat transfer coefficient between the largest (0.78 mm) and the smallest (0.95 mm) one was only 4%. This result is significantly different from the experimental data shown in Fig. 8 and Eq. (14). More supporting data are needed to clear this issue. One thing to be mentioned is that the present multi-port tubes had different number of channels and different channels aspect ratio as shown in Table 1. Number of channels or channel aspect ratio may affect the heat transfer performance by many ways. Flow non-uniformity between channels may occur in multi-port tubes, and the amount may be affected by number of channels. This may be true if the number of channels is significantly different. For the present study, every effort was made to provide uniform flow to each channel, for example, by providing a long (150 mm) smooth transition section at inlet of the sample tube. However, flow mal-distribution may have occurred, which degrades the heat transfer performance. Next, channel aspect ratio will affect the condensation heat transfer performance of a multi-port tube. The aspect ratio of the present samples ranged from 1.71 to 2.41. Thus, number of channels and channel aspect ratio may also be responsible for the difference in the heat transfer coefficient of the present study in addition to the hydraulic diameter.

Pressure drop correlations may also be divided into two categories - those for macro-channel tubes and those for mini-channel tubes. Macro-channel correlations include Cavallini et al. [39], Friedel [44], Jung and Radermacher [43], Muller-Steinhagen and Heck [45]. Mini-channel correlations include Cavallini et al. [46], Kim and Mudawar [42], Sun and Mishima [47], Lopez-Belchi et al. [17], Zhang and Webb [7]. All these correlations are applicable to annular flow regime. The present frictional pressure drop data are compared with the correlations, and the results are shown in Fig. 12, and RMS errors are summarized in Table 4. Fig. 12a shows that, in general, macro-channel tube correlations overpredict the data. Fig. 12b shows that some mini-channel tube correlations reasonably predict the data. Especially, Cavallini et al. [46] correlation predicts the data with 28.5% RMS error.

Fig. 12
figure 12

Present frictional pressure gradients compared macro- and mini-channel tube correlations

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5 Conclusions

R-410A condensation tests were conducted in four different smooth multi-port tubes having a range of hydraulic diameter (0.78 ≤ Dh ≤ 0.95 mm). The test range covered the mass flux from 100 to 400 kg/m2 s and the heat flux was fixed at 3 kW/m2, which are typical operating conditions of a residential air conditioner. The saturation temperature was 45o C. Listed below are major findings.

  1. 1)

    Both the heat transfer coefficient and the pressure drop increased as the hydraulic diameter decreased.

  2. 2)

    The effect of hydraulic diameter on condensation heat transfer was much larger than the predictions of existing correlations for the range of investigation.

  3. 3)

    Both the macro-channel tube correlation [39] and the mini-channel correlation [6] reasonably predicted the heat transfer coefficient.

  4. 4)

    Only mini-channel correlation, e. g. Cavallini et al. [46], adequately predicted the pressure drop. Macro-channel correlations highly overpredicted the data.

  5. 5)

    Number of channels and channel aspect ratio may also be responsible for the difference in the heat transfer coefficient of the present study in addition to the hydraulic diameter.